Cenco Physics

Errors

bpearson — Thu, 04 Mar 2010 19:19:21 +0000

I. INTRODUCTION: Observations are taken in the laboratory and from these observations certain conclusions are drawn. Since no observation or series of observations is absolutely accurate, it is often desirable to check the dependability of the conclusions by a study of the errors in the experiment.

Suppose that an experiment on the relation between the pressure and volume of a gas is performed in the laboratory and that the conclusion is the statement of the law that the volume is inversely proportional to the pressure. The experiment does not prove that the law is absolutely accurate but only that within certain limits, determined by the accuracy of the experiment, it has been found to be true. Small departures from the law will always be found and it should be possible to determine whether these departures indicate that the law is not exactly true or whether they are due to unavoidable experimental errors. Even if in this experiment no significant departures were found, observations with more refined apparatus might show conclusively that the law was only an approximation to the truth.

II. SIGNIFICANT FIGURES: In most experiments a detailed study of the probable error is not required. Usually it is sufficient to indicate roughly how accurate the result is. In elementary work all the sure figures and one (but only one) of the estimated figures are recorded so that in merely writing down an observation an estimate of its accuracy is indicated. In Table I are recorded five observations each of the length L, width W, and thickness T, of a block of wood. The first observation of T is 3.57cm. The first two figures are known but the third figure 7 is doubtful. Although the 7 is doubtful it does have significance. We feel reasonably sure that the correct value is between 5 and 9, say, and we have, therefore, three significant figures. The location of the decimal point has nothing to do with the number of significant figures. Whether written as 3.57cm, 35.7mm, or 0.0357m, this item has three significant figures. When a zero serves merely to locate the decimal point it is not a significant figure. However the zero in the third observation of W is the first doubtful figure and is significant. To omit this zero would be wrong, for that would indicate that the preceding 2 was doubtful.

Each value of T has three significant figures and since there is considerable variation in the third figures, three significant figures should be used in expressing the average. In other words, the second 5 in the average is doubtful and the average is known to three significant figures. However, in the observations of W the variations in the third figure are so small that the third figure 1 in the average can hardly be called doubtful and in the average we are justified in keeping four significant figures.

The product LWT is 313.9735020cm³. However, this does not correctly represent the volume. It indicates that all the figures are known except the final zero, and, of course, this is far from true. Since an error of 1%, say. in anyone of the factors will cause a 1% error in the result, the volume cannot be determined to any greater degree of accuracy than the least accurate of the factors. Although it is difficult to make hard and fast rules about significant figures we may say that, in general, in multiplication and division the result should have as many significant figures as the least accurate of the factors. In some cases the answer should have one more significant figure than the least accurate of the factors. For example, in the equation 9.8 x 1.28 = 12.5, if the answer is to be as accurate as the least accurate of the factors, it must have three significant figures although the least accurate factor has only two. An inspection of the equation should make clear why this is true. The rule must be supplemented by the judgment of the experimenter. The volume of the wooden block should, therefore, be recorded as 314cm³. Do not carry worthless figures through a series of computations only to discard them at the end. To save time keep only one or two doubtful figures throughout the computations. This will not affect the accuracy of the result. In addition and subtraction the case is entirely different. Suppose that a certain metal rod is 126.73cm long at 20°C and that experiment shows that when heated to 100°C the increase in length is 0.2138cm. The new length is 126.73cm + 0.2138cm = 126.94cm. Since the numbers to which the 3 and 8 are to be added are unknown (there is no reason to believe they are zeros) the sum is known to 2 decimal places only. From the above illustration it should be clear that when numbers are properly arranged in columns for a computation in addition or subtraction, if any digit in a column of digits is unreliable, the answer in that column is likewise unreliable.

III. CLASSIFICATIONS OF ERRORS: An error that tends to make an observation too high is called a positive error and one that makes it too low a negative error. Errors can be grouped in two general classes, systematic and random. A systematic error is one that always produces an error of the same sign, e. g., one that would tend to make all the observations of some one item too small, say. A random error is one in which positive and negative errors are equally probable. Systematic errors may be subdivided into three groups: instrumental, personal, and external. An instrumental error is an error caused by faulty or inaccurate apparatus. Personal errors are due to some peculiarity or bias of the observer. External errors are caused by external conditions (wind, temperature, humidity, vibration, etc.)

To understand better these various kinds of errors, let us study the errors in a particular experiment. In this experiment two observers, in an attempt to measure the velocity of sound, use a pistol and a 1/20sec stop watch to measure the time required for sound to travel between two points. The first observer fires the pistol and the second observer uses the stop watch to measure the time required for the sound to reach him. This simple experiment may be used to illustrate all the types of errors listed above. It should be possible to determine the sources of error. analyze these errors, and assign each to its proper classification. This is done in the paragraphs immediately following.

(A) Systematic Errors.

(1) Instrumental Error. Since the stop watch is the only instrument used in this experiment, all instrumental errors must come from faults in the watch. A stop watch that did not run at the proper rate would cause an instrumental error. If the watch ran too fast all the observed times would be too high. Readings taken by a stopwatch are also subject to another type of systematic error. It takes an appreciable time merely to start or stop the mechanism of a watch. If the lag at starting is not equal to the lag at stopping, error will result. To reduce instrumental errors the instrument should be checked with an accurate standard and the necessary corrections applied to the observations. If a high degree of accuracy is called for, the instrument may be sent to the National Bureau of Standards for calibration.

(2) Personal Error. There are several ways in which the bias of the observer or his particular method of taking data might produce systematic errors. Two of these will be considered. Owing to the difference in the way he reacts to the flash of the pistol and the report that he hears, an observer might tend always to get time readings which are too large, say. It also might be true that having been warned by the flash of the gun he will be more alert and react more quickly in stopping the watch than in starting it. Personal errors maybe minimized by taking the observations under various conditions and by using several observers working independently.

(3) External Errors. External errors are usually caused by conditions over which the observer has no control. Therefore they cannot be eliminated, but necessary corrections may be applied. In this experiment the error due to wind might be quite serious. To keep this error small the experimenters might follow any of the following procedures: choose a time when there is very little wind, measure the wind velocity and make the necessary corrections, or change positions with each other so that when the results are averaged the errors tend to cancel out.

(B) Random Errors. In this experiment 26 observations of the time required for sound to travel between two points are taken. These observed times are recorded in column II of Table II. Since a systematic error would affect each observation in the same way, the variations in this column are not due to systematic errors. We believe that each of these errors is due to a large number of factors each of which adds its own small contribution to the total error. Since these factors are unknown and variable it is assumed that the resulting error is a matter of chance and therefore positive and negative errors are equally probable. Such errors are called random errors; some authors prefer to call them accidental errors. Owing to the fact that random errors are subject to the laws of chance, their effect on the experiment maybe made quite small by taking a large number of observations. It should be clear why increasing the number of observations has no effect on systematic errors.

IV. PROBABLE ERROR: Since the variations in the observed times t (column II, Table II) are governed by chance, one may apply the laws of statistics to them and arrive at certain definite conclusions about the magnitude of the errors.

No attempt will be made to derive these statistical laws, but the ones that are pertinent to this discussion will be simply stated. Along the horizontal axis of Fig. 1 are plotted observed times and each dot represents one observation. For example, three of the twenty-six observations of time gave 1.60sec. It is clear from this figure that the data tend to cluster about a certain mean value. What value is the one having the highest probability of being correct? To answer this question the methods of statistics are used and although the proof is rather difficult the conclusion is quite simple. It indicates that the best average is obtained by dividing the sum of the t’s by the number of observations n. This is the simple method of averaging with which the reader is already familiar and an average obtained in this way is known as the arithmetic mean a.m. In other words, the arithmetic mean, obtained by dividing the sum of the observed values by the number of observations taken, represents the best value obtainable from a series of observations. The a.m. in this experiment is found to be 1.540sec and is represented by the vertical line in Fig. 1.

The difference between an observation and the arithmetic mean a.m. is called the deviation d and the average deviation a.d. is a measure of the accuracy of the experiment. Obviously, the average deviation is the sum of the deviations divided by the number of observations, a.d. = Σd/n. Deviations of the observations from the a.m. are recorded in column III, in which negative deviations are placed on the left side of the column and positive deviations on the right. Adding separately it is seen that the sum of the positive deviations is approximately equal to the sum of the negative deviations. Ideally they should be exactly equal.

To study the distribution and significance of the deviations, these deviations are plotted in Fig. 2 in much the same way that observations were plotted in Fig. 1. Each dot represents the deviation of one observation. Let us divide the observations into groups by the vertical lines a, b. c, etc., which divide the figure into slices each 1/10 sec wide with zero deviation at the midpoint of the central slice. In the figure the number of observations in each slice is represented by across (X) at the midpoint of the slice and the best smooth curve is drawn through these points. The curve therefore represents the relation between the magnitude of the deviations and the frequency with which they occur. From this graph the following general rules may be inferred:

(1) Positive and negative deviations are equally probable.
(2) Small deviations occur more frequently than large deviations.

Theory indicates that the relation between the probable error p.e. of a single observation, the sum of the deviations Σd (added without regard to sign) and the number n of observations is given by the equation

If this equation is applied to the data in Table II, we find that

This does not mean that no observation will deviate by more than 0.075 sec from the mean. It does mean that the chances are 50 to 50 that the error of a single observation will not exceed 0.075 sec; or. what amounts to the same thing, if a large number of observations are taken, half of these observations will have errors less than this amount. In Fig; 2 vertical lines P and P’ are drawn at d = -0.075 and d = +0.075. It can be seen that halt of the observations lie within these limits. Obviously, the probable error P.E. of the a.m. is less than the probable error p.e. of a single observation. P.E. may be computed by the equation

remembering that a.d. = Σd/n. which in this experiment gives

and one may write as the result of the twenty-six observations of time t that t = 1.540 ± 0.015. Again this does not mean that one can be sure that the correct value is between 1.525 sec and 1.555 sec but that the chances are even that it lies between these limits.

Comparing Eq. (1) with Eq. (3) it is found that the probable error of the mean of n observations is 1/√n times the probable error of a single observation. P .E. = p.e./√n. For example, the result obtained by averaging 9 observations is three times as reliable as a single observation and 81 observations are three times as reliable as 9. Since the accuracy increases as the square root of the number of observations taken, it is evident that an observer is not justified in spending the time required to take a very large number of observations. For most experiments 5 or 10 observations should be sufficient.

V. PROBABLE ERROR-ANOTHER METHOD: The probable error may also be computed from the sum of the squares of the deviations Σd². The squares of the deviations are given in column IV, Table II. This method is more tedious and only slightly more accurate than the method discussed in Section IV. The equations used in this method and the numerical results for this experiment are given below. where the symbols have the same significance they had in Eqs. (1). and (3).

It is seen that this method gives substantially the same probable error as the one used in Section IV.

It was stated in Section IV that the best average of a series of observations is the a.m. It can also be shown that the best average is the value which makes Σd² a minimum. For this reason the branch of mathematics which has been employed in the study of errors is often called The Method of Least Squares.

VI. PERCENTAGE ERROR: In a great many cases one is not so much interested in the numerical error as in the percent of error. For example, in Section IV it was found that the probable error of a single observation is 0.075 sec. It is often desirable to express the error in percent of the thing being measured. The probable percentage error p.p.e. of a single observation is (.075/1.54) 100% or 4.9%; the odds are even that an observation will not deviate more than 4.9% from the mean. Similarly, the probable percentage error P.P.E. of the mean is (.015/1.54) 100% or .97%.

VII. PROPACATION OF ERRORS: So far this discussion has been limited to a study of the errors in a group of observations all measuring the same thing, namely, the time t required for sound to travel between two points. Since in this experiment the observers are interested in the velocity of sound, it will be necessary to measure the distance S between the two points and compute the velocity v from the measured values of t and S. What effect do errors in t and S have upon v? Can general laws be formulated governing the effect of errors in the several items on the computed result? We shall consider only two cases.

Addition and Subtraction. The probable error of the result is the square root of the sum of the squares of the probable errors of the separate items. For example, (12.15 ± 0.03)cm + (8.63 ± 0.04)cm – (6.15 ± 0.05)cm = (14.63 ± .07)cm, since √ (0.03² + 0.04² + 0.05²) = 0.07

Multiplication and Division. When the result is obtained by multiplication and division its probable percentage error is determined by the application of the following two rules:

(1) The P.P.E. of the result is the square root of the sum of the squares of the P.P.E.’s of the factors.
(2) In case a factor is raised to the nth power its P.P.E. should be multiplied by n.

For example, let us assume that the density D of a certain cylinder is computed from the equation D=Mπr2h and that the P.P.E.’s of M, r, and h are 2%, 3%, and 5%, respectively. Then the P.P.E. of D is 8% since ‘√(2² + (2 x 3)² + 5² = 8. From this it is clear that a 3% error in the radius r will more seriously affect the result than a 5% error in the height h. Having determined the P.P.E. of the result, the P.E. may readily be computed.

VIII. PROBABLE ERROR WHEN SYSTEMATIC ERROR IS PRESENT: In the discussion of probable error above only random errors were considered. Systematic errors will always be present and in some cases will be sufficiently large to affect the reliability of the result. The probable value of the systematic error may be determined from a separate experiment or, with an experienced observer, may be estimated. Calling the probable error of a single observation due to random errors r, the probable systematic error r₁, the probable error of the result due to both causes r_o, and the number of observations n, it can be shown that

If r and r₁ cannot be expressed in the same units, probable percentage errors must be used.

Eq. (7) shows that if n is large it is the systematic error that determines the reliability of the result and a very large number of observations would not be justified. In many experiments r₁ is small and only random errors need be considered.

IX. GENERAL: In the study of errors we have applied certain statistical laws to 26 observations. Actually these laws apply accurately only when the number of observations is quite large. It is unusual for 26 observations to agree as well with theory as the ones given in Table II. To those interested in a more rigorous and more complete treatment of errors the following books are recommended:

H. M. Goodwin, Precision of Measurements and Graphical Methods, McGraw-Hill, 1920;

D. Brunt, The Combination of Observations, Cam-bridge University Press, 1931;

G. V. Wendell and W. L. Severinghaus, A Manual of Physical Measurements, Privately Printed, 1918.

J. W. Mellor, Higher Mathematics for Students of Chemistry and Physics, Longmans Green & Co., 1929.

From: Cenco Physics Selected Experiments in Physics (No. 71990-001), Copyright, 2003, Sargent-Welch Scientific Company.

Charles’ Law – Experiment One

bpearson — Tue, 02 Mar 2010 18:12:36 +0000

OBJECT: To study the expansion of gases, to check Charles’ law and to measure the temperature coefficient of pressure increase of dry air at constant volume.

METHOD: Charles’ law for the expansion of gases is studied by the use of a simple form of constant-volume air thermometer. A fixed volume of dry air is subjected to certain measured temperatures and the corresponding pressures observed. From the resultant pressure-temperature curve the temperature coefficient of pressure increase at constant volume is determined. By extrapolating this curve the value of “absolute zero” is determined approximately.

THEORY: When the temperature of a confined gas is changed, the gas win change in volume if the pressure upon it is kept constant, or it will exert different pressures if the volume is kept constant. The present experiment is restricted to a study of the variation in pressure of dry air when its temperature is changed and its volume is kept constant.

Pioneer workers in this field were the Frenchmen Jacques Charles and L. J. Gay-Lussac. The law of the expansivity of gases was independently discovered by them and is variously known by each of their names. In this experiment the more common usage is followed by referring to it as Charles’ law. These workers-and independently, John Dalton- found that the pressure of a gas kept at constant volume changes linearly as the temperature of the gas is varied. If the pressure is plotted against temperature, a curve similar to that shown in Fig. 1 is obtained. The equation of the straight line may be written

P_t = P₀(1 + β_vt)

where P_t represents the pressure at the temperature t and P_o is the pressure at some standard initial temperature, usually taken at 0°C. The quantity represented by βv is called the temperature coefficient of pressure variation at constant volume. It is defined by the equation

β_v = (P_f − P₀)/P₀t

or, in words, the temperature coefficient of pressure variation of a gas at constant volume may be defined as the fractional change in pressure per unit temperature change, the initial pressure being measured at zero degrees centigrade. The slope of the pressure-temperature curve divided by P_o (the pressure when t = 0°C) offers a convenient method for determining the coefficient of pressure variation.

It is a fact of extraordinary interest that the experimental value of βv for most common gases turns out to be approximately 1/273 per degree centigrade. This means that for every degree centigrade change in temperature above or below zero degrees centigrade, the pressure changes by 1/273 of the pressure which the gas exerts at zero degrees centigrade (the volume being kept constant). Hence if the temperature were lowered by 273°C below 0°C, the change of pressure would be 273 x 1/273 of P0 or the change of pressure would equal the initial pressure at 0°C and the final pressure would be zero! This irreducible minimum of temperature is called absolute zero, i.e., the temperature of an ideal gas at which molecular activity ceases and the pressure consequently is zero. Its value is roughly checked in this experiment by extrapolating (projecting beyond the measured values) the observed pressure-temperature curve until it intersects the axis of zero pressure, as in Fig.2. This should occur at a place where t = -273°C = 0°K (degrees Kelvin are the units of temperature on the absolute scale of temperatures).

A careful distinction should be drawn between a linear relation and a direct proportion in the present and many similar cases. The pressure here varies linearly with the temperature in degrees centigrade, as indicated by Eq. (1) and Fig. 1. It is also true that the pressure is directly proportional to the temperature in degrees Kelvin (absolute), but the pressure is not directly proportional to the temperature expressed on any other temperature scale. This may be more clearly seen by substituting the value 1/273 for, βv in Eq. (1) and obtaining

P_t = P₀(1 + (t/273)) = P₀((273 + t)/273) = (P₀/273)T = CT (3)

where T = temperature in degrees Kelvin and C is a constant for any given case. From Eq. (3) and Fig. 2 it is apparent that the pressure is directly proportional to the temperature only if the latter is measured on the absolute scale.

APPARATUS: The Charles’ law apparatus, or constant-volume air thermometer, is shown in Fig. 3. It consists essentially of a metal bulb to contain the gas under investigation, connected to an iron reservoir containing mercury in which a vertical glass tube is held by a tight stuffing box. The volume of the gas is kept constant by adjusting the mercury level until it is always brought to a fixed line etched on a short section of glass tubing through which the bulb is connected to the mercury reservoir.

A copper jacket surrounds the gas bulb proper; it is provided with tubulures so that heating water may be added or removed as desired to control the temperature of the confined air. The water may be brought to and maintained at any particular temperature by an electric immersion heater, if desired. Adjustment of pressure between the gas and the mercury column of the open tube at the right is accomplished by turning a small-pitch screw by means of a large milled head. The end of the screw presses against a corrugated steel diaphragm which forms one side of the mercury reservoir and whose movement forces the mercury up into the manometers. *

*In one common modification of this type of Charles’ law apparatus (Fig. 4) an additional vertical tube is inserted in the mercury reservoir for use when the apparatus is to be used in a study of Boyle’s law. During the Charles’ law experiment this portion of the apparatus may be ignored, as it will have no effect on the validity of the other readings.

To facilitate reading the pressure a horizontal line has been placed on the meter stick at the side of the open tube, the line being at the same height as the etched line on the tubing of the gas bulb. When, therefore, the mercury level on the closed tube is adjusted to this etched line, the actual pressure on the gas is merely the barometric pressure plus the difference between the height of the mercury at the top of the open tube and the height of the index line.

Since Charles’ law does not hold for vapors it is essential that the air introduced into the gas bulb be perfectly try.

Never allow the level of the mercury in the tubes to come below the lower end of the meter stick, as to do so will often allow air to enter the closed bulb and thus to vitiate the results.

As auxiliary apparatus there are required a l00°C thermometer, a steam generator and a Bunsen burner or an electric heater, two metal vessels for water and ice, a pinch clamp and suitable rubber tubing.

PROCEDURE:
Experimental: A series of readings will be taken of the pressure of the air in the closed bulb when kept at constant volume and adjusted to various measurable temperatures ranging from 0°C to 100°C. The temperatures will be read by a mercury thermometer and the pressures will be obtained from the barometric height and the open-tube manometer.

Fill the water jacket with a mixture of chipped ice and water. After equilibrium is attained, adjust the pressure until the mercury is brought to the line etched on the glass in the open portion of the metal tubing. Measure the actual pressure P on the gas as given by

P = B + (M − I) (4)

where B is the atmospheric or barometric pressure, M is the height of the mercury in the open tube and 1 is the height of the index line.

Heat the ice water to about 20°C by means of the electric heater (or run it off and introduce other water at room temperature), adjust the volume to the fixed value as before and again read the pressure and temperature. Repeat this process by approximately 20°C intervals until the water is near the boiling temperature. A final convenient value to observe is the one in which steam is passed to the upper tubulure from a steam generator. Arrange a rubber tube from the lower tubulure to a suitable vessel to catch the condensed water vapor.

WARNING! As the steam enters the jacket the mercury in the gas tube will descend and may go below the lower end of the meter stick and admit air to the bulb. This should be prevented by raising the pressure to keep the level of the mercury on the gas-bulb tube at the index line. Throughout the entire experiment one observer should continually watch the pressure and keep adjusting the screw to maintain the mercury levels at the desired values. When the hot water is removed from the vessel or when the steam is discontinued, the pressure will greatly decrease and the mercury might run up into the closed bulb if the adjusting screw were not manipulated to lower the level of the mercury in the closed tube.

The temperature of the steam is best obtained from a table of boiling points against pressure.

Interpretation of Data: Plot a curve to show the variation of the pressure with the temperature in degrees centigrade. Explain clearly the significance of the shape of this curve.

Determine the slope of the curve, choosing points near its ends, and divide the slope by Po. This should give the value of βv; determine the percentage difference between this value and the standard value, 0.003663 per degree centigrade.

Plot a curve from the observed data to show the variation of pressure with absolute temperature. Start the pressure scale with zero and the temperature scale with -273°C or 0°K. Include the values of the temperature in both degrees centigrade and degrees Kelvin on the temperature scale. Extrapolate by a dotted line the observed data to the axis of zero pressure. Compare this temperature intercept with the standard value for absolute zero. What is the ratio of the slope of the curve and the pressure at 0°C? Explain in the report the significance of the curve.

QUESTIONS:

The gas bulb of the Charles’ law apparatus is made of cast iron. What happens to its volume when the temperature is changed from 0°C to 100°C? What effect does this have upon the “constant volume” assumption? Is the error serious? Why?
In the Charles’ law apparatus as used, a portion of the air bulb is connected to the mercury reservoir by a glass capillary tube, the level of the mercury always being brought to a fixed point on this tube. Hence some of the air above this line but below the metal bulb is not subject to the same temperature as the air inside the bulb. Will this introduce a serious error into the results? Why?
The level of the mercury in the open arm of a Charles’ law apparatus stands 5cm below the index when the volume bulb is surrounded by ice water, and 20cm above the index when the bulb is surrounded by steam at normal barometric pressure. Calculate the temperature coefficient of pressure increase at constant volume from these data.
The pressure of a gas is measured as 100cm of mercury at 50°C and 114.3cm of mercury at 100°C, volume constant. What value of absolute zero is obtained from these data?
An automobile tire gage registers a pressure of 35lb./sq. in. at the start of a trip. After 10 miles of driving the same tire gage registers 37lb./sq. in. on the same tire although no air has been added and the barometric pressure is unchanged. Explain what pressure the tire gage reads and account for the difference in readings by some physical law. Assume the tire gage reads perfectly.

Equipment List

From: Cenco Physics Selected Experiments in Physics (No. 71990-341), Copyright, 2003, Sargent-Welch Scientific Company.

Centripetal Force by Graphical Induction

bpearson — Sun, 28 Feb 2010 16:24:16 +0000

OBJECT: To determine empirically the functional relationship of the centripetal force (a) to the mass of the moving body, (b) to its tangential velocity and (c) to the radius of the circular arc in which the mass moves.

METHOD: The centripetal force acting on a pendulum bob, when the bob passes across its rest position, is measured directly with an equal arm balance. Data are taken to show how this force is dependent on the mass of the bob, on the radius of the arc, and on the tangential velocity of the bob. The centripetal force equation is derived from these data and the graphs are drawn.

THEORY: In 1687 Sir Isaac Newton announced his famous three laws of motion. The second law, F = ma, expressed the fact that a force is required to accelerate a given mass. Acceleration is defined as the ratio of change in velocity to time involved. Since velocity is a vector quantity, this change in velocity may be either one of magnitude (speed), or of direction, or a combination of both.

When the path of motion of a mass is constrained to a circular arc (Figure 1), the direction of motion is continuously changing. The force producing this directional change acts along the radius of the arc and points toward the center of rotation. This force, therefore, is designated the radial, or the centripetal, force.

In Figure 1, the mass m has a velocity of v1 tangent to the circular arc at that point. Assume that a moment later, after traversing the small angular displacement θ, the mass had the velocity of v₂. When v₂ has the same magnitude as v₁ the change in velocity Δv is a direction change only. Thus the force constraining the mass to the circular arc is the centripetal force

F_r = m (Δv/Δt)

The moving object of Fig. 1 may be the bob of a simple pendulum swinging in the circular arc ABC or radius OA, depicted in Fig. 2. The speed of the bob varies from zero at the end positions to a maximum velocity vB at its lowest or rest position. At the rest position there is momentarily no force to change the tangential velocity of the bob. At this position the string must provide the supporting weight force of the bob and the centripetal force associated with the circular arc path. Thus the string tension force less the weight force of the bob is the acting centripetal force.

The variables which might influence the magnitude of the centripetal force are (a) the mass of the bob, (b) the radius of the arc, and (c) the instantaneous velocity of the bob at its bottom position.

To the degree that a swinging pendulum has a negligible air resistance to dissipate energy, and this resistance is ordinarily accepted as negligible, the sum of the kinetic energy plus the potential energy of the bob remains constant throughout the bob’s swing. At the end positions A and C the energy is all potential and is equal to E_p = mgh. At the lowest position it is all kinetic and is equal to E_k = 1/2(mv_B²). Therefore, equating the two,

1/2(mv²) = mgh

v_B = √2gh

where v_B is the tangential velocity of the bob.

The object of this experiment is to find the functional relationships of m, v, and r to the centripetal force F_r and, thereby, obtain the laboratory equation. Measurements will be made of the relationship of F_r to each of these factors in turn, holding the other two factors constant. The data will then be graphically analyzed.

To facilitate these measurements an equal arm balance is used. See Fig. 3. A pendulum, consisting of a mass tied to a cord, is attached to one end of the balance arm. A balancing load W may be applied to the opposite end of the arm to measure the tension force in the pendulum cord. A light L in the electrical circuit glows as long as the weight force of the balancing load W can close the electrical contact at P. Thus, the pendulum bob can be weighed by finding the minimum value of W needed to close the contact when the pendulum hangs at rest.

When the pendulum bob is placed on the starting platform S and then released by pulling the starting platform to the left, the bob will swing on the arc of radius r. The velocity of the bob at B is dependent on the initial height h through which the bob drops. The light will then blink each time the bob passes B because of the centripetal force, which adds to the weight of the bob. The minimum load which must be added to W to stop the blinking gives the value of the centripetal force producing the circular motion.

APPARATUS: Centripetal force balance (Fig. 4), three different mass bobs, measuring stick and attached starting platform, weight hanger and set of weights ranging from one to five grams, battery for the electrical circuit.

PROCEDURE: Arrange the apparatus as shown in Fig. 3. The desired value of r is obtained by adjusting the position of the plate containing the string guide hole. The radius r is measured by means of the starting platform and its meter-stick support.

Note that the pin set through the bob locates its center of gravity. This pin rests on the starting platform S. See also

Fig. 5. When the starting-platform tripod is pulled gently to the left, the bob is released to swing through its predetermined arc. Thus force readings may be made repeatedly. Care must be taken to minimize vibrations which would cause a false measurement of the balance force.

EXPERIMENT:
Centripetal force as a function of v; r and m constant. Hang a pendulum bob on the balance. Adjust the plate containing the string guide hole so that the pendulum string centers in the hole and radius r has a value of about 70cm. Measure r using the starting-platform meter stick. Determine the weight of the pendulum bob.

Using a total load of approximately twice the weight of the bob, find the minimum value of h (use starting platform as shown in Fig. 5), which causes the light to flash only on the first two swings of the pendulum. Record F_r and h.

Set the starting platform, in turn, for heights which are 3/4, 2/4 and 1/4 of the value of h previously determined. Find the corresponding values of F_r, using the method stated, for each of these heights of fall.

Compute the values of v and v₂ for each h value and plot F_r vs. v and F_r vs. v₂.

What must be the relationship of the tangential velocity to the centripetal force?

Centripetal force as a function of r; m and v constant. Using the same pendulum as in the previous experiment, adjust the starting platform for a drop h of about 15cm. Record h and measure F_r.

Move the plate containing the string guide hole to positions, in turn, at which r = 3/4, 2/4 and l/4 of the value just used. Find for each r the corresponding value of centripetal force.

Plot a graph of F_r vs. r.

What conclusion can be drawn from this curve?

Centripetal force as a function of m; r and v constant. Adjust the pendulum as used in Fig. 1. Set the starting platform at a fixed height h of about 30cm. Record h. Measure, in turn, the centripetal force for each mass bob provided.

Plot F_r vs. m.

What conclusion can be drawn from these data?

Relationship of Variables. Relate the three variables by a single expression, namely, Fr in as much? Using your data find the value of the proportionality constant and write the laboratory equation.

QUESTIONS:

A car traveling on a circular track has a constant speedometer reading of 30 miles per hour. Is the car accelerated? State reason for the answer given.
The car in question 1 weighs 3200 pounds and the track has a radius of 1000 feet. What is the centripetal force acting on the car when its speed is 30 miles per hour?
By what factor must the centripetal force of question 2 be multiplied to give the new value of the centripetal force when
(a) the mass of the car is doubled?
(b) the radius of the track is doubled?
(c) the speed of the car is doubled?
Assume that in Fig. 3m = 200gm, r = 60cm, and h = 20cm. What is the tension in the cord immediately after the bob is free from the starting platform? What is the magnitude of the centripetal force?
Using the values m, r, and h of question 4 compute the tension in the cord when the mass has descended a distance h/2. (Suggestion-construct the several force vectors involved.)
Suppose that a rigid horizontal pin were placed at a point h/3 directly above B (Fig. 3) so that the cord of the released pendulum would strike the pin. Describe the resulting motion.
A mass of x kg placed on a frictionless horizontal surface is constrained by a cord to move in a circular path at a constant speed. The centripetal force is 180 newtons. How much work is done during the time the mass moves a distance of 50 meters? Support your answer with an explanation.
Suppose an astronaut were placed in orbit inside a spinning hollow sphere having a radius of 20 feet. At what speed must the sphere spin to give him his earth “weight” effect? Describe his experiences as he moves about inside the sphere.
Assume that the earth is a perfectly uniform revolving sphere with a radius of 4000 miles. A spring balance registers a chunk of gold as “1 pound” at the geographic North Pole. What “weight” reading would the spring balance give for the gold chunk at the equator?

Equipment List

From: Cenco Physics Selected Experiments in Physics (No. 71990-140), Copyright, 2003, Sargent-Welch Scientific Company.

Centripetal Force

bpearson — Fri, 26 Feb 2010 15:25:43 +0000

OBJECT: To make a study of the motion of a body traveling with constant speed in a circular path, and to verify the expression for centripetal force.

METHOD: By means of an electrically driven rotator a body of known mass is rotated about a vertical axis in such away as to produce a definite extension of a spiral spring. From the mass of the body, the radius of the circular path, and the speed of rotation, the centripetal force is computed and compared with the gravitational force necessary to produce the same extension of the spring.

THEORY: Consider the motion of a body of mass m about a point O in a circular path of radius r (Fig. 1). At any given

instant the body at A has a linear velocity v1 tangent to the circle and it would retain this linear velocity, in accordance with Newton’s first law of motion, if not acted upon by external forces. The application of a force along the path (tangent to the circle) would increase or decrease the speed of the body, while a force acting normal to the path (along a radius) would produce a constantly changing direction of motion leaving the speed unaffected.

Suppose the body in Fig. 1 to be traveling in its circular path with constant speed. It has then no acceleration along its path but, since the direction of travel is changing constantly, the velocity is non-uniform and it must have an acceleration normal to path to account for the changing velocity. This radial acceleration is directed toward the center of the circular path and is called centripetal acceleration.

According to Newton’s second law, the force f required to impart to a mass man acceleration a is

F = kma (1)

where k is a constant of proportionality, the value of which depends upon the units used. In this experiment, the c.g.s. absolute system will be used and k will be unity. The centrally directed force f on the mass m is called centripetal force. In accordance with Newton’s third law, the mass m exerts an equal and opposite force upon its restraints; this force is referred to as centrifugal force. It must be clearly understood that the centripetal force and the centrifugal force do not act upon the same body. An expression will now be derived for the centripetal force. In a small interval of time Δt, let the body move along the arc AB a distance Δs=v·Δt. The linear velocity has the same magnitude at A and B but in going from A to B the direction of motion has changed, the vector difference between v₁ and v₂ representing the change in velocity, Δv=a·Δt. This vector difference is shown in the vector diagram of Fig. 1. Then by similar triangles

Δv/Δs = (a·Δt)/(v·Δt) = v/r
whence
a = v²/r (2)

The approximation involved in taking the chord AB equal to the arc disappears as the time interval ?t is made vanishingly small. Furthermore, as ?t approaches zero the vector ?v becomes more nearly parallel to r. Hence the acceleration, the direction of which is in the direction of the change in velocity, is directed toward the point O. Substitution of the value of a from Eq. (2) in Eq. (1) yields the following expression for the centripetal force

F_c = m(v²/r) (3)

In Eq. (3) the centripetal force is expressed in terms of the linear velocity of the body. It may be expressed in terms of angular velocity ω by substituting the relationship v = ωr. Thus

F_c = mω²r (4)

ω being measured in radians per second. Usually the frequency of rotation n is more readily observable than either the linear or the angular velocity. Substituting v = 2πnr in Eq. (3) yields

F_c = 4π²n²rm (5)

In accordance with Hooke’s law the external force F necessary to produce a deformation din an elastic body is given by the equation

F = kd (6)

where the “force constant” k is defined numerically as the force necessary to cause unit deflection. The value of k depends upon the geometry of the body and the material of which it is composed. For example, the force constant of a coil spring depends upon the dimensions of the coil, the size of the wire, and the elastic property of the material.

APPARATUS: The apparatus required for the performance of this experiment consists essentially of two units: a specially designed centripetal force apparatus (Fig. 2) and an electrically driven, variable speed rotator (Fig. 3). The complete assembly with the centripetal force apparatus mounted on the spindle of the rotator is shown in Fig. 4. The centripetal force apparatus consists of a metal frame Y within which is mounted a cylindrical mass m attached to a coil spring Z, the entire assembly being rotated about a vertical axis through its center of gravity. The tension in the spring is adjusted by a threaded collar K to which the spring is fastened. In some models the position of the collar is indicated by a millimeter scale S attached to the frame. By means of three guide rods G the cylindrical body is constrained to move only along the axis of the spring, which axis intersects the axis of rotation at right angles. While at rest the cylinder is held by the spring against a stop.

When the apparatus is rotated about a vertical axis the mass moves outward producing an extension of the spring. The situation is represented diagrammatically in Fig. 5 in which the identifying symbols correspond to those in Figs. 2 and 4.

A specially designed pointer P is loosely pivoted at O and is so shaped that when the cylinder presses against it at Q its tip moves upward through a range of about 5mm. At the middle of this range is a fixed index I. In operation the speed is adjusted until the pointer is opposite the index. Since the index is practically on the axis of rotation, the position of the pointer can be seen clearly while the apparatus is rotating.

The frequency of rotation is determined by counting the number of revolutions occurring in a given interval of time. This observation is facilitated by means of a revolution counter C attached to the frame of the rotator by means of a steel spring which normally holds the counter disengaged from the rotating spindle. By pressing with the finger on the end of the spring the counter gear is engaged with an identical gear on the spindle. The speed of rotation of the spindle is controlled by adjusting the point of contact of the friction disk D with the driving disk W. Turning the milled head H of the screw J carries the friction disk in or out along the radius of the driving disk.

As auxiliary apparatus there are needed a stopwatch or a clock with a sweep seconds hand, a weight holder, miscellaneous weights, a vernier caliper, and rods and clamps for suspending the centripetal force apparatus as shown in Fig. 6.

PROCEDURE:
Experimental: By means of the threaded collar adjust the spring to minimum, or nearly minimum, tension. Mount the centripetal force apparatus securely upon the rotator spindle taking care that the axis of rotation is vertical. Set the friction disk so that it is near the center of the driving disk and start the motor. With the eyes on a level with the index, adjust the speed control device until the pointer is just opposite the index. With one hand constantly on the control, practice regulating the speed until sufficient skill is acquired to keep the pointer vibrating about the index as its mean position S K Z M Y I P G Fig. 2. Centripetal Force Apparatus Fig. 4. Centripetal Force Apparatus mounted on Variable Speed Rotator. Fig. 3. Variable Speed Rotator 2 the pointer vibrating about the index as its mean position with as little oscillation as possible.

When some skill in manipulation has been acquired a run may be made. Record the reading of the revolution counter and, keeping one hand on the speed control, engage the counter with the other. When two experimenters are working together it is a good plan for one to regulate the speed while the other manipulates the counter and observes the time. At the end of one minute disengage the counter and record the reading. The counter may be prevented from spinning after it is disengaged by applying the finger lightly to the gear as a brake at the instant of release. Make this observation for 5 one-minute intervals, taking care that the counter reading is not altered between observations. This data yields 6 readings of the counter spaced at one-minute intervals. Record these 6 readings consecutively in a column.

The second part of the experimental procedure consists in determining the gravitational force necessary to produce the same extension of the spring as that caused by rotating the apparatus. To make this determination remove the centripetal force apparatus from the rotator and suspend it with the mass down as in Fig. 6. Attach a weight holder and add weights until the pointer is again brought to the index. The mass is then in the same position in the frame and hence the force in the spring is the same as when the apparatus was rotating. Record the total weight sustained by the spring including that of the weight holder and of the cylindrical body. The mass of the latter is stamped on it. With the vernier caliper measure the distance r between the axis of revolution (indicated by the scribed line L1 on the frame) and the center of gravity of the cylinder (indicated by the line L2). Repeat this measurement several times and record the values. Change the tension of the spring and repeat the entire experiment.

Calculations: In computing the average value of the frequency of rotation, divide the data into two parts consisting, respectively, of the first three and the last three readings of the counter. The difference between the 4th and the 1st is the number of revolutions occurring in three minutes. Likewise the differences between the 5th and 2nd and between the 6th and 3rd represent three-minute intervals. The data thus represents three over-lapping three- minute time intervals. Note that this method of averaging makes use of all the data, whereas, if the simple arithmetic mean of the 5 one-minute intervals had been taken, all readings except the first and the last would have dropped out. Express the average value of the frequency in revolutions per second. Compute the average value of r from the values obtained in the second part of the experiment. Substitute the values of mass, frequency, and radius in Eq. (5) and compute the centripetal force in absolute units. Compute the percentage difference between this force and the weight (also in absolute units) required to produce the same extension of the spring.

ALTERNATIVE PROCEDURE:

Supplementary Theory: The relationship between the force in the spring and the frequency of rotation can be shown graphically. Solving Eq. (5) for n²

n² = (1/(4π²mr))·F_c (7)

in which the force F_c satisfies Eq. (6). In this experiment the radius r is kept constant, the extension of the spring being taken up by the collar K. Since the pitch of the screw is constant, the force in the spring is directly proportional to the number of turns N of the collar. Thus

F_c = F_o + k’N (8)

where F_o is the initial force in the spring corresponding to the (arbitrary) “zero” setting of the collar, and k’ is a constant for the spring, namely, the force in the spring for one turn of the collar. Note that k’ in Eq. (8) and k in Eq. (6) are simply related, being merely two equivalent ways of expressing the force constant of the spring. Substituting F_c from Eq. (8) in Eq. (7)

n² = (1/(4π²mr)) (F_o + k’N) (9)
or
n² = C(F_o + k’N) (10)
where
C = 1/(4π²mr) (11)

Consideration of Eq. (10) shows that a graph of experimental values of n² versus N yields a value of the force constant k’ of the spring. This might be called a dynamical method of determining k’. A direct static method of determining k’ consists in measuring the weight necessary to balance the elastic force in the spring for each turn of the collar.

Experimental: Having learned how to manipulate the apparatus according to the procedure described above, adjust the spring for (nearly) minimum tension, and identify the initial setting of the collar K either by observing its position on the scale S, or by measuring the distance between it and the end of the frame. Determine the frequency by observing the number of revolutions recorded by the counter in 2 minutes. Since the results are to be determined from a graph of the data involving several independent observations, it will be sufficient to take the average value of the frequency over a two-minute interval. Make four such determinations, each time increasing the spring tension by giving the collar a certain number (say 5) complete turns. Tabulate the data.

Remove the centripetal force apparatus from the rotator and suspend it as shown in Fig. 6. Attach a weight holder of known mass. With the same initial spring setting as before, add weights to the holder until the pointer is opposite the index. Increase the tension by giving the collar a certain number (say 5) whole turns and again observe the total weight W (including the holder) necessary to bring the pointer to the index. In this manner obtain several corresponding values of W and N.

Analysis of Data: Plot curve I, as shown in Fig. 7, taking the square of the frequency n² as the ordinate and the total number of turns N of the collar (measured from the initial position) as the abscissa. Plot curve 2 (not illustrated) taking the weight W as ordinate and number of turns N as abscissa. Consideration of Eq. (10) shows that when N = 0,

F_o = n_o²/C (12)

where n_o is the frequency corresponding to the initial setting of the collar. Thus, the n2-intercept of curve 1 yields the initial force F_o in the spring. The initial force in the spring may also be obtained from the W-intercept of curve 2. Compare the two values so determined, remembering to express F_o and W_o in the same units. Again referring to Eq. (10), note that when n = 0

N_o = -F_o/k’ (13)

where N_o is the (negative) number of turns from the initial setting necessary to cause zero force in the spring when the mass m is in the operating position. Thus, the N-intercept of curve), together with the value of F_o determined from Eq. (12), yields the force constant k’ of the spring. Compare the value so determined with that obtained from the slope of curve 2.

QUESTIONS:

Which would be the more serious, a 1 per cent error in observing the time, or a 1 per cent error in measuring the radius? Why?
What are the advantages of using the same value of r and varying the spring tension for various speeds rather than keeping a constant spring tension and observing the values of r at different speeds?
Could a horizontal axis of rotation be used? Explain.
What was the angular velocity in rad/sec of the rotating body in each case?
How is the centripetal force on a rotating body affected by (a) doubling the radius, keeping the linear velocity constant; (b) doubling the radius, keeping the angular velocity constant?

Equipment List

From: Cenco Physics Selected Experiments in Physics (No. 71990-136), Copyright, 2003, Sargent-Welch Scientific Company.

Boyle’s Law – Experiment 3

bpearson — Wed, 24 Feb 2010 19:08:53 +0000

OBJECT: To study Boyle’s law, at moderate pressures above and below atmospheric, by both analytical and graphical methods.

METHOD: A fixed mass of air confined in a glass tube is kept at room temperature and subjected to various pressures, ranging from half to double atmospheric pressure. A series of corresponding pressures and volumes are observed and Boyle’s law is checked by noting the constancy of their products. The data are plotted in several graphical forms the interpretation of which also indicates the validity of Boyle’s law.

THEORY: The relation existing between the pressure exerted by a confined gas and its volume is given by what is usually known as Boyle’s law, namely: The temperature remaining constant, the volume V occupied by a given mass of gas is inversely proportional to the pressure P to which it is subjected. In symbols

V ∝ 1/P
or
V = k × 1/P

whence PV = k (1)

where k is (numerically) a constant under given conditions.

It is apparent that Eq. (1) is an equation of the second degree, since the left side is the product of two variable quantities. When the pressure is plotted as a function of the volume, an equilateral hyperbola, as shown in Fig. 1, is obtained.

The actual pressure P may be thought of as consisting of the atmospheric or barometric pressure B plus an added pressure p, the algebraic sign of the added pressure depending upon whether the actual pressure is above or below atmospheric pressure. Eq. (1) may therefore be written

(B+p)V = k (2)
or
B + p = k × 1/V (2a)

By placing 1/V = x, Eq. (2) becomes

B + p = kx (3)
or
p = kx – B (4)

Since B is numerically constant (for any given case), it will be seen that Eq. (4), being of the first degree, is the equation of a straight line. If, then, not the actual pressure but merely the added pressure p may be plotted as ordinates and the reciprocal of volume, 1/V or x, be plotted as abscissas, the resultant curve should be a straight line (Fig. 2). If the curve is produced downward until it intersects the pressure axis, i.e., when x or 1/V equals zero, the intercept on the p-axis gives immediately the negative of the barometric pressure at the time of the experiment. This is obvious from Eq. (4), for if x is placed equal to 0, then

p = -B (4a)

While the present experiment is designed ostensibly to furnish a check upon the approximate validity of Boyle’s law, it also furnishes a splendid example of a study of both graphical and analytical representation and interpretation of experimental data. In addition it offers an excellent illustration of the processes used in logical scientific reasoning.

When this experiment is performed with this point of view in mind and the differences in the apparatus, procedure and precision are considered, it will be seen that the present experiment is far from a mere repetition of one of the commonest experiments of elementary-school physics.

APPARATUS:
The Barometer:
The pressure of the atmosphere may be most accurately determined by the mercurial barometer illustrated in Figs. 3 and 4. The glass tube G, closed at its upper end, projects into the mercury cistern w. The metric scale S and the British scale S’ have their zeros at the lower end of the ivory tip I. Consequently, it is necessary that the upper level of the mercury in the cistern always be brought to this point before a reading is taken. This is done by adjusting a screw at the bottom of the barometer. A vernier reading device V makes possible accurate settings and readings at the top of the mercury column.

The following procedure is used in making a measurement of atmospheric pressure by means of a mercurial barometer:

Another way of making the setting is to hold apiece of white paper behind the well as a background, keeping the eye at the level of the ivory point and adjusting the mercury surface until the light between the point and the mercury is just cut off.

2. Move the vernier up until the top of the mercury column appears below its lower edge. Tap the barometer lightly to permit the mercury to form a free meniscus. Then move the vernier downward until the sighting edges A and A’ are in line with the uppermost point of the meniscus.

3. Read the scale in millimeters and determine tenths with the aid of the vernier.

The Boyle’s Law Apparatus: The apparatus used in studying Boyle’s law is shown in Fig. 5. The mass of air on which the measurements are made is confined in the calibrated glass tube T which has a stopcock K at its upper end. The stopcock tube T and another glass tube T’ form the opposite ends of what might be designated an adjustable U-tube. The glass tubes are connected through suitable metal couplings by means of heavy, flexible tubing and this adjustable U-tube is filled with the proper amount of mercury. By means of the metal couplings C and C’ the glass tubes are supported in clamps which can be moved vertically on the support rods G and held in any desired position. Midway between the two support rods is the vertical millimeter scale B, graduated on metal. By means of this scale and a suitable reading device R, the height of mercury column in either of the two glass tubes may be ascertained within 0.1mm.

The reading device consists of a sleeve which slides freely along the graduated square tube and to which is attached the mirror M and the vernier V. A fine horizontal line etched on the mirror permits setting without parallax on the mercury columns, and after the setting has been made the reading is taken by means of the vernier. The reading device is held in any desired position on the graduated tube by means of spring friction. The clamp which holds the open tube is provided with a micrometer adjusting screw S by means of which a setting of the mercury level in either tube may be made with precision. The entire apparatus is supported on a massive tripod base A provided with leveling screws.

When the tubes are turned in toward the mirror, great care must be observed not to strike them against the top of the support.

PROCEDURE:
Experimental:
Place the apparatus in good light where the scale may easily be read. With the stopcock open, adjust the levels of mercury until a volume of air is enclosed which occupies about half the total volume of the closed tube. Level the apparatus by adjusting the leveling screws in the base until the mercury levels in the two glass tubes coincide with the horizontal line on the mirror. Then close the stopcock tightly and keep it closed throughout the progress of the experiment. (While adjusting the stopcock, steady the top of the closed tube with the hand – otherwise the glass tube may be snapped off. Fasten the stopcock with a rubber band. Keep it well lubricated with stopcock grease.)

Test the system by lowering the open tube or raising the closed tube to decrease the pressure as far as the apparatus will conveniently permit. Allow it to remain in this condition for a few minutes and note whether there is any change in the mercury levels. The barometer may be read during this interval.

Take a series of ten or twelve readings of volumes at various pressures ranging from the lowest to the highest attainable. The volumes are read directly on the graduated glass scale of the closed tube. The pressure is determined by reading the height of the mercury columns on the open and closed tubes and subtracting these values to obtain the “added” pressure p. The actual pressure P is B + p when the level in the open tube is above that in the closed tube.

In tabulating the date the following should be recorded: (a) the reading of the mercury level in the open tube; (b) the reading of the level in the closed tube; (c) the “added” pressure p; (d) the actual pressure B + p; (e) the volume V; (f) the product PV; (g) percentage difference between the observed PV and the means of all the values; (h) 1/V.
In varying the pressure it is convenient to set the mirror index on the closed tube at 1/2cc reduction in volume each time and to raise the open tube until the pressure is properly adjusted to give the desired volume. Final adjustment of the levels is made by the use of the slow-motion screw on the open tube. When the open tube is near the top of the frame, the closed tube may be lowered to produce the same effect as further raising of the open tube.

Make all changes slowly, to avoid changing the temperature, and wait a few minutes before taking the readings. Do not handle the closed tube after the preliminary adjustments are completed. (Why?) While changing the positions of the tubes be careful to avoid spilling the mercury out of the open tube.

Interpretation of Data: Calculate the various PV products and take their average. Determine the percentage variation between the individual values of PV and their mean. What is the physical significance of the constancy of the various values of PV?

Carefully interpret in the report the significance of these curves. From curve (2) determine the barometric pressure by Fig. 4. Detail of Mercury Barometer Fig. 5. Boyle’s Law Apparatus. The insert shows a “close-up” of the upper end of the tubes. 3 4 extrapolating the observed portion of the curve (use a dotted line for extrapolated portion) back to the intercept where 1/V = 0. Compare this pressure-intercept value of B with that observed by the barometer.

QUESTIONS:

1. Show by dimensional reasoning that the constant k in the equation PV = k is not a mere proportionality constant, i.e., one having no unit, but that it has the dimensions of work. On what does the value of k depend?
2. Considering the accuracy of the observations of the volume in the present experiment, was it necessary to measure the pressures to tenths of millimeters in order to secure accurate values of PV? Would more precise readings of the mercury levels have given more accurate values of PV?
3. A certain automobile tire is labeled, “Inflate to 35lb. or 2.5kg.” What does this mean? Is the statement clear? Is 35lb. equal to 2.5kg?
4. The moving coil of an ammeter weighs 1/2 gram and is supported in a jewel bearing by a needle-point pivot which is rounded off to have a radius at the tip of 75/100,000 inch. Calculate the pressure, in pounds per square inch, which the pivot exerts on the bearing.
5. The mercury stands at a height of 74cm in a barometric tube. The top of the tube is 6cm above the top of the mercury column and the cross-sectional area of the tube is 0.6cm2. A quantity of air is introduced above the mercury which then falls to 64cm. What was the volume of the air before it was introduced into the tube?
6. The closed end of uniform U-tube containing mercury has an air space 20cm long. The mercury in the open arm stands 5cm lower than in the closed arm. If more mercury is poured into the open arm until its level in this arm is 10cm higher than in the closed arm, how long will the air space be? The barometric height is 75cm of mercury.
7. A barometer having a little air in the top of the tube has the mercury 70cm above that in the cistern. If the level of the mercury in the cistern is raised so that the volume of the air space at the top is half as great as before, the mercury column stands 67cm above the level of that in the reservoir. What is the barometric pressure at the time?

Equipment List

From: Cenco Physics Selected Experiments in Physics (No. 71990-332), Copyright, 2003, Sargent-Welch Scientific Company.

Boyle’s Law – Experiment 2

bpearson — Mon, 22 Feb 2010 19:08:51 +0000

OBJECT: To study Boyle’s law, at moderate pressures above and below atmospheric, by both analytical and graphical methods.

METHOD: A fixed mass of air confined in a glass tube is kept at room temperature and subjected to various pressures, ranging from half to double atmospheric pressure. A series of corresponding pressures and volumes are observed and Boyle’s law is checked by noting the constancy of their products. The data are plotted in several graphical forms the interpretation of which also indicates the validity of Boyle’s law.

V ∝ 1/P
or
V = k × 1/P

whence PV = k (1)

where k is (numerically) a constant under given conditions.

(B + p)V = k (2)
or
B + p = k × 1/V (2a)

By placing 1/V = x, Eq. (2) becomes

B + p = kx (3)
or
p = kx – B (4)

p = -B (4a)

The following procedure is used in making a measurement of atmospheric pressure by means of a mercurial barometer:

1. Adjust the mercury level to the tip of the ivory point by turning the well upward with great care until the tip of the ivory point just touches the mercury surface. When the mercury is clean this contact may be judged by noting apparent contact between the tip of the point and its image in the mercury. When the mercury has become contaminated, the mercury level is adjusted upward until the point has made the slightest depression discernible in the mercury surface. Another way of making the setting is to hold apiece of white paper behind the well as a background, keeping the eye at the level of the ivory point and adjusting the mercury surface until the light between the point and the mercury is just cut off.

3. Read the scale in millimeters and determine tenths with the aid of the vernier.

The Boyle’s Law Apparatus: The Boyle’s law apparatus, Fig. 5, consists of two vertical glass tubes, one open at the top and the other closed by a stopcock, held at the lower end by stuffing boxes in an iron reservoir. The reservoir is mounted on a tripod base and is provided with a screw-operated diaphragm for varying the height of the mercury in the tubes and so changing the pressure and volume of the gas confined in the closed tube.*

*In one common modification of this form of Boyle’s law apparatus (Fig. 6) an additional metal-bulb reservoir is provided for use in another experiment (Charles’ law). When the apparatus is being used to study Boyle’s law, this bulb is sealed off by tightly closing the needle valve provided for that purpose. Thereafter this portion of the apparatus may be ignored. To see that there is no leak into this bulb during the progress of the Boyle’s law experiment, it is wise to have the level of the mercury in the metal-bulb side of the tube just at the index mark on the glass window and to observe from time to time that this level does not change.

The large, milled-head screw has a small pitch and it presses against the corrugated steel diaphragm to form one side of the mercury reservoir. Readings of the mercury levels to measure the corresponding pressures and volumes are taken from a metric scale mounted vertically between the tubes. A sliding glass cursor provided with a horizontally Fig. 3. Mercury Barometer 2 etched hairline makes it conveniently possible to read the mercury levels with satisfactory precision. The air admitted to the closed tube should be carefully freed from moisture by a method described later.

PROCEDURE:
Experimental: Place the apparatus in good light where the scale may easily be read. Before beginning the experiment, the instructor or student should be sure that the air in the closed tube is perfectly free from moisture. Introduction of dry air into the tube is effected by connecting a Dessigel S drying tube to the stopcock tube and running the mercury up and down in the tube, pumping dry air in and out and thereby removing water vapor. The stopcock tube is finally tightly closed at a place where the volume of the entrapped air is about one half the total volume of the closed tube when the levels of the mercury in both tubes are the same.

Test for leaks by turning the milled head until the mercury is near the top of the open tube and observing it for a few minutes to see that the level remains constant. Check this also by having the mercury in the open tube near the bottom of the tube. To be sure that no air bubbles are present, turn the adjusting wheel back and forth several times to move the mercury in the open tube from the top to the bottom, meanwhile watching both tubes for bubbles. While this is being done the barometer may be read, as described above.

Never allow the level of the mercury in the tubes to come below the lower end of the meter stick, as to do so will often allow air from the reservoir to enter the closed tube and thus to vitiate the results. During the experiment never adjust or open the stopcock on the closed tube, as to do so will change the mass of air, admit moist air and make necessary a complete drying of a new mass of air.

As the volume of the closed tube is not calibrated directly in cubic centimeters, the volume of the gas will be measured here in terms of the length of the tube above the mercury, since for a uniform bore the volume is proportional to the length. Since it is impossible to seal the glass stopcock to the end of the tube and still have a uniform bore right up to the barrel of the stopcock, an etched line on the tube is placed at such a point that the volume in the capillary between that line and the stopcock barrel represents exactly the volume of 1cm of uniform capillary bore. Hence the corrected scale reading for the top of the enclosed air column is obtained by adding 1cm to the reading of the scale just opposite the etched line on the tube below the stopcock. This corrected value will be designated R_t.

Tabulation of Data: Tabulate the following: (a) R_o, the reading of the open-tube mercury level; (b) R_c, the reading of the closed-tube mercury level; (c) p, the added pressure, i.e., R_o – R_c; (d) P, the pressure, or B + p; (e) V, the volume or R_t – Fig. 4. Detail of Mercury Barometer Fig. 5. Boyle’s Law Apparatus. The insert shows a “close-up” of the upper end of the tubes. Fig. 6. Combination Boyle’s Law and Charles’ Law Apparatus. 3 4 Rc; (f) 1/V; (g) PV; (h) percentage variation of PV from the average value of all PV’s.

Note that when the closed-tube readings exceed those on the open tube the confined gas is below atmospheric pressure and the values of p must be subtracted from B.

Interpretation of Data: Calculate the various PV products and take their average. Determine the percentage variation between the individual values of PV and their mean. What is the physical significance of the constancy of the various values of PV?

Curves: Plot the following curves: (1) P vs. V (begin both axes at zero); (2) “added” pressure p vs. reciprocal of volume 1/V. Choose the axis of p near the center of the page and be sure that -p extends as far as 770mm below the axis. In laying off the scale for 1/V, begin with 1/V = 0 at the origin.

Carefully interpret in the report the significance of these curves. From curve (2) determine the barometric pressure by extrapolating the observed portion of the curve (use a dotted line for extrapolated portion) back to the intercept where 1/V = 0. Compare this pressure-intercept value of B with that observed by the barometer.

QUESTIONS:
1. Show by dimensional reasoning that the constant k in the equation PV = k is not a mere proportionality constant, i.e., one having no unit, but that it has the dimensions of work. On what does the value of k depend?
2. How would a P vs. V curve for data taken (a) at a higher temperature, (b) at a lower temperature, compare with the curve as actually obtained?
3. A certain automobile tire is labeled, “Inflate to 35lb. or 2.5kg.” What does this mean? Is the statement clear? Is 35lb. equal to 2.5kg?
4. The moving coil of an ammeter weighs 1/2 gram and is supported in a jewel bearing by a needlepoint pivot which is rounded off to have a radius at the tip of 75/100,000 inch. Calculate the pressure, in pounds per square inch, which the pivot exerts on the bearing.
5. The mercury stands at a height of 74cm in a barometric tube. The top of the tube is 6cm above the top of the mercury column and the cross-sectional area of the tube is 0.6cm2. A quantity of air is introduced above the mercury which then falls to 64cm. What was the volume of the air before it was introduced into the tube?
6. The closed end of uniform U-tube containing mercury has an air space 20cm long. The mercury in the open arm stands 5cm lower than in the closed arm. If more mercury is poured into the open arm until its level in this arm is 10cm higher than in the closed arm, how long will the air space be? The barometric height is 75cm of mercury.
7. A barometer having a little air in the top of the tube has the mercury 70cm above that in the cistern. If the level of the mercury in the cistern is raised so that the volume of the air space at the top is half as great as before, the mercury column stands 67cm above the level of that in the reservoir. What is the barometric pressure at the time?

Equipment List

From: Cenco Physics Selected Experiments in Physics (No. 71990-331), Copyright, 2003, Sargent-Welch Scientific Company.

Boyle’s Law

bpearson — Sat, 20 Feb 2010 16:22:13 +0000

OBJECT: To investigate the relationship between the pressure and the volume of a confined mass of gas at Constant temperature.

METHOD: Amass of dry air is trapped above a column of mercury in a closed tube which forms one arm of a mercury manometer. The pressure upon the confined air can be regulated by means of a plunger which controls the height of the mercury columns, and its value is determined from the difference between the mercury levels in the open and closed arms. The volume of the confined air is measured by the length of the closed tube above the mercury level. From a series of determinations of pressure and volume, curves are plotted showing the relation between pressure and volume.

THEORY: In 1661 the British scientist Robert Boyle made an exhaustive experimental study of the variation of the pressure and volume of a given mass of gas under constant temperature. As a result of his experiments he concluded that: Temperature remaining constant, the volume of a confined gas varies inversely as the pressure to which it is subjected. This relationship, discovered experimentally by Boyle and since shown to be true from theoretical considerations, is known as Boyle’s law. Mathematically the law may be written

V = k(1/P) (t constant) (1)
or
PV = k (t constant) (2)

Thus Boyle’s law may be stated as follows: At constant temperature the product of the pressure and volume of a given mass of gas is constant. The numerical value of the constant k depends upon the mass of the gas used and the temperature of the gas.

APPARATUS: The Boyle’s law apparatus represented diagrammatically in Fig. 1 and illustrated in Fig. 2 consists of three vertical tubes A, C and D connected together at their lower ends. The tubes A and Care open at the top while the upper end of D is closed. The large tube A serves as a reservoir for mercury, and the mercury levels in the tubes C and D can be controlled by means of a wooden plunger E inserted in the reservoir. When mercury is poured into A some air is trapped in the closed arm D. The volume V of the confined air is proportional to the length of the air column (since the cross section is uniform) and the pressure is measured by the difference in height h of the mercury columns in C and D (Fig. 1).

The values of V and h are measured by a scale S mounted beside the tubes. The pressure on the surface of the mercury in the open tube C is atmospheric and can be determined merely by reading a barometer. The pressure on the air confined in the closed tube D differs from atmospheric pressure by an amount equal to the pressure exerted by the column of mercury h (Fig. 1); it is greater or less than atmospheric pressure depending upon whether the level in C is above or below that in D. Thus, if B represents the barometric reading expressed in centimeters of mercury, the pressure P of the confined air is

P = B ± h (3)

The necessary auxiliary apparatus consists of a mercury barometer, a series of drying tubes, and an aspirator pump.

PROCEDURE:

Experimental: Before undertaking to perform the experiment the apparatus must be filled with dry air, since Boyle’s law applies to a gas but not to a vapor near its condensation point. A convenient way of introducing dry air is illustrated in Fig. 3. The drying process should be carried out only under the direct supervision of the instructor.

A series of drying tubes containing calcium chloride, or other drying agent, is connected to the open tube C. The air is exhausted from the apparatus by means of an aspirator pump connected to the tube A. Close the pinchcock K and start the pump. After pumping a minute or two, open the pinchcock and permit air to be drawn in through the drying tubes. In this manner flush the apparatus with dry air two or three times before beginning the experiment.

Read the laboratory barometer. Pour about 25cc of clean mercury into the reservoir A. Insert the plunger E and note the rise of the mercury in the tubes C and D as the plunger is depressed. The quantity of mercury should be such that the level in C is near the top of the tube when the plunger is pushed to the bottom of the reservoir. Under these conditions it will be noted that the level in D is below that in C, indicating that the confined air is under pressure somewhat above atmospheric. Tilt the apparatus and manipulate the plunger so as to permit some of the trapped air to escape, leaving the mercury level in D a few centimeters above the level in C.

With the plunger withdrawn, take initial readings of the levels in C and D. Push the plunger into the reservoir until the mercury rises four or five centimeters in the open tube C and again read the levels in C and D while holding the plunger in position; Continuing in this manner take a series of eight or ten readings up to the maximum attainable pressure. Record the data as shown in Table I. Take a second reading of the laboratory barometer and record the average value over the duration of the experiment.

Interpretation of Data: By means of Eq. (3) compute the values of the pressure P and enter in column V of the table. Compute the products of corresponding values of P and V and enter in column VI. Plot Curve 1 of pressure versus volume with P as the ordinate and V as the abscissa. On the same sheet of graph paper plot Curve 2 of pressure versus reciprocal volume with P as the ordinate and 1/V as the abscissa.

QUESTIONS:

1. Show how Curve 1 conforms with Boyle’s law. What type of geometrical figure is it?
2. Describe the appearance of Curve 1 at extremely large and at extremely small values of the pressure.
3. What does Curve 2 indicate about the consistency of the results? Explain.
4. Does the diameter of tube C have to be uniform? Explain.
5. What would the value of h have to be in order to subject the confined air to a pressure of three atmospheres?
6. Show how the apparatus may be used to measure the barometric pressure by applying Boyle’s law to any two different settings.

TABLE I

I	II	III	IV	V	VI
Level in C	Level in D	Volume V	Pressure Difference h	Pressure P	Product PV

Equipment List

Boyle’s Law Apparatus, Adjustable Form

From: Cenco Physics Selected Experiments in Physics (No. 71990-330), Copyright, 2003, Sargent-Welch Scientific Company.

Boyle’s and Charles’ Laws

bpearson — Thu, 18 Feb 2010 19:14:14 +0000

OBJECT:To study Boyle’s law and Charles’ law, as applied to air at moderate temperatures and pressures.

METHOD: To study Boyle’s law a fixed mass of air confined in a glass tube is kept at room temperature and subjected to various pressures, ranging from half to double atmospheric pressure. A series of corresponding pressures and volumes are observed and Boyle’s law is checked by noting the constancy of their products. The data are plotted in several graphical forms the interpretation of which also indicates the validity of Boyle’s law. Charles’ law for the expansion of gases is studied by the use of a simple form of constant-volume air thermometer. A fixed volume of dry air is subjected to certain measured temperatures and the Corresponding pressures observed. From the resultant pressure-temperature curve the temperature coefficient of pressure increase at constant volume is determined. By extrapolating this curve the value of “absolute zero” is approximately measured.

THEORY:
Boyle’s Law: The relation existing between the pressure exerted by a confined gas and its volume is given by what is usually known as Boyle’s law, namely: The temperature remaining constant, the volume V occupied by a given mass of gas is inversely proportional to the pressure p to which it is subjected. In symbols

V ∝ 1/P
or
V=k×1/P

whence

PV=k (1)

where k is (numerically) a constant for the given conditions.

(B+p)V = k (2)
or
B+p = k x 1/V (2a)

By placing 1/V = x, Eq. (2a) becomes

B + p = kx (3)
or
p = kx – B (4)

Since B is numerically constant (for any given case), it will be seen that Eq. (4), being of the first degree, is the equation of a straight line. If, then, not the actual pressure P but merely the added pressure p be plotted as ordinates and the reciprocal of volume, 1/V or x, be plotted as abscissas, the resultant curve should be a straight line (Fig. 2). If the curve is produced downward until it intersects the pressure axis, i.e., when x or 1/V equals zero, the intercept on the p-axis gives immediately the negative of the barometric pressure at the time of the experiment. This is obvious from Eq. (4) for if x is placed equal to 0, then

p = -B (4a)

While this portion of the present experiment is designed ostensibly to furnish a check upon the approximate validity of Boyle’s law, it also furnishes a splendid example of a study of both graphical and analytical representation and interpretation of experimental data. It also offers an excellent illustration of the processes used in logical scientific reasoning. When this experiment is performed with this point of view in mind, and when the differences in the apparatus, procedure and precision are considered, it will be seen that the present experiment is far from a mere repetition of one of the commonest experiments of elementary-school physics.

Charles’ Law: When the temperature of a confined gas is changed, the gas will change in volume if the pressure upon it is kept constant, or it will exert different pressures if the volume is kept constant. The present experiment is restricted to a study of the variation in pressure of dry air when its temperature is changed and its volume is kept constant.
Pioneer workers in this field were the Frenchmen Jacques Charles and L. J. Gay-Lussac. The law of the expansivity of gases was independently discovered by them and is variously known by each of their names. In this experiment the more common usage is followed by referring to it as Charles’ law. These workers- and independently, John Dalton-found that the pressure of a gas kept at constant volume changes linearly as the temperature of the gas is varied.
If the pressure is plotted against temperature, a curve similar to that shown in Fig. 3 is obtained. The equation of the straight line may be written

P_t = P_o(1 + β_vt) (5)

where P_t represents the pressure at the temperature t and P_o is the pressure at some standard initial temperature, usually taken at 0°C. The quantity represented by, βv is called the temperature coefficient of pressure variation at constant volume. It is defined by the equation

β_v = (P_t – P_o) / P_ot (6)

or, in words, the temperature coefficient of pressure variation of a gas at constant volume is the ratio of the fractional change in pressure per unit temperature change, the initial

pressure being measured at a temperature of zero degrees centigrade. The slope of the pressure-temperature curve divided by P_o (the pressure when t = 0°C) offers a convenient method for determining the coefficient of pressure variation.

It is a fact of extraordinary interest that the experimental value of, β_v for most common gases turns out to be approximately 1/273 per degree centigrade. This means that for every degree centigrade change in temperature above or below zero degrees centigrade the pressure changes by 1/273 of the pressure which the gas exerts at zero degrees centigrade (the volume being kept constant). Hence if the temperature were lowered by 273°C below 0°C, the change of pressure would be 273 x 1/273 of P_o or the change of pressure would equal the initial pressure at 0°C and the final pressure would be zero! This irreducible minimum of temperature is called absolute zero, i.e., the temperature of an ideal gas at which molecular activity ceases and the pressure consequently is zero. Its value is roughly checked in this experiment by extrapolating (projecting beyond the measured values) the observed pressure-temperature curve until it intersects the axis of zero pressure, as in Fig. 4. This should occur at a place where t = -273°C = 0°K (degrees Kelvin are the units of temperature on the absolute scale of temperatures)

A careful distinction should be drawn between a linear relation and a direct proportion in the present and many similar cases. The pressure here varies linearly with the temperature in degrees centigrade, as indicated by Eq. (5) and Fig. 3. It is also true that the pressure is directly proportional to the temperature in degrees Kelvin (absolute), but the pressure is not directly proportional to the temperature expressed on any other temperature scale.
This may be more clearly seen by substituting the value 1/273 for, β_v in Eq. (5) and obtaining

P_t = P_o(1 + t/273) = P_o((273+t)/273) = (P_o/273)T = CT (7)

where T is the temperature in degrees Kelvin and C is (numerically) a constant for any given case. From Eq. (7) and Fig. 4 it is apparent that the pressure is directly proportional to the temperature only if the latter is measured on the absolute scale.

APPARATUS: The principal piece of apparatus, Fig. 5, is designed so that it may be used to study independently either Boyle’s law or Charles’ law. When Boyle’s law is being studied, the metal-bulb reservoir shown at the left is tightly sealed off by closing a needle valve provided for that purpose and hence this portion of the apparatus may thereafter be ignored. The Boyle’s law apparatus proper consists of two vertical glass tubes, one open at the top and the other closed by a stopcock, held at the lower end by stuffing boxes in an iron reservoir. The reservoir is mounted on a tripod base and is provided with a screw-operated diaphragm for varying the height of the mercury in the tubes and so changing the pressure and volume of the gas confined in the closed tube. The large, milled-head screw has a small pitch and it presses against the corrugated steel diaphragm which forms one side of the mercury reservoir. Readings of the mercury levels to measure the corresponding pressures and volumes are taken from a metric scale mounted vertically between the tubes. A sliding glass cursor provided with a horizontally etched hairline makes it conveniently possible to read the mercury levels with satisfactory precision. The air admitted to the closed tube should be carefully freed from moisture by a method described later. The apparatus may be leveled by having the tripod adjusted on the apparatus so that the leg having the leveling screw is in the plane of the glass tubes. A slight rotation of the screw will then suffice to level the apparatus.

The Charles’ law portion of the apparatus consists essentially of a metal bulb to contain the gas (air) under investigation, connected by a short section of glass capillary tubing to the mercury reservoir and manometer previously described. The volume of the gas is kept constant by adjusting the mercury level with the screw until the level is always brought to a fixed line etched on the glass tubing beneath the gas bulb. A copper jacket surrounds the bulb proper; it is provided with tubulures so that heating water may be added or removed as desired to control the temperature of the confined air. The water may be brought to and maintained at any desired temperature by an electric immersion heater, or hot or cold water may be introduced from outside.

As auxiliary apparatus there are required a mercurial barometer (Fig. 6), a 100°C thermometer, a steam generator and a Bunsen burner or an electric heater, two metal vessels for water and ice, a pinch clamp and suitable rubber tubing.

PROCEDURE:
I. Boyle’s Law:
Experimental:Close tightly the needle valve which seals off the Charles’ law gas bulb from the Boyle’s law apparatus. Place the apparatus in good light where the scale may easily be read. Before beginning the experiment the instructor or student should be sure that the air in the closed tube is perfectly free from moisture. Introduction of dry air into the tube is effected by connecting a Dessigel S drying tube to the stopcock tube and running the mercury up and down in the tube, pumping dry air in and out and thereby removing water vapor. The stopcock tube is finally tightly closed at a place where the volume of the entrapped air is about one half the total volume of the closed tube when the levels of the mercury in both tubes are the same.

Test for leaks by turning the milled head until the mercury is near the top of the open tube and observing it for a few minutes to see that the level remains constant.

Check this also by having the mercury in the open tube near the bottom of the tube. To be sure that no air bubbles are present, turn the adjusting wheel back and forth several times to move the mercury in the open tube from the top to the bottom, meanwhile watching both tubes for bubbles. While this is being done the barometer may be read, as described below.

Never allow the level of the mercury in the tubes to come below the lower end of the meter stick, as to do so will often allow air from the reservoir to enter the closed tube and thus to vitiate the results of the experiment. During the experiment, never adjust or open the stopcock on the closed tube, as to do So will change the mass of air, admit moist air and make necessary a complete drying of a new mass of air.

As the volume of the closed tube is not calibrated directly in cubic centimeters, the volume of the gas will be measured here in terms of the length of the tube above the mercury, since the volume is proportional to the length for a uniform bore. Since it is impossible to seal the glass stopcock at the end of the tube and still have a uniform bore right up to the barrel of the stopcock, an etched line on the tube is placed at such a point that the volume in the capillary between that line and the stopcock barrel represents exactly the volume of 1cm of uniform capillary bore. Hence the corrected scale reading for the top of the enclosed air column is obtained by adding 1cm to the reading of the scale just opposite the etched line on the tube below the stopcock. This corrected value will be designated R_f.

When the apparatus is properly adjusted, take a series of ten or twelve readings of corresponding pressures and volumes, choosing the values so that the open-tube readings vary by 70mm intervals over the entire scale. Take one reading when the mercury levels are the same on the open and closed tubes. Tabulate the following: (a) R_o, the reading of the open-tube mercury level; (b) R_c, the reading of the closed-tube mercury level; (c) p, the added pressure, i.e., R_o – R_c; (d) P, the actual pressure, or B + p; (e) V, the volume or R_t – R_c; (f) 1/V; (g) PV; (h) percentage variation of PV from the average value of all PV’s. Note that when the closed-tube readings exceed those on the open tube the confined gas is below atmospheric pressure and the values of p become negative.

Curves: Plot the following curves: (1) P vs. V (begin both axes at zero); (2) “added” pressure p vs. reciprocal of volume 1/V. Choose the axis of p near the center of the page and be sure that -p extends as far as 770mm below the axis. In laying off the scale for 1/V, begin with 1/V = 0 at the origin. Carefully interpret in the report the significance of these curves. From curve (2) determine the barometric pressure by extrapolating the observed portion of the curve (use a dotted line for extrapolated portion) back to the intercept where 1/V = 0. Compare this pressure-intercept value of B with that observed by the barometer.

II. Charles’ Law:
Experimental: A series of readings should be taken of the pressure of the air in the closed bulb when kept at constant volume and adjusted to various measurable temperatures ranging from 0°C to 100°C. The end temperatures will be the standard values for the freezing and boiling temperatures of water, while the intermediate values are to be read by a mercury thermometer. The pressures are obtained from the barometric height and the open-tube manometer.

WARNING! As the steam enters the jacket, the mercury in the gas tube will descend and may go below the lower end of the meter stick and admit air to the bulb. This should be prevented by raising the pressure to keep the level of the mercury on the gas-bulb tube at the index Line. Throughout the entire experiment one observer should continually observe the pressure and keep adjusting the screw to maintain the mercury levels at the desired values. When the hot water is removed from the vessel or when the steam is discontinued, the pressure will greatly decrease and the mercury might run up into the closed bulb if the adjusting screw were not manipulated to lower the level of the mercury in the closed tube.

The temperature of the steam is best obtained from a table of boiling points against pressure.

Interpretation of Data: Plot a curve to show the variation of Fig. 6. Mercurial Barometer, for accurately determining the atmospheric pressure. The glass tube G, closed at its upper end, projects into the mercury cistern W. The metric and British scales S and S’ have their zeros at the lower end of the ivory tip I. In reading the barometer, adjust the mercury level by turning the well upward with great care until the tip just touches the mercury surface. Move the vernier V up until the top of the mercury column appears below its lower edge. Tap the barometer lightly to permit the mercury to form a free meniscus. Then move the vernier downward until the sighting edges are in line with the uppermost point of the meniscus. Read the scale in millimeters and determine tenths with the aid of the vernier. 4 the pressure with the temperature in degrees centigrade. Explain clearly the significance of the shape of this curve.
Determine the slope of the curve, choosing points near its ends, and divide the slope by P_o. This should give the value of, β_v; determine the percentage difference between this value and the standard value, 0.003663 per degree centigrade.

Plot another curve from the observed data showing the variation of pressure with temperature, but starting the pressure scale with zero and the temperature scale with -273°C (0°K). Include on the temperature scale the values in both degrees Kelvin and degrees centigrade. Extrapolate by a dotted line the observed data to the axis of zero pressure. Compare this temperature intercept with the standard value for absolute zero. What is the ratio of the slope of the curve and the pressure at 0°C? Explain in the report the significance of the curve.

QUESTIONS: 1. Show by dimensional reasoning that the constant kin the equation PV = k is not a mere numerical constant, i.e., one having no unit, but that it has the dimensions of work. On what does the value of k depend?
2. A certain automobile tire is labeled, “Inflate to 35lb. or 2.5kg.” What does this mean? Is the statement clear? Is 35lb. equal to 2.5kg?
3. The gas bulb of the Charles’ law apparatus is made of cast iron. What happens to its volume when the temperature is changed from 0°C to 100°C? What effect does this have upon the “constant volume” assumption? Is the error serious? Why?
4. In the Charles’ law apparatus as used, a portion of the air bulb is connected to the mercury reservoir by a glass capillary tube, the level of the mercury always being brought to a fixed point on this tube. Hence some of the air above this line but below the metal bulb is not subject to the same temperature as the air inside the bulb. Will this introduce a serious error into the results? Why?
5. The level of the mercury in the open arm of a Charles’ law apparatus stands 5cm below the index when the volume bulb is surrounded by ice water, and 20cm above the index when the bulb is surrounded by steam at normal barometric pressure. Calculate the temperature coefficient of pressure variation at constant volume from these data.
6. The pressure of a gas is measured as 100cm of mercury at 50°C and 114.3cm of mercury at 100°C, volume constant. What value of absolute zero is obtained from these data?

Equipment List

From: Cenco Physics Selected Experiments in Physics (No. 71990-345), Copyright, 2003, Sargent-Welch Scientific Company.

Basic Optical Properties of Mirrors, Prisms, and Lenses (Optical Disk)

bpearson — Tue, 16 Feb 2010 17:23:59 +0000

OBJECT: To study the basic optical properties of mirrors, prisms, and lenses by the use of an optical disk.

METHOD: A light source sends parallel rays of light to an optical disk. The paths of the rays are observed after they strike various optical devices, such as mirrors, prisms, and lenses, which are mounted on the disk. From these observations the fundamental principles that govern the reflection and refraction of light are studied.

THEORY: A beam of light can be represented by a single line, a ray, parallel to the direction in which the light is propagated. When a beam of light strikes a surface of a different medium some of the light may be reflected, some may be transmitted, and the remainder is absorbed. The light which is reflected from a smooth surface and that which is transmitted through the medium obey certain elementary laws. Those laws will be studied in this experiment.

A. Reflection of Light. The angle between an incident ray and the normal to the surface is called the angle of incidence i. The angle between the reflected ray and the normal is called the angle of reflection r. By the law of reflection the angle of incidence is equal to the angle of reflection, as shown in Fig. 1. When the reflected beam, formed by the diverging rays BC and B₁C₁, is viewed by the eye of an observer, it seems to converge at the point A₁, and this virtual image appears to the observer as if it were the source of the light.

The position of the image of an extended object formed by a plane mirror may be readily located by a ray diagram, following the law of reflection, Fig. 2. The image is virtual, as far behind the mirror as the object is in front of the mirror, and the same size as the object. A plane mirror does not change the curvature of the wave front of the light beam incident upon it.

If the reflecting surface is curved, the same law of reflection holds at each point as for a plane mirror, but the size and position of the image are quite different. Spherical mirrors are classified as concave or convex when the reflecting surface is on the inside or outside, respectively, of the spherical shell, Fig. 3.

The center of curvature C is at the center of the sphere. The radius of curvature r is the radius of the sphere. A line connecting the vertex V and the center of curvature C is called the principal axis. In Fig. 3 there are also shown incident light rays parallel to the principal axis of the mirrors. From the law of reflection it follows that these rays will converge through a common point F after reflection from a concave mirror, or will diverge after reflection from a convex mirror, as though they originated from a common point F behind the mirror. Such a point is called the principal focus of the mirror. The distance of the principal focus from the mirror vertex is called the focal length f. It can be shown that r = 2f.

Simple geometrical constructions are available for locating images formed by spherical mirrors, as shown in Fig. 4.

Two rays from any point O, whose directions after reflection can readily be predicted, are drawn: first, the ray parallel to the principal axis, which after reflection passes through F; and second, a ray from O in the direction OC through the center of curvature, which strikes the mirror normally and is reflected back upon itself. The intersection of these two rays at I locates the image of O. Another similar pair of rays could be drawn from O’ to locate I’. The image depicted in Fig. 4a is real, since it is formed by converging rays that could form an image on a screen at II’. In Fig. 3b the reflected rays are diverging, as if they came from II’; such an image is said to be virtual. The image of any real object formed by a convex mirror (Fig. 5) is always virtual, erect, and diminished.

There is a simple relation between the distance p of an object from the mirror, the distance q of the image from the mirror, and the focal length f; this relationship makes it easy to locate images in spherical mirrors. This mirror equation is

1/p+1/q=1/f (1)

This equation can be used for both concave and convex mirrors if the following convention with regard to the algebraic signs of the quantities in Eq. (1) is used: f is positive for concave mirrors, negative for convex mirrors; p is positive for real objects arid negative for virtual objects; q is positive for real images and negative for virtual images.

Only the rays which are parallel to the principal axis and which are close to the axis are focused at the principal focus. The extreme rays farther from the axis in a mirror of large aperture (Fig. 6) cross the axis closer to the mirror than do the rays reflected nearer the center.

This imperfection of spherical mirrors is called spherical aberration. It results in a blurring of the image. The trace of the surface formed by the intersecting rays is called the caustic curve of the mirror. Spherical aberration can be minimized by using a diaphragm in front of the mirror to block off the rays far from the axis.

B. Refraction of Light. When light passes obliquely from one medium into a second medium, it undergoes an abrupt change in direction if the speed of light in the second medium differs from the speed in the first medium. This bending of the light path is called refraction. If the speed of light in the second medium is less than the speed in the first medium, the second medium is said to have a greater optical density and the ray is bent toward the normal. When light travels from an optically more dense medium into one of lesser optical density, the ray is bent away from the normal. The angle between the refracted ray and the normal to the surface is called the angle of refraction r.

In Fig. 7 a ray of light is shown passing from air to glass. The angle of refraction r is less than the angle of incidence i. There is an experimental relationship between these two angles known as Snell’s law of refraction, namely: The ratio of sin i to sin r is a constant (for a given wave length of light and pair of substances). This constant is known as the index of refraction n of the second medium relative to the first. Expressed in equation form

sini / sinr = n (2)

It may also be shown that

n = v₁/v₂ (3)

where v₁ and v₂ are the speeds of the light in the first and second mediums, respectively.

When light passes from an optically dense medium, such as glass, into a medium of lesser optical density, such as air (Fig. 8), the angle of refraction r is greater than the angle of

incidence i, and r increases more rapidly than does i. The value of r approaches the limiting value of 90°, beyond which no light is refracted into the air. The angle of incidence in the denser medium, for which the angle of refraction is 90°, is called the critical angle of incidence ic. From Snell’s law of refraction it follows that, when the medium of lesser optical density is air,

sinic=1n (4)

When the angle of incidence is increased beyond its critical value, the light is totally reflected, making the angle of reflection r’ equal to the angle of incidence i. Total reflection can take place only when the light in the optically denser medium is incident on the surface separating it from the medium of lesser optical density, and the angle of incidence is greater than the critical angle.

C. Dispersion. The index of refraction of light depends in a non-linear manner upon the wavelength of the light and the nature of the refracting substance. Blue-producing light is refracted more than red-producing light. The separation of complex light into its differently colored rays is called dispersion.

D. Prisms. A transparent body bounded by two plane faces which are not parallel is called a prism. It often is desirable to use prisms for the deviation or the dispersion of light. In a triangular prism the angle of deviation is determined by the angle of the prism, its index of refraction and the angle of incidence of the entering ray. When light passes through a glass prism in air the ray is bent toward the thicker part of the prism.

E. Lenses. A transparent body with regularly curved surfaces ordinarily produces changes in the path of light; such a body is called a lens. The most common forms of lenses are those in which at least one surface is a part of a sphere. As shown in Fig. 9a, rays parallel to the principal axis, after passing through a convex (converging) lens, will converge to the principal focus F; rays parallel to the principal axis, after passing through a concave (diverging) lens, diverge from the principal focus, at which a virtual image is formed, Fig. 9b. As in the case of spherical mirrors, rays not near the principal axis are focused closer to the lens than are the central rays. This spherical aberration is minimized by use of a diaphragm to decrease the aperture of the lens. This small aperture produces a sharper image but reduces its brightness.

APPARATUS: Hartl optical disk, with accessories, Figs. 10 and 11; illuminator, Fig. 12; ruler; protractor, compass; cross- ruled paper.

The optical disk consists of an aluminum disk surrounded by a sheet metal screen. These two parts turn independently upon the same horizontal axis. The disk is etched in degrees

marked on a circular scale around the circumference and two diameters are marked upon it. The screen has as its center a square opening over which slotted plates can be attached. The optical accessories, Fig. 11, consist of a set of four concave and convex lenses, two prisms and a plane, a concave and a convex mirror. Any one of the mirrors may be attached to the face of the disc by means of thumbscrews. The lenses and prisms are frosted on one side so that the path of the light in the glass may be seen just as it is seen on the face of the disc.

So that lenses and mirrors may be placed in the proper position for use, their outlines are marked upon the disc. These outlines are so positioned that the necessary angles, such as the angles of incidence, reflection, refraction, etc., may be obtained from the graduated circle.

A steady, intense beam of parallel rays of light may be obtained either from a carbon arc illuminator or from a concentrated filament incandescent lamp illuminator, Fig. 12.

PROCEDURE: I. Preliminary Adjustments. 1. Place the disk so that a beam of parallel rays from the illuminator strikes the edge of the disk. Turn the screen so that it is between the illuminator and the disk. Adjust the illuminator until the beam covers most of the opening in the screen and traces its path across the face of the disk. Put the three-slot plate in place and cover the two slots on each side of the central slot. Adjust the disk and screen until the beam of light crosses along the zero axis of the disk. The angle of incidence of this beam upon any optical device fastened to the disk may be varied at will by leaving the screen stationary and rotating the disk.

II. Plane Mirrors.
2. Fasten the plane mirror to the disk so that the face of the mirror coincides with the 90-90 diameter of the disk (see Fig. 10). Arrange the screen so that the beam passes along the 0-0 axis of the disk (Fig. 10) and strikes the mirror exactly at the center of the disk, Fig. 13. Turn the disk so that several angles of incidence and reflection can be observed and recorded. Do these data obey the law of reflection?

3. Substitute the seven-slot plate and check to see if parallel rays are still parallel after reflection from a plane mirror (Fig. 14).

III. Spherical Mirrors.
4. Arrange the slot covers to obtain only a single beam of light. Fasten the concave mirror to the center of the disk. (It is sometimes desirable to put a piece of paper under the back edge of the mirror to tip it slightly forward.) Rotate the disk until the single light ray crosses the center of the disk in the axis of the mirror. Leave the disk stationary and rotate the screen slightly backward and forward to move the ray parallel to itself across the face of the mirror. The reflected ray will be seen always to pass through one point on the axis of the disk. Mark this point. What is it called?

5. Attach a piece of cross-ruled paper to the disk with masking tape and sketch the mirror, rays, principal axis, and principal focus. Measure f and compare it with the radius of curvature of the mirror as determined by a compass and ruler.

6. Adjust the screen and disk so that the ray passes through the center of curvature of the mirror. Record the way in which this ray is reflected.

7. Use the seven-slot plate to check the location of the principal focus of the mirror (Fig. 15).

8. Remove the slotted plate and use the full opening. Note the spherical aberration and the caustic curve.

9. Turn the disk so that the convex side of the mirror faces the opening, and repeat the appropriate observations as in Step 4. Use the seven-slot plate to trace with dotted lines the reflected rays back to the virtual principal focus (Fig. 16).

IV. Refraction of Light.
10. Attach the semi cylindrical glass plate to the disk so that the straight edge coincides with the 90-90 diameter (Fig. 17). Adjust the screen so that the single ray touches the flat edge at the 0-0 axis.

Sketch the reflected and refracted rays. Compare the angle of incidence with the angle of reflection and the angle of refraction. Read the angles of refraction for several angles of incidence, such as 30°, 40°, and 55°. Compute the corresponding values of the index of refraction and comment on the results.

11. Rotate the disk through 180° and send the light through the glass in a direction opposite to that of Step 10 (Fig. 18). Note the reflected and refracted rays. Compare the angle of incidence with the angle of reflection; with the angle of refraction. What difference is seen as compared with Step 10? Note the new angles of refraction when the same angles of incidence are used as were observed in Step 10. Do the observations indicate that the light paths are reversible?

V. Total Internal Reflection.
12. Vary angle i and note that as i is increased, the intensity of the refracted ray is decreased. Turn the disk carefully near an angle of incidence of about 41.5° and note the value of the critical angle for which the refracted ray is parallel

to the surface (Fig. 19). Measure the critical angle for white, red, and blue light, obtained by covering the slit with colored glass. Comment on the values of the index of refraction.

13. Insert the 90° prism so that the long face is on the vertical 90-90 axis, with the apex of the prism toward the right (Fig. 20). Use two differently colored rays to show the paths of the rays that are totally internally reflected. Turn the disk through 45° so that the rays fall perpendicularly on one leg face and trace the totally internal-reflected rays. This is an example of the use of such a prism in modern field glasses.

VI. Refraction Through a Parallel Plate
14. Arrange the trapezoidal glass plate on the disk in such a position that a ray enters one of the parallel surfaces and emerges at the other (Fig. 21). Note the relative directions of the incident and emergent beams. Repeat for several angles of incidence and comment on the results.

VII. Refraction by a Prism.
15. Rearrange the trapezoidal prism so that first the 45° and then the 60° angle can be bisected by the 90-90 diameter (Fig. 22). Use a colored glass over the slit to obtain a monochromatic beam and, thus, avoid dispersion.

Let the single beam fall on the edge of the glass so that half the beam passes through the glass and the other half passes by the edge undeviated. Read the deflection caused by the glass. Record the deviation both for the 45° angle and the 60° angle. Comment on the variation of the deviation with the refracting angle of the prism.

16. With the single beam of white light, hold a prism against the disk behind the square opening of the screen. Note the dispersion of the white light into a prismatic spectrum. Record the colors in their order of deviation.

VIII. Refraction by Lenses.
17. Fasten the convex lens to the disk parallel to the 90-90 diameter (Fig. 23). Send a single beam along the 0-0 axis passing through the center of the lens. Rotate the disk through a small angle on either side of the initial position and observe the incident and emergent rays. When such a ray passes through the optical center of a lens, how does it emerge?

18. Turn the disk so that the 0-0 diameter (axis of lens) is parallel to the incident ray. Rotate the screen about its horizontal axis and note that the refracted ray turns about one point on the disk. What is this point called? Record this location of the principal focus F of the lens. Show the location of F by using the plate with seven slits (Fig. 24). Remove the slotted plate and shine the light through the square opening. Is the light sharply focused? Sketch the caustic curve.

19. Attach the concave lens in place of the convex lens. Use the seven parallel rays to trace the paths of the diverging rays and to locate the virtual principal focus of the lens (Fig. 25). Is this focus sharply identified? Explain?

QUESTIONS:
1. Show geometrically that the image of an object formed by a plane mirror is as far behind the mirror as the object is in front of the mirror. Use the spherical-mirror equation to justify this same fact.
2. If light waves are to converge to a point after reflection from a plane mirror, what must be their form before reflection? Explain by a sketch.
3. Justify geometrically the fact that when a plane mirror is rotated, the beam of light reflected from it rotates through twice the angle turned by the mirror.
4. Is it truthful to say that the image seen in a plane mirror is located behind the mirror?
5. An observer moves toward a mirror at a speed of 5mi/hr. At what speed does his image move toward him?
6. Mention a number of practical uses of (a) concave and (b) convex mirrors.
7. What effect is produced on the curvature of a plane wave by a plane mirror? By a concave mirror? By a convex mirror? By a concave lens?
8. Mention several conditions under which a double convex glass lens will produce diverging rays.
9. A glass concave lens (n = 1.52) has a focal length of 15 cm in air. How would this focal length be affected if the lens were placed in water (n = 1.33)? In carbon disulfide (n = 1.64)?
10. A glass prism is placed first in air and then in water. Trace the paths of a beam of light for these two cases. Repeat for an “air prism,” that is, a hollow prism made from plane glass sides.

Equipment List

From: Cenco Physics Selected Experiments in Physics (No. 71990-515), Copyright, 2003, Sargent-Welch Scientific Company.

Archimedes’ Principle

bpearson — Sun, 14 Feb 2010 15:07:04 +0000

OBJECT: To study Archimedes’ Principle and to apply this principle to determine the density of solids and liquids.

METHOD: A body is weighed in air and then weighed when submerged in a liquid. The apparent loss of weight is, by Archimedes’ Principle, equal to the weight of the liquid displaced by the body. From these measurements, the density and specific gravity of the solids and liquids used in the experiment may be determined.

THEORY: The fact that an object immersed in a fluid, liquid or gas, should be “buoyed up.’ by a force equal to the weight of the fluid it displaces was deduced by Archimedes (287-212 BC). This principle, called Archimedes’ Principle applies to any object in any fluid, for example, a submarine in water or a dirigible in air.

If a free body remains at rest when totally immersed in a fluid, there must be no resultant force acting and, hence, the weight of the body must be equal to the weight of the displaced fluid. The body will sink if its weight exceeds that of the displaced fluid and. conversely, rise if lighter. Thus a block of cork released in water, bobs up to the surface and floats like a ship.

A ship will adjust to a depth in water for which its weight just equals the weight of water (and air) displaced. The contribution of the air to the buoyant force of a ship is a negligible amount and, therefore, disregarded. The dirigible is, however, totally supported by the buoyant force supplied by air displacement.

Proof of Archimedes’ Principle: Let the irregular outline of Fig. 1 contain any desired portion of a fluid at rest. The arrows are representative forces acting against the bounding surface. Each force is perpendicular to its element of surface and the resulting pressure has a magnitude dependent on the depth of the fluid at that point.

Since the fluid is at rest there is no unbalanced force in any direction. The components of the forces pushing towards the right on the bounding surface must balance those pushing towards the left. The vertical components upward on the surface must support the downward forces on the surface plus the weight of the designated portion of fluid. That is. the resultant upward force F_y must equal the weight mg of the fluid contained inside this surface.

When this portion of fluid is replaced by a solid body of exactly the same shape, the pressure at every point is exactly the same as before. Hence, the fluid must exert an upward or buoyant force on the object immersed in it, a force which is again just equal to the weight of the fluid displaced by the object. If the buoyant force is greater than the weight of the object, the object is lifted to the surface. If the weight of the object is greater than the buoyant force, the unbalanced downward force causes the object to sink like a rock in water.

Application of Archimedes’ Principle: Archimedes’ Principle is experimentally applied to obtain the density p of a substance, that is, its mass per unit volume,

p=m/V (1)

Where m is the mass and V is the volume of the object.

A body of weight w = mg has an apparent loss of weight when immersed in a fluid. This loss is equal to the weight of the volume of fluid displaced by the object or wf = mfg. Hence, the apparent weight of the object m’g submerged in the fluid is given by

m’ g = mg-m_fg (2)

Since m = pV (see Eq. 1) one may replace m_f of Eq. (2) with p_fV where p_f is the density of the fluid and V the volume displaced. But the volume of fluid displaced is also the volume of the submerged object, V = m/p. Thus, m_f = p_f (m/p) and Eq. (2) becomes,

m’g = mg – p_f(m/p)g
or
w ‘ = w – (p_f/p)w
and
p = (w/w – w’)p_f (3)

If the density of the body to be measured is less than the density of the liquid, that is, p/p_f, it is necessary to fasten a sinker to the body so that the two together will sink in the liquid. If this sinker (see s of Fig. 2 (b)) is made a part of the weighing apparatus for all the readings, the value of p again may be computed from the two readings required for Eq. (3). Note that the sign of w’ is now negative. Equation 3 states that the

density of object = (weight of object/loss of weight when submerged in fluid) × density of the fluid

When the fluid used is water, the portion of this equation within the brackets may be rewritten as

sp.gr. = weight of object/weight of equal volume of water

where sp. gr. is called the specific gravity of the substance. Mercury has a specific gravity of 13.6; this means that a given volume of mercury is 13.6 times heavier than the same volume of water. To obtain the density of mercury, its specific gravity value must be multiplied by the density of water, i.e., 1gm/cm³ or 62.4lb/ft³. Thus, the density of mercury is 13.6gm/cm3 or 849lb/ft³. Note that specific gravity is a ratio only and has no units. Density has the units of mass/volume. In the British system the units pounds-per-cubic-foot give a “weight density”.

This experiment deals only with solids and liquids which have a very high density compared to air. Thus, the “weight of the object” is accepted as approximately its weight in air.

The relative densities of liquids may be obtained by finding the weights of a given dense object in the various liquids. Suppose that the apparent loss of weight of the object immersed in a liquid of a density p_f1 is m₁g and when it is immersed in a liquid of density p_s2 the apparent loss of weight is m₂g. By Archimedes’ Principle, m₁g = Vp_f1g and m₂g = Vp_f2g where V is the volume of the object. Hence,

V = m₁/p_f1 = m₂/p_f2
or
p_f1 = (m₁/m₂)p_f2(4)

The ratio m₁/m₂ may be measured directly from the corresponding weights. The value of p_f1 may be computed, provided p_f2 is known. It is common practice to use water as the comparison liquid since its density (see Table I) is very accurately known.

APPARATUS: Trip scale on a supporting stand, Fig. 3; set of masses; overflow can; cylinder or block of wood having a specific gravity of less than 1: irregular metal object (Fig. 4); vernier caliper; liquids of different densities. A dense solution of salt water may be used.

PROCEDURE: 1. Study of Archimedes’ Principle:
A. Floating object. Fill the overflow can with water. Catch the resulting overflow when an object which floats in water is placed in the can. Measure the weight of the overflow water. Weigh the object and compare its weight with that of the overflow. Draw a conclusion.

Note: See that no air bubbles adhere to the objects used in these experiments. The objects must be dry when weighed in air. Some of the data obtained here will be used later.

B. Dense metal object. Fill the overflow can with water. Catch the resulting overflow of water when the dense metal object is immersed in it. Weigh the overflow water.

Weigh the object in air. Next weigh it in the water by attaching a light string between the subject and the underside of one pan of the trip scales (see Fig. 2(a)). Compare the apparent loss of weight of the object with the weight of the overflow water. Draw a conclusion.

II. Density and Specific Gravity of Solids:
A. Determine the volume of the floating object used in part 1 (A) from (a) measurements of its dimensions and (b) the weight of the overflow for complete immersion in water.

Compute the density of this object by using Eq. (1). Express the density in both metric and British units. What is the specific gravity of this object?

B. Compute the density of the metal object used in part I (B). Use Eq. (3).

C. Obtain the density of the object measured in I (A) using the method shown in Fig. 2 (b). The metal object may serve as the sinker s.

III. Density of a Liquid: In part I (B) the loss of weight of the metal object in water was measured. Replace the water with a liquid of different density and find the buoyant force on the object in this liquid.

What is the density of the second liquid?

QUESTIONS: 1. Why is it important that no air bubbles adhere to the objects during measurements? See Note I (A).
2. The specific gravity of gold is 19.3. What is the mass in grams of a cubic centimeter of gold? What is the weight in pounds of a cubic foot of gold?
3. A block of metal having a density of 9.00gm/cm³ has an apparent weight of 180gm in water and 135gm when submerged in a liquid. What is the density of the liquid?
4. A block of wood and a block of lead of the same mass are weighed carefully on a trip scale using brass weights. Are the weights of these blocks the same? Explain.
5. A submarine floats at rest 50ft deep in water. Without using the propeller how can the captain make it surface? Make it sink?
6. A rock has a specific gravity of 1.30. A man can lift 120 pounds. How many cubic feet of this rock can he lift in air? In water?
7. Oil having a density of 0.80gm/cm³ floats on water of density 1.0gm/cm³. A solid object which has a specific gravity of 0.90 is dropped into the container. Locate its exact position of rest.

Equipment List

_{From: Cenco Physics Selected Experiments in Physics (No. 71990-185-1), Copyright, 2003, Sargent-Welch Scientific Company.}