PHYSICS M20A LAB MANUAL MOORPARK COLLEGE BY: PROFESSOR H. FRED MEYER

Transcription

1 PHYSICS M0A LAB MANUAL MOORPARK COLLEGE BY: PROFESSOR H. FRED MEYER

2 LIST OF EXPERIMENTS Measurements: Mass, Volume, Density Errors and Error Propagation Height of a Building The Simple Pendulum Acceleration of Gravity The Force Table Cantilever Beam Friction Center of Mass Ballistic Pendulum Rotational Motion and the Moment of Inertia Young s Modulus and Torsion Modulus Archimedes Principle Hooke s Law and the Effective Mass of a Spring

3 MEASUREMENTS LAB DETERMINING VOLUME, MASS AND DENSITY USING MICROMETERS, VERNIER CALIPERS AND A LABORATORY BALANCE INTRODUCTION Instructional Objectives: Learn how to use calipers, micrometers and a laboratory balance. Determine the least count of an instrument. Determine the number of significant figures in a measurement. Determine the calculated percent error in a measurement from the errors in the measurements. Experimental Objectives: Measure the dimensions and mass of various items use these measurements to determine the density of these objects. The densities will be compared with accepted values to see if the experimental values are within the predicted margin of error. THE VERNIER CALIPER Note the various measurement types listed below. The Vernier Caliper The Vernier Caliper Inside Dimensions English Units (inch) Depth Gauge Vernier Scales Metric Units (cm, mm) Outside Dimensions

4 Looking at the lower scale, note how measurements are read. The least count is the smallest subdivision reading that can be read without estimating. Note that the least count and determines the precision of the instrument. The least count shown above is 0.01 cm. and the reading would be recorded as 3.47cm±0.01cm on the data sheet. The ±0.01cm is called the absolute error in the measurement. MICROMETER

5 The spindle of an ordinary metric micrometer has threads per millimeter, and thus one complete revolution moves the spindle through a distance of 0.5 millimeter. The longitudinal line on the frame is graduated with 1 millimeter divisions and 0.5 millimeter subdivisions. The thimble has 50 graduations, each being 0.01 millimeter (one-hundredth of a millimeter). Convince yourself that the reading shown on the above micrometer is 1.93mm. (Note that the thimble is 0.43mm past the 0.50mm mark on the sleeve. The reading would be recorded as 1.94mm±0.01mm on the data sheet. When using the micrometer, turn the ratchet (also called the friction clutch) until it slips. This provides the proper torque on the thimble. Also, after using the micrometers, make sure to leave the jaws open so they don t get sprung with temperature changes. Whole millimeter marks..01 millimeter marks ½ millimeter marks. Notice that the least count on the micrometer is 0.01mm. The above readings on the instruments assumed that they read zero when closed. Before taking a reading, close the measuring device completely and take a zero reading. This will give you the zero correction. If the zero correction is positive, this value must be subtracted from all readings. If the zero correction is negative, this value must be added to each reading. THEORY The mass density of an object ρ is defined as MASS/VOLUME. The volumes of various shapes are given by:

6 V SPHERE = V CYLINDER = πr L THE EXPERIMENT Caliper Measurements DATA TABLE 1: Two measurements per partner. Least count Zero correction Reading 1 Reading Reading 3 Reading4 AVERAGE Sphere Radius Cylinder Radius Cylinder Length Least count Reading 1 Reading Reading 3 Reading4 AVERAGE Sphere Mass Cylinder Mass DATA TABLE -two measurements per partner Micrometer Measurements -repeat the measurements with the micrometer using a different size sphere and different size cylinder. Least count Zero correction

7 Reading 1 Reading Reading 3 Reading4 AVERAGE Sphere Radius Cylinder Radius Cylinder Length Reading 1 Reading Reading 3 Reading4 AVERAGE Sphere Mass Cylinder Mass ANALYSIS Using the equations given for volume and the data calculate the volume of each sphere and each cylinder. ( be sure the results are stated with the proper number of significant figures and include the units of either grams per cubic centimeter or kilograms per cubic meter) Using the equation for density, calculate the density ρ of each object.

8 We are now going to use the estimated absolute error to calculate the predicted error in the density. Add the absolute error (estimated from least count) to each of the measurements and calculate the density again. Call this value ρ +. Now subtract the absolute error from each measurement and calculate the density of each object. Call this value ρ -.

9 Calculate the percent predicted error using the equation converted to percent. Look up the accepted value of the density of each object. Calculate the percent discrepancy using the equation converted to percent. SUMMARY TABLE CALIPER VALUES MICROMETER VALUES ρ CYLINDER ρ SPHERE PREDICTED % ERROR (CYL) PREDICTED % ERROR (SPHERE) % DISCREPANCY (CYL) % DISCREPANCY (SPHERE) DISCUSSION: If the % discrepancy is less than the % predicted error, then the results are within the MARGIN OF ERROR. Note whether or not your results are within the margin of error. If the results are not within the margin of error, can you give a reason why? Note possible sources of error in the measurements and give suggestions how the error might be reduced.

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11 ERRORS AND ERROR PROPAGATION INTRODUCTION: Laboratory experiments involve taking measurements and using those measurements in an equation to calculate an experimental result. It is also necessary to know how to estimate the uncertainty, or error, in physical measurements and to know how to use those uncertainties to calculate the error in the experimental result. TYPES OF EXPERIMENTAL ERRORS Experimental errors can generally be classified into three types: personal, systematic, and random. Personal Errors These errors arise from personal bias of carelessness in reading an instrument, in recording data, or in calculations, and parallax in reading a meter. Of these, only parallax errors can be estimated and used in error propagation. Effort should be made to eliminate experimental errors. (When looking at non-digital meter, there is a small distance between the needle and the scale. As a result, the reading will change as the observer s eye position changes from side to side. This apparent change in reading, due to the change in position of the observer s eye, is called parallax.) Systematic Errors Errors of this type result in measured values which are consistently too high or too low. Conditions which lead to systematic errors are as follows: 1. An improperly calibrated instrument such as a thermometer which consistently reads 99ºC in boiling water instead of 100ºC.. A meter, micrometer, vernier caliper, or other instrument which was not properly zeroed or for which the zero correction factor was not considered. 3. Theoretical errors due to a simplified mathematical model for the system which consistently gives a calculated value different from the calculated value predicted from a more accurate mathematical model. Random Errors Random errors result from unknown and unpredictable variations in experimental measurements. Possible sources of random errors are: 1. Observational, e.g. errors when reading the scale of a measuring device to the smallest division.

12 . Environmental- unpredictable fluctuations in readings beyond the experimenters control. Such errors can be determined statistically or can be estimated by the experimenter. STATISTICAL DETERMINATION OF RANDOM ERRORS When there are many measurements of the same quantity, the average or mean value is defined by _ N 1 x i N i1 x where x i is the i th measured value and N is the total number of measurements. There are two ways to statistically calculate the uncertainty in the measured value. One method is to calculate the deviation from the mean or mean deviation d N xi x i d 1 N It is common to express the experimental value of the measurement as: Measured value of x = x d where d a statistical estimate of the uncertainty in the measured value. As can be observed, the mean deviation is a measure of the spread on the data. Another method used to calculate the random error is by calculating the standard deviation, (s.d.) xi x i1 s. d. N The measures value of x can then be expressed as: N Measured value of x x s.d. The statistical methods above will be used in selected lab exercises to follow such as THE SIMPLE PENDELUM and MOMENT OF INERTIA where several measurements of time are needed and an average or mean is calculated. ESTIMATION OF RANDOM ERRORS An easier method to determine random error is to estimate the random error by utilizing the accuracy of the instrument and the judgment of the experimenter. The error in a given instrument is determined by the smallest division on that instrument or least count. For example, the smallest division on a meter stick is 1mm or 0.1cm. This is the least count for the meter stick. In most measurements the smallest division represents the rightmost digit in the

13 value of that measurement and the estimated error is the measurement is the least count. For example, a measure value may be 78.cm 0.1cm. Sometimes a measurement may be made with an estimated error less than the least count. For example, an experimenter may estimate reading on a meter stick as 78.5cm by noting that the reading was about half way between 78.cm and 78.3cm. The experimenter may represent the value as 78.5cm 0.05cm. Keep in mind that rightmost digit must be estimated by the experimenter and is thus doubtful. Sometimes the estimated error is larger than the least count. For example, when measuring the distance between the two spots below, the experimenter would need to estimate where the center of each spot would be located. The error in the measured distance would be larger than the least count and the amount of the estimated error would be up to the judgment of the experimenter. Note how much the error estimates depend on the judgment of the experimenter. There may be errors in judgment; however, to avoid stating a result more accurately than you probably measured it, one should try to avoid being too conservative in estimating errors. ERROR PROPAGATION PARTIAL DERIVITIVES Before we can perform error propagation calculations, we must know how to take what are called partial derivatives of a function with many variables. Some may already know how to do this; you can help the others. Suppose we have a function f where f=f(x,y,z). The partial derivative of f with respect to x is found by taking the ordinary derivative while treating y and z as constants. The notation for this f derivative is. Likewise, the partial derivative of f with respect to y is found by taking the x f ordinary derivative while treating x and z as constants and is written as and the partial y derivative of f with respect to z is found by taking the ordinary derivative while treating x and y f as constants and is written as. z As an example, let 3 f 3 f 5x yz. Then 5 x yz x x = x x 3 3 5yz 10xyz Convince yourself that f y 3 5x z and that f z 15x yz.

14 ABSOLUTE AND RELATIVE ERRORS Absolute Error: When an error is estimated in a measured value of x it will be designated as x (delta x). x has the same units as x and is called the absolute error in x. For example, if cm cm x , the absolute error is cm x Relative Error: The ratio of the absolute error x to the measured value x, x x, is called the relative error. It is usually represented as a percent. For example, the relative error in the above example is % cm cm (note, there are times when it is necessary to from relative error back to absolute error: x error x relative ) COMPUTATION OF ERROR For a function z y x f f,,, the absolute error in f, f, is defined as: z z f y y f x x f f The relative error in f would thus be 1 z z f y y f x x f f f f EXAMPLE Using the function we used as an example for partial derivatives, we would have z yz x y z x f xyz f thus z yz x yz x y yz x z x x yz x xyz f f which when simplified becomes

15 f f x x y y 3z z Note that the quantities in the parentheses are just the percent errors multiplied by the exponent for that particular variable. Suppose we have the experimental values for x, y, and z as: x 3.0cm 0. 1cm, y 5.cm 0. 1cm, and z.4cm 0. 1cm. We would thus have the percent error in f as: % f f Note that the % error is rounded up to the nearest whole number. Since it is just an estimate, we can not justify more accuracy in the error. ANOTHER EXAMPLE.4 = Suppose V 3a 5b where a cm, b cm, and c cm c Thus V V a a V b b V c c where V 6a V 10b V 3a 5b,, and a c b c c c ; or, 6a 10b 3a 5b V a b c c c c V Notice that the negative sign in does not matter since it is squared. c

16 Now c c b a c b a b c b a c b a c b a c a V V Or, V V =5.5% 6% The final results would be given as % 81 6 cm V PERCENT DISCREPANCY Once an experimental value and percent error are calculated, the percent discrepancy is defined as: percent discrepancy in X= accepted erimental accepted X X X exp There will be agreement between the accepted value and the experimental value if the percent discrepancy is less than the predicted percent error in the experimental value as determined by error propagation. In other words, the experimental value is within the margin of error. This should be addressed in your conclusion. If there is not agreement, some sources of error may be present which may not have been accounted for and some reasonable explanation should be included in the conclusion of your report c c b b a b a b a a V V

17 ERROR PROPAGATION EXERCISES Determine the calculated value using the given values in the given equations. Be sure to include the units in your answer. Using the error propagation method described above, calculate the percent error in the calculated value. For this exercise, your percent error is to be given to two significant figures. Hand in this answer sheet. Work the problems neatly on scratch paper and staple your work to this sheet. 1. A=xy, x 3.0cm 0. 1cm, y 4.0cm 0. 1cm. f=x+y, for x and y given in problem # 1 3. f=x-y, for x and y given in problem # 1 4. z=3x+y, for x and y given in problem # 1 h 5. g for h.00m 3%, t 0.630s 4% t 6. T M 100N, M.5Kg 6%, k % k m d ML cm g, M 30.0g %, L cm z x y, 3.0cm % x, y 4.0cm % 3 5a cm b z, a.0cm 1%, b 3.0cm 1%, C 11.0cm % C 10. h d sin, d 1.00m 0. 05m, 10 1

18 HEIGHT OF A BUILDING INTRODUCTION: Using an angle meter and other simple measuring devices, the student will take measurements which will be used to calculate the height of the South East corner of the Physical Science building at Moorpark College relative to the walkway directly below the corner. The diagram below shows the basic geometry involved: PS BUILDING METER STICK θ METER STICK H X A meter stick will be placed in a meter stick holder directly at the point below the overhanging corner. This meter stick will be used by the whole class as a measurement reference. Each group of students will have an angle meter, a ring stand and rod with a meter stick clamp and meter stick attached to the rod, and a long tape measure. INSTRUCTIONAL OBJECTIVES ARE TO GAIN COMPETANCE IN THE FOLLOWING: 1. Planning an experiment. Deciding what data needs to be taken, how much data needs to be taken, and over what range.

19 3. Organizing a data table. 4. Error estimation and minimization of systematic errors. 5. Calculation of averages and standard deviations. 6. Comparing results with accepted values. EQUIPMENT: Ring stand, meter stick clamp, meter stick, angle meter, long measuring tape. PLANNING: Go outside and look at the height building height H which is to be determined. Notice that the sidewalk and lawn are sloping away from the building and that θ is measured from the horizontal. The reason for the meter stick on a stand directly under the building corner is so when θ is at 0º, you have a reference point on the meter stick, (call it h) which can be added to a calculated distance Y, (using θ and X) from this reference point, to determine the building height H. You need to figure out the details for doing this. Decide what range of values you will use to get 10 different angles and baseline values for X. Hint: you will not get accurate results for small angles or for large angles approaching 90º. Using a straight edge, organize a data sheet with appropriate rows and columns for your data. Label each row and column with the appropriate variable and units. Draw a neat diagram on your data sheet with all the variables labeled. EXPERIMENTAL PROCEDURE: 1. Place a meter stick on a stand directly beneath the building corner with the zero end touching the walkway. (Your instructor or lab tech will do this.). Attach your meter stick to your meter stick clamp and then to your ring stand and place the ring stand at selected distances X, then measure the corresponding angles θ. 3. Each time a new distance X and angle θ are used, a new reference height measurement must be made at θ = 0º on the meter stick below the building corner. ANALYSIS: Calculations must be done neatly in this section and the details must be shown.

20 THE SIMPLE PENDULUM INTRODUCTION: The Simple pendulum consists of small mass, m, suspended by a string. The period of the pendulum, T, is the time for the mass (bob) to go from one extreme position to the other and return (the time for one complete swing). The length, L of the pendulum is the distance from the point of suspension to the center of the bob. The amplitude, θ, of the pendulum s swing is the angle between the vertical position and the extreme position. The experimental objective is to determine if the period, T, depends on the variables θ, L, and m, and then find an empirical equation for the period, T, as a function of those variables. The empirical equation will then be compared with the theoretical equation. The acceleration of gravity will also be determined from the slope of the period graph. INSTRUCTIONAL OBJECTIVES: Give the student practice in 1. Organizing a data sheet. Taking data when many variables are involved 3. Error estimation 4. Minimizing systematic errors 5. Representing data in graphic form 6. Obtaining equations from graphs 7. Comparing experimental results with accepted values EQUIPMENT: Large wooden protractor, monofilament string, pendulum clamp, triple beam balance, table clamp and rod, aluminum, lead, and steel spheres, meter stick, stop watch. EXPERIMENTAL PROCEDURE: 1. Using a straight edge, organize a data sheet consisting of columns and rows. All of the variables should be listed at the top of the column. Note: Reaction time is a significant source of error. To minimize error due to reaction time, let the pendulum swing ten times. You will thus need 10T at the top of a column as well as T.. Estimate the absolute error in each measurement and list it at the top of each column. 3. Set up the pendulum using the listed equipment. 4. Using a string length of about one meter and using the three masses, determine if the period depends on the mass.

21 5. Keeping the length at about one meter and using the lead mass, determine if the period depends on the angle, θ. Measure 10T for several amplitudes from 5 degrees to 80 degrees. 6. Using the lead mass and an angle of about 0 degrees, vary the length in 5cm increments starting at one meter and decreasing the length to as small as possible as allowed by the protractor. ANALYSIS: From each set of data, for the three variables, find, within experimental error, whether or not the period depends upon the variable. If the period depends on the variable, plot a graph of T verses the variable. (T goes on the y axis.) From the graph of T vs. L, with L in meters, find the empirical equation of T as a function of L. Determine the proportionality constant and the units of the constant. The theoretical equation for small angles is T L. Find the % discrepancy of your g proportionality constant compared to. From your proportionality constant, determine a g value for g and compare it (% discrepancy) with the accepted value of g. GRAPHING BY HAND: Plot a graph of period T versus L. Take L = 0m to be at the origin. If this graph gives a straight line, this shows that T depends on L. When drawing your graph, choose appropriate scales for the x and y axes so the graph comes close to filling the page while, at the same time, keeping an easy to read scale. The graph needs to have a descriptive title, the x and y axes need to be labeled with the variables you are graphing and the units need to be on each axis also. Each point on the graph should be just a small point with a protective circle around it. Your line should represent a visual best fit straight line (try to draw a line which represents an average with about as many points on one side of the line as the other). Draw a large slope triangle on your graph using easy to read grid points, and from this, determine the slope of the line. This slope represents. Draw maximum and minimum slope lines on your graph and from maximum g and minimum slopes on your graph, determine the % error in the slope. QUESTION: (put calculations and answer in analysis section)

22 From an advanced mathematical solution for the period of the simple pendulum, the theoretical L expression for the period is: T 1 sin sin... g 4 64 Where g is the acceleration of gravity and the terms in parentheses are part of an infinite series. For small angles, the θ terms in the series are small and a good approximation for the period is found by taking only the first term in the series. The approximation is L T. For an angle θ of 60º, how many additional terms must be used in g order for the theoretical period to exceed this approximation by 7%? REPORT: For this experiment your report will contain only the following two sections of a formal report. ANALYSIS: This section includes a sample calculation of each type with error propagation if appropriate. Any graphs needed should also be in this section, as well as the calculation of percent discrepancies. CONCLUSION: This section includes a summary of the results along with the estimated error. Error may be the random error as calculated from the standard deviation, or it could be an error calculated from error estimates in your measurements. When a physical quantity is measured, as in this lab, include in the conclusion a comparison of the measured value with the accepted value (% discrepancy.) Note whether or not the % discrepancy is greater or less than the predicted uncertainty; if it is greater, try to give a reasonable explanation as to why. Comments or suggestions for improving the results are appropriate here also.

23 ACCELERATION OF GRAVITY INTRODUCTION: The value of g, the acceleration of gravity, is to be determined using the Behr free-fall apparatus. Measurements of distance intervals and time intervals will be used to calculate speeds at various times. A graph of speed versus time will be used to calculate the magnitude of the acceleration of a freely falling body. EQUIPMENT: Behr free-fall apparatus, vernier calipers, meter stick. INSTRUCTIONAL OBJECTIVES 1. Practice using the calipers 3. Organizing a data sheet 4. Experience drawing graphs on graph paper 5. Estimating error using maximum and minimum slopes 6. Learn to graph using Microsoft Excel 7. Become familiar with writing an ABSTRACT and an INTRODUCTION to a formal report EXPERIMENTAL PROCEDURE Since a free-falling body acquires a fairly large velocity in a short time interval, a special apparatus is required to measure its position at short time intervals. The apparatus we will use is called the Behr Free-fall Apparatus. It consists of a projectile which is dropped between two vertical wires and a waxed tape. A high voltage spark timer produces sparks at the rate of 60 pulses per second between the two wires and leaves small holes in the waxed tape each time it pulses. We thus have a waxed paper strip which has marks on it at time intervals of 1/60 th of a second between each mark.

24 BEHR FREE-FALL APPARATUS The tape appears as shown in the following figure with the points numbered etc. The time interval between each point is, of course, 1/60 th of a second.

25 Operate the apparatus and obtain one tape for each partner group. Number the points on the tape 0, 1,, etc. Do not start your numbering at the first points since the first few points are too close to permit accurate measurements. Start well down the tape where the points are at least one cm apart. (Remember that you will be drawing a graph of v vs. t and the zero point for time is arbitrary; it just means that the beginning velocity is not zero). Take measurements of Δx values which will be used to calculate instantaneous velocities at various times. For example, the instantaneous velocity at point 1 is v 1 = Δx 0, /Δt. This would be the velocity at time t 1 =1/60 th sec. v 3 = Δx,4/Δt would be the velocity at time t 3 = 3/60 th sec. Δt= 1/30 th of a second for each interval. As you can see, the measurements for Δx that you need to make are distances between points (0,), (,4), (4,6) etc. which will be used to calculate v 1, v 3, v 5, v 7, etc. until you obtain at least ten points for a graph of v vs. t. The average velocity over an interval, such v 1 = Δx 0, /Δt gives the instantaneous velocity v 1 at the mid-time point of the interval. Using a straight edge, construct a data table with columns and rows and label the top of the columns with Δx (i,j), Δt, t, v i. Measure the various Δx values as accurately as possible using vernier calipers. ANALYSIS If the instantaneous velocities at various times are plotted on the y-axis verses the time on the x- axis, a straight line should be obtained and the slope of the line will be the acceleration. GRAPHING BY HAND: ( see handout on graphing) Plot a graph of your velocities against the time. Take time t = 0sec to be at the origin. When drawing your graph, choose appropriate scales for the x and y axes so the graph comes close to filling the page, while, at the same time, keeping an easy to read scale. The graph needs to have a descriptive title, the x and y axes need to be labeled with the variables you are graphing and the units need to be on each axis also. Each point on the graph should be just a small point with a protective circle around it. Your line should represent a visual best fit straight line (try to draw a line which represents an average with about as many points on one side of the line as the other). Draw a large slope triangle on your graph using easy to read grid points, and from this, determine the slope of the line. This slope represents g. Draw maximum and minimum slope lines on your graph and from maximum and minimum slopes on your graph, determine the % error in g. Using Microsoft Excel (see handout) plot v vs. t and on your graph. Include a TITLE, UNITS and VARIABLE NAMES, EQUATION, and the CORRELATION COEFICIENT.

26 REPORT In addition to an analysis section and conclusion as included in the last lab report, you are to write and ABSTRACT and INTRODUCTION in your report for this experiment. ABSTRACT: On your cover sheet to the lab, you should have the class name and time, your name and your partner s name and the experiment title. In addition, the cover sheet should have an ABSTRACT. For complete instructions on writing an abstract, refer to For our purposes the abstract should be one paragraph stating the experimental objective, how the experimental objective was met, and the results of the experiment including any comparison with accepted values. The purpose of an abstract is to enable the reader to basically have a summary of the experiment to see if the paper is about a topic the reader is researching. The goal is to make the abstract as complete but as concise as possible. INTRODUCTION: This consists of one or two sentences describing the experimental objectives of the laboratory and what you are going to do to accomplish those objectives. An example would be By varying the length, mass and amplitude of a simple pendulum, how the period of the pendulum depends upon those variables will be determined. Notice that statements such as To learn how to analyze data or To learn how to organize a data sheet or other Instructional Objectives do not belong in the introduction or anywhere else in the report. These comments will be found only in the laboratory manual.

27 FORCE TABLE INTRODUCTION: The force table is an apparatus that allows the experimental determination of the resultant of force vectors. It consists of a large aluminum disk with the rim graduated in degrees. Forces are applied to a central ring by means of strings passing over pulleys and attached to weight hangers. The magnitude of the vector is varied by adding masses to the weight hangers and the direction of the vector is changed by moving the pulleys along the rim at different angles. Vectors will be added experimentally using the force table. The vectors will also be added graphically and analytically. The magnitude of the experimental resultant will be compared to the magnitude of the resultant as found from the analytical method. EQUIPMENT: Force table apparatus, four pulleys, weights and weight hangers, string, protractor, ruler, bubble level, graph paper.

28 THEORY: Review the analytical method using components and also the graphical methods of adding vectors as found in your text. You may also go to: EXPERIMENTAL PROCEDURE: When two or more forces are applied to the ring, their vector sum, or resultant R, can be found by finding the additional force needed to balance the applied force. For example, if two forces are applied, the resultant, or vector sum, is (1) F 1 + F = R The magnitude and direction of R can be found by finding a third force E such that F 1 + F + E = 0 or, () F 1 + F = -E E is called the equilibrant and we can see when comparing equations (1) and () that -E = R Remember that taking the negative of a vector is just reversing its direction by 180º. Thus, to find the resultant, just find the equilibrant and add 180º to the angle. 1. Level the force table.. Place a pin in the hole at the center of the table. 3. Attach strings, and weight hangers to the ring. Make sure the strings are free to slide on the ring. 4. Attach the pulleys to the disk at the angles given in the vector problem. 5. Add the masses given in the vector problem (be sure to include the weight hanger in the total.) 6. Find the equilibrant needed to center the ring around the pin. (Note: You can find the proper direction of the equilibrant by just pulling on the string with your hand while trying different angles until you find the right angle to center the pin. You can then just add weights) Using the above method, find R for the following vectors. Note: If we assign a direction to a given mass on the weight hanger, we then have a magnitude and direction for the mass and can hence treat it as a vector. It will not then be necessary to multiply the mass by g to get force. We shall just do everything in mass units.

29 VECTOR PROBLEM I: MAGNITUDE ANGLE F 1 00g 30º F 00g 10º E I R I VECTOR PROBLEM II: MAGNITUDE ANGLE F 1 500g 0º F 300g 90º E II R II VECTOR PROBLEM III: MAGNITUDE ANGLE F 1 00g 30º F 100g 90º F 3 300g 170º E III R III ANALYSIS: Using the polygon method, draw each vector to scale on graph paper. Use one sheet of paper per problem. You do not need to draw the equilibrant, just the given vectors and the resultant. Choose a scale so the vector diagram fills up most of the sheet (this gives more accuracy.) Make sure each vector has arrows on the tip and show all the angles. Show the scale calculation for the resultant.

30 Using the diagrams from the graphical method, break each vector into x and y components and use the component method to find R. Use a different color to show the components. REPORT: Hand in your initialed data sheet, the graphical and analytical calculations, and a summary table. Calculate the percent discrepancy comparing the resultant from the force table with the resultant as calculated using the analytical method. When there are several results to report, a summary table should always be included in the conclusion. Since this is your first lab with several results, a sample summary table is shown (use excel). PROBLEM FORCE TABLE GRAPHICAL ANALYTICAL % DISCREPANCY R θ R θ R θ R I II III QUESTIONS: 1. Why is it necessary to include the mass of the weight hangers? Since they have the same mass shouldn t it cancel out?. What are the sources of error? List them. What is the largest source of error? 3. Why is it necessary to let the string slip on the ring? 4. What would be the effect of a more massive ring?

31 CANTILEVER BEAM INTRODUCTION: A wooden meter stick will be used as a cantilever beam as shown in the diagram. As mass M is added to the end of the beam, the deflection d will increase. The deflection d for a given mass M also depends upon the length of overhang L. By measuring d for varying L while keeping M constant, and by measuring d for varying M while keeping L constant, two sets of data can be obtained which can then be graphed. One graph will yield an equation of d as a power function of M, d = k 1 M a and the other graph will yield an equation of d as a power function of L, d = k L b. These two equations can then be combined mathematically to yield a single equation, d = k 3 M a L b. The experimental objectives are to determine the values of k 3, a, and b. INSTRUCTIONAL OBJECTIVES: This Exercise is designed to give the student practice in designing an experiment, determining the order in which to take data, analyzing data using graphical techniques, and combining two equations to get one joint variation equation. EXPERIMENTAL PROCEDURE: 1. Determine what you are going to use as a reference for the measurements of the deflection d. Remember that d is the distance the meter stick deflects with a mass M placed on it compared to its position with no mass.. Clamp the meter stick to the table, and, using a ruler, measure the deflection d for various lengths L using a fixed mass M (you need to figure out what to use as a reference for measuring d). You also need to determine what mass to use but the mass M should not be large enough to cause a deflection of more than 10 cm when the stick is out at an L of 90 cm. If the deflection is more than 10% of the length, the meter stick may break and it no longer behaves like a cantilever beam. (You decide on the order of data for L values. Should you start with large or small values of L?) 3. Now vary M for a fixed L. You need to choose a value for L. Keep in mind that small lengths do not give much deflection and hence d would be hard to measure. Also, values of L that are too large give a lot of deflection for very little mass and thus limit the range of values you can use for M.

32 ANALYSIS: Graph your data using EXCEL. Force each graph thru (0, 0) and use power function trend line for d vs. L and linear trend line for d vs. M. Have each equation shown on each graph. Compare the equation on the graph of d vs. M to the equation d = k 1 M a, and determine the values of k 1 and a. Compare the equation on the graph of d vs. L to the equation d = k L b, and determine the values of k and b. Now comes the tricky part. Remember the final equation is: d = k 3 M a L b, and when you varied M, L was a constant. Let s call this constant value L 0. For this value of k 3, the equation becomes : d 3 k M a L. Note that the value of k 1 b = k3 L0, and by knowing the value of k 1 and L 0, k 3 can be calculated. Calculate the value of k 3, including units and write the final equation with your values of a and b included. b o TABLE L d M. REPORT: Write a complete report with all the necessary sections as listed in The Simple Pendulum lab. Pay particular attention to the analysis section and show clearly how you obtained the final equation.

33 FRICTION INTRODUCTION: The coefficient of kinetic friction between a block of wood and a wooden plane will be determined by measuring the friction force while varying the normal force and by measuring the angle of repose. The effect contact area of the sliding surface will also be investigated. THEORY: When relatively smooth solid surfaces slide over each other, the force of kinetic friction, f k, is proportional to the normal force F N and is directed opposite to the displacement. The coefficient of kinetic friction is define by the equation: re-arranging the equation gives (1) plotted on the x-axis, the slope of the straight line is k. k f f F k k N F. When f k is plotted on the y-axis and F N is k N Before the lab period, the student shall derive the equation for the angle of repose, θ R, for kinetic friction (the angle for which the block slides down the plane at constant speed). x y Draw a force diagram, use F 0 and F 0 to derive the equation: () k tan R EQUIPMENT: Plane board, wood block, weight hangar, pulley, string, weights, table clamps, angle meter, right angle clamps, rods, triple beam balance. EXPERIMENTAL PROCEDURE: Part 1: Mass the block. Place the plane board flat on the table, attach a pulley to one end and place the block on the plane. Run a light string from the block, over the pulley, and attach a weight hangar. The pulley should be attached to the end of the block without the hole thru it. This will insure the grain of the wood is the same direction as in part of this experiment. Notice that the block has two sliding surfaces, one wide and one narrow. Start with the wide side down. Notice that the pulley is adjustable in height. Make sure the pulley is adjusted so the string is parallel to the sliding surface. The wooden blocks should only be held by the sides. Touching the sliding surface with sticky or greasy hands will change the coefficient of friction. Add weight to the weight hangar until the block slides with constant speed when started in motion with a light tap. Add masses to the block in 00g increments up to 100g. For each mass M A added to the block, record the total mass m, hanging on the string, which makes the block move with constant speed.

34 Repeat part 1 with the narrow side of the block as the sliding surface. Part. Slide a small rod thru the hole in the end of the wood plane and attach a right angle clamp to the rod. Attach the angle clamp to a vertical rod and then attach the vertical rod to a table clamp. Keeping the grain of the block and plane the same as in part 1, place the wide side of the block on the plane, change the angle between the plane and the table until the block slides down the plane at constant speed when given a small tap. Measure this angle with the angle meter. This angle is the angle of repose. Repeat part for the narrow side of the block. ANALYSIS:

35 Part 1. The friction force is equal to mg where m is the mass attached to the string when the block slides at constant speed. The normal force is equal to Mg where M is the mass of the block plus added masses. Thus, equation (1) becomes mg Mg or m M. Thus, we do not need to multiply the masses by g to get weight and we can just graph m on the y-axis and M on the x-axis. The slope of the line will be µ. Plot graphs of m verses M for the wide side and the narrow side. Draw the best fit line thru the points for each graph and find the slope of these lines. The slope will be the coefficient of friction. Put error bars for m on the graph points. Draw maximum and minimum slopes on the graph and determine the slope of the max and min slopes. The percent error in the slope is given max slope min slope by slope and this is the % error in µ. slope Part. From the equation k tan R, determine the coefficient of friction for the wide and the narrow sides. Find the % difference for the wide side using the results from parts 1 and. Find the % difference for the narrow side using the results from parts1 and. Note: percent difference = x x 1 x avg Using the equation k tan R and differential error propagation to determine the predicted error in µ. REPORT: Unless asked by your instructor to do a formal report, your report shall include the following sections: THEORY ANALYSIS CONCLUSION (summarize the results and state whether or not the contact area makes a significant difference in the coefficient of friction? In other words, is the difference greater than predicted by the % error in the value of µ? State sources of error and how the results could be improved) APPENDIX (answers to questions and original data sheet) QUESTIONS: 1. If the tangential force of the string just balances the frictional force, why does the block move?. How does friction in the bearings of the pulley change the coefficient of friction?

36 3. (problem) Using the mass of your block and using the coefficient of friction for the wide side, determine how much force parallel to the plane would be required to move the block up the plane at a constant speed? Assume the plane angle is 30º

37 CENTER OF MASS INTRODUCTION: Using five nearly identical meter sticks, you are to find the best systematic procedure for stacking the sticks lengthwise, one on top of the other, out over the edge of the table such that end of the top stick is at a maximum distance D from the edge of the table. Once the proper method of stacking is found, measurements of the displacement of each stick, relative to the stick immediately below it will be made and a sequence, for this displacement, determined from these measurements. This sequence will be summed to find a series which can be used to predict the theoretical value of the distance D. This sequence will also be used to predict the location of the center of mass of the stacked sticks. PROCEDURE: By trial and error, determine a systematic method for stacking the sticks. Should you start with the bottom stick, the top stick, the middle stick, or what? We shall refer to the top stick as stick number one, the second from the top as stick number two etc. In general, we can have n sticks, n = 5 for our case. If the proper method of stacking has been discovered, the distance D of the end of the top stick from the edge of the table will be over 110 cm. Once the proper method of stacking has been determined, take measurements of D, d 1, d, d 3, d 4, d 5 as shown in the following diagram. Systematically shuffle the sticks and repeat the measurements five times. d 3 d d 1 d 5 d 4 D

38 ANALYSIS: Let L represent the length of the sticks. 1. Find the closest fractional value of d 1, d, d 3, d 4, d 5 in terms of the length L. For example, d 1 = L/. You find the rest. These values of d n represent a sequence for the distance of the front edge of each stick relative to the stick below it. Now find a general term for the sequence. An example of a general term would be d n = L/n. This, of course, is not the correct general term.. Represent the distance D as the sum of the sequence, D = d 1 + d + d 3 + d 4 + d 5 using the general term determined from part 1 of the analysis. The sum of a sequence is called a series. Use this series to predict the value of D and compare it to the measured value of D. For example, 5 L using the sample sequence above, the series would be D 1 n 3. Using the definition for the center of mass, find the location of the center of mass of the five sticks using the sequence from part 1 of the analysis. Choose a coordinate system with the origin located at the edge of the table. Let x 1 be the coordinate of the number 1 stick (top stick), x the coordinate of stick number etc. Now x 1 =d 5 +d 4 +d 3 +d. Now substitute the value for d 5, d 4, etc. using your sequence. Do the same for x, x 3, etc. Substitute these values of x n into the equation 5 5 Mi x Mixi 1 1 where x is the center of mass and then solve for x. Hint, assume all masses are equal and simplify the above expression by solving for x before doing any substitutions. Another hint is that x =d 5 +d 4 +d 3 +d L/. Where does your intuition tell you the center of mass is located? 4. Extra credit: How far from the table would the far end of the top stick be if the number of sticks approached? Or, if you let the number of sticks get larger and larger, say approaching, what does the value of D approach? Could D approach? In other words, prove that the series diverges. Use the integral test. REPORT: In your report, be sure to include the stacking method, detailed analysis for each section above, and a conclusion which summarizes your results and compares your measured results of D with the calculated value.

39 BALLISTIC PENDULUM AND PROJECTILE MOTION One period for part 1 and One period for part 3 INTRODUCTION: The initial speed of a projectile fired from a spring gun will be determined by two methods: Part 1 will use the ballistic pendulum to find the initial speed. In part, the measurement of the projectile range, when it is fired horizontally from a known height, will be used to calculate the initial speed. In part 3, the known initial speed from parts 1 or or both will be used to calculate the range of the projectile when it is fired at an angle θ from the table and lands on the floor. The projectile will be fired and its actual range will be compared to the calculated value. Also in part 3, the projectile will be fired at various angles at ground level and the range versus the angle will be plotted on a graph. EQUIPMENT: Ballistic pendulum apparatus (Record the equipment number on your data sheet.) Triple beam balance Table clamp Tape Measure Meter stick Plumb bob Target paper Carbon paper THEORY: (to be completed by the student before class) Part 1 Ballistic pendulum: The ball is shot horizontally form a spring gun with speed v 0 and is caught by the pendulum bob. The ball and the pendulum rise to an angle θ, the highest point. Let m be the mass of the projectile, L is the pendulum length, M is the mass of the pendulum bob and shaft, h is the maximum height to which both will rise, v 0 is the initial speed of the ball (this is what we want to find.) V represents the speed of the ball and pendulum immediately after impact.

40 The speed after impact, V, can be found by using conservation of mechanical energy: (Kinetic Energy of pendulum and ball immediately after impact) = (Potential Energy of ball and pendulum at angle θ). From this equation the speed, V, can be found in terms of the vertical rise h. Note that masses cancel out in this equation. Write down the equation and solve it for V. Call this Equation (1). Next, we need to get v 0 in terms of V. Using conservation of momentum for the completely inelastic collision of m with M we have: (momentum of the ball, m, before the collision) = (total momentum of the ball and pendulum, M+m, after the collision) Write down this equation and solve it for v 0 in terms of V, m, and M. Call this equation (). Combine equations 1 and and solve for v 0 in terms of M, m, h, and g. Call this equation (3). Next, it is necessary to get h in terms of the pendulum length L and the angle θ. From the diagram and using some triangle trigonometry, show that h = L(1 - cosθ). Substitute this into equation (3) to get the final equation.

41 Part : Calculation of v 0 from the measurement of the range. The pendulum is moved up at 90º and locked in position so it is out of the way and the projectile is fired horizontally from a known height H above the floor. The range X is measured, and from this known range and height, the initial speed v 0 can be calculated. Using the equations of motion from lecture, derive the equation for v 0 in terms of H, X, and g. Derive this before class and include the derivation in your report. Hint, use the y motion to find t and then substitute this time into the x equation. Let y=0 where it lands and call y 0 =H Part 3: The spring gun will now be set at an angle θ with respect to the horizontal. Using the known velocity v 0, derive an equation for the range R of the projectile. The equations you will use are x x v cos t and y y0 v0 sin t gt The equation for R will be in terms of v 0, θ, H, and g. Derive this equation before coming to class. Hint, use the y motion to find t and then substitute this time into the x equation. Let y=0 where it lands and call y 0 =H. The solution of the y equation (quadratic) is a bit tricky and is subject to algebraic errors (especially sign errors.) Work independently of your partner so you can check each other for errors. Compare your final equation with your partner s. For the theory regarding the range versus the angle, you can look this up in your text.

42 PROCEDURE Part 1: Initial velocity from the Ballistic Pendulum. Place the pendulum apparatus on the table. Put a table clamp behind the launcher as a backstop (do not clamp the base to the table as it will crack the base). Remove the ball from the launcher and mass it. Do the same for the pendulum and rod as a unit. The length L of the pendulum is 8.5cm. Put everything back together and put the ball back in the launcher. Use the ram rod to cock the launcher to the desired spring tension (there are three levels). Set the angle meter to zero and fire the projectile into the pendulum. Read the angle and repeat this at least 10 times. Calculate the value of v 0 using the equation derived in the theory section. Part : On the launcher, there is a white circle with a cross on it. This is the reference point for all height and distance measurements. Place the launcher next to the edge of the table so you can drop a plumb bob from this point to the floor. Put a table clamp behind the launcher as a backstop. Drop your plumb bob from the cross in the circle and put a mark on the floor, where it touches the floor as a reference for your range measurements. Cock the launcher with the same spring tension as in part 1 and do a test firing. Tape an 8 1/ X 11 inch piece of paper on the floor, centered where your test firing landed. Place a piece of carbon paper over this paper and fire the launcher at least 10 times. Remove the carbon paper and measure to the center of the grouping. This will be your value of X. Measure the distance from the floor to the cross on the launcher. This will be your value of H. Calculate v 0 using the equation derived in the theory section. Part 3: The value of θ will be provided by your instructor. Remove the launcher from its mount, rotate it 180º and reconnect it to the launcher at the angle given. Measure the new height H from the floor to the cross. Using equation derived in the theory section and the value you obtained for v 0, (you decide whether to use the average value of v 0 obtained from parts 1 and or the value of v 0 from part 1 or part ) calculate the range R. Again, work independently of your partner and compare results. After calculating what R should be, tape a sheet of target paper to the floor, lengthwise along the line of fire, centered at the calculated distance R. Place a sheet of carbon paper on top of the target paper. Do not shoot the ball until your instructor has seen your calculated distance and is present to watch the shot. Fire the ball several times to get a grouping of points. Remove the carbon paper and measure the distance to the center of the grouping. Compare this distance to the calculated distance R. Points will be assigned depending on what region of the target the ball lands. Now fire the projectile at ground level for 5º intervals starting at 30º and ending at 60º. Graph the range versus the angle. Is the shape of this graph as expected? Discuss in light of the theory from your text.

43 REPORT: One report for both parts. In the theory section, be sure to include all of the derivations for each part. In the calculation section, write down the equation you derived in the theory section. In the next step, substitute the numbers showing just one sample of each type of calculation along with appropriate error propagation. In the conclusion, be sure to indicate whether or not the percent discrepancy in part 3 is less than the predicted percent error. QUESTIONS: (put answers in the appendix) 1. An experimenter measures X and H in part. Suppose that, unknown to the experimenter, the floor slopes down at 1º in the direction of firing. The experimenter calculates v 0 using his measurements, assuming that the floor is level. Will the calculated value of v 0 be larger or smaller than the v 0 calculated from measurements made on a level floor? Justify your answer.. Suppose the gun were mounted on frictionless rollers instead of being firmly anchored to the table. Would the velocity of the ball be smaller or larger than it would be if the gun were anchored? Justify your answer.

44 ROTATIONAL MOTION AND THE MOMENT OF INERTIA Introduction: When a sphere, solid cylinder, and a hollow cylinder are released at the top of a triple track, one finds that the sphere arrives at the bottom first, the solid cylinder arrives second, and the hollow cylinder arrives last. The objective of this experiment is to measure the times it takes for each object to travel down the track and compare these times with those predicted from theory. Equipment: Triple beam balance, Stop watch, Angle Meter, Meter stick, Vernier calipers, Triple track, Table Clamp, Rods (long and short), Rod Connector Rolling objects (sphere, solid and hollow cylinder) Drawing tools ( T-Square, Triangles, ruler, etc.) Theory: From the definition of the moment of inertia shown that r dm it can be I hollow cyl = mr (thinned walled) 1 I hollow cyl = r r m (thick walled) 1 I solid cyl = ½ mr (derive this) I sphere = mr 5

45 Notice that each of the above moments of inertia, except the thick walled cylinder, can be expressed as I = kmr where k=1 for the thinned walled cylinder, k=1/ for the solid cylinder, and k= /5 for the sphere. To find an expression for the time it takes each object to travel down the ramp, use conservation of energy to first find the final speed at the bottom of the ramp. mgh 1 mv 1 I where H is the height of release above the bottom of the ramp. Substitute the expression for I kmr in the above equation and solve for v. Once v has been found, it can be substituted into the equation for average speed L v avg to find the time where L is the length t of the ramp. Show that: Note that k 1 t L. gh H=Lsinθ

46 Note that the equation for I of a thick walled hollow cylinder cannot be expressed as I=kmr and therefore the equation shown for t is not as accurate as one derived using 1 I hollow cyl = r r m (thick walled) 1 Where r 1 is the inside radius and r is the outside radius. Using this equation for I, derive a more accurate equation for t. (does v=r 1 ω or does v= r ω?) L H θ Procedure: 1) Adjust the incline to θ = 5º ) Make ten measurements of t for each object 3) Measure L, H, m, θ, r outside for all objects, and r inside for the hollow cylinder.

47 4) Include as part of your data, an estimate of the errors for the above measurements. 5) Repeat all measurements for θ = 10º Analysis: Using the equation k 1 t L, calculate the time it takes for gh each object to travel down the ramp for each angle. (Use measured values of H). Using error propagation and the error estimates in your data, calculate the % predicted error in t when θ = 10º for the solid cylinder. Use estimated errors in H and L to find % error in t. Compare your measured values with the calculated values. Use the more accurate thick walled cylinder equation derived for the time t, to calculate new values of t for the hollow cylinder. Report: This will be a formal report. Be sure to include all of the sections necessary for a formal report. All of the report must be done with a word processor including using equation editor for all of the mathematics. Nothing except your data should be in handwriting. The theory section must include the derivation of the moment of inertia for a solid cylinder, using integration, as well as the derivation of the equation for t. In your conclusion, you should include a summary table showing the average of the measured values of t for each angle and the calculated value of t for each angle along with the % difference.

48 YOUNG S MODULUS AND TORSION MODULUS INTRODUCTION: The experimental objectives of this lab are to find values of Young s modulus and torsion (shear) modulus for steel. EQUIPMENT: Young s modulus apparatus with dial indicator Torsion modulus apparatus and torsion rod Bubble level Micrometers Meter stick Assorted large and small weights and hangars Bow calipers THEORY: Young s Modulus For the description of the elastic properties of linear objects like wires, rods, columns which are either stretched or compressed, a convenient parameter is the Young's modulus of the material. Young's modulus can be used to predict the elongation or compression of an object as long as the stress is less than the yield strength of the material. F L Young modulus is defined as the stress,, divided by the strain, A L where F is the force stretching the material, A is the cross-sectional, ΔL is the elongation, and L is the original length of the material. In this lab, a wire will be stretched by adding weights to a weight hangar. Thus, F=mg and Young s modulus becomes F Y A = L L F A L L or Y mg A L L If we re-arrange this equation, we can write YAL m. gl YA A graph of m vs. ΔL will then have a slope of from which Y can be determined. gl Torsion Modulus Usually the sheer or torsion modulus is measured by applying a torque to one end of the rod which is fixed at the other end and measuring the angular rotation φ. It can be shown, by using the definition of shear modulus and integrating, that the torsion modulus is given by:

49 where l M (you do not need to derive this equation) 4 r M is the shear modulus l is the length of the rod r is the radius of the rod φ is the angle of twist or rotation in radians A weight mg will be attached to a wheel of radius R to give a torque τ = mgr and the above lmgr equation then becomes M. 4 r Re-arrange this equation as was done with Young s modulus so a graph of m vs. φ gives a straight line. PROCEDURE: Young s modulus: 1. Place one kilogram on the weight hangar initially to eliminate kinks. Leave this on throughout the experiment and do not count it as part of the load. The elongation which the initial weight produces will not enter into your measurements since you have not zeroed your reading of the wire length.. Level the stand using the bubble level 3. Measure the diameter of the steel wire using micrometers 4. Measure the length of the wire which is subject to stretching.

50 5. Take a zero elongation reading of the dial indicator 6. Add weights one kilogram at a time and record the dial indicator reading each time.

51 7. Once you have added as many weights as possible and taken your dial indicator readings, remove the weights on at a time and again take readings each time a weight is removed. Note: Be sure to record the absolute error in all of your measurements since this will be needed for error propagation in your analysis. Torsion Modulus The apparatus will be set up by the lab tech at the back of the lab. When you finished, just leave the apparatus as you found it. 1. Adjust the vernier to zero when an initial mass of 00 grams is suspended from the strap. Do not count this initial weight in your calculations.. Add masses 0.50kg at a time and record the angle reading after each addition. Do not twist the rod excessively. Do not exceed a total of 5.0kg 3. Measure the rods length and diameter. 4. Measure the diameter of the wheel on which the masses are hung. Note that the vernier for measuring the angle is in tenths of a degree.

52 ANALYSIS: Young modulus: Plot a graph of m verses ΔL and from the slope, calculate a value of Y and using error propagation, calculate the predicted % error in Y. Compare your value to the accepted value of 00 GN/m. Remember, compare means to find the % discrepancy and see if it is within the margin of error as determined by error propagation. Torsion modulus: Using the analysis of Young s modulus as a guide, decide what quantities to graph and perform an analysis similar to that, described above, for Young s modulus. Report: For your report, include all of the sections (abstract, introduction, etc.).

53 ARCHIMEDES PRINCIPLE INTRODUCTION: Archimedes principle will be used to determine the density of a liquid, a solid that has a density greater than that of water, and a solid with a density less than that of water. EQUIPMENT: (for a group of ) Beakers 1 Triple beam balance Table clamps and rods 1 Vernier caliper 1 Metal cube with hook 1 wood block (for class use) Monofilament line Graduated cylinder filled with alcohol Hydrometer (placed in graduated cylinder) Analytical balance Jug of alcohol mixture THEORY: The mass density ρ, of a substance, is the mass per unit volume or, m Eqn. (1). V The mass density of water is 1.00 g/cm 3. The specific gravity (s.g.) is defined as s. g.. water Since the denominator has a numerical value of one when using the density of water as 1.00 g/cm 3, the specific gravity will have the same numerical value as the density of the substance but will have no units. Physics students know that w=mg and is in units of dynes or newtons (chemistry students may not know this.) However, since our balances weight things in grams, we will use grams as though it is force and not multiply any masses by g. We will call it gram force. Archimedes principle states that the buoyant force on a body immersed in a fluid has a buoyant force (upward force in grams) equal to the weight of the fluid that the body displaces. This can be expressed as: Eqn. () buoyant force(gram force) = V fluid displaced ρ fluid (COMPLETE THE THEORY FOR PARTS 1,, AND 3 BEFORE THE CLASS MEETING)

54 Theory, part 1: Measurement of the density of a solid cube. If we suspend a solid body (metal cube) from a balance and completely submerge it in a liquid, the sum of the gram forces can be written as: Eqn. (3) F scale + buoyant force m = 0 Combine equations (1), (), and (3) and derive an equation for the volume of the cube and the density of the cube in terms of m, buoyant force, and ρ water.. Theory, part : Measurement of the density of an unknown liquid. If we now suspend the submerged cube of known volume and mass m from the balance and measure the new buoyant force, the density of the unknown liquid can be determined. Derive the expression unk m F unk water where F stands for scale reading. m Fwater Theory, part 3: Measurement of the density of a wood block using Archimedes principle by using a sinker of known volume. Use the metal cube of mass m from part 1 to submerge the wood block and suspend the assembly in a beaker of water. Summing the forces in the y direction gives: Eqn. (4) F scale + buoyant force (m cube +m wood )=0 Combine equations () and (4) and derive an equation for the volume of the wood block in terms of F scale, m cube, m wood, ρ water, and V cube. Combine this derived equation with Eqn. (1) to get an equation for the density of the wood. PROCEDURE: Part 1: Density of metal cube Measure the weight of the metal cube. Using the vernier calipers, measure the dimensions of the cube. Obtain a beaker of de-ionized water. Fasten a 1 cm diameter rod to a table clamp and mount the triple beam balance on this vertical rod (see Figure 1 below.)

55 Using monofilament line, hang the cube from a notch provided in the lever arm underneath the balance. Fig. 1 Record the scale reading while the cube is completely submerged in the water. Do not let the cube touch the bottom or sides of the beaker. Take at least two readings it is difficult to get accurate readings because of the damping effect on the submerged weight. Use Eqn. (3) to calculate the buoyant force. The buoyant force can also be measured by weighting the beaker and water without the cube suspended in the liquid and then weighting the beaker and the water with the cube suspended as shown. Fig. Convince yourself using Newton s third law that the buoyant force is the difference between these readings. Take two readings using this method, and average the buoyant forces with those from the previous method. Remember that error estimates need to be included with all of your data.