Calculating Uncertainties and Forces in Physics Experiments, Lecture notes of Physics

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Introductory Physics Laboratory Manual Course 20300 Contents Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . 2 Measurements and Uncertainty . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . 3 Graphical Representation of Data . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . 7 The Vernier Caliper . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 The Micrometer Caliper . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 Angle Scale Verniers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 Vectors - Equilibrium of a Particle. . . . . . . . . . . . . . . . . . . . . . . . 12 Air Track . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 Atwood's Machine . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . 25 Centripetal Force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 Linear Momentum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 Elasticity and Simple Harmonic Motion . . . . . . .. . . . . . . . . . . . . . . . . . . . 34 Buoyancy and Boyle's Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 Introductory Physics Laboratory Manual Introduction The aim of the laboratory exercise is to give the student an insight into the significance of the physical ideas through actual manipulation of apparatus, and to bring him or her into contact with the methods and instruments of physical investigation. Each exercise is designed to teach or reinforce an important law of physics which, in most cases, has already been introduced in the lecture and textbook. Thus the student is expected to be acquainted with the basic ideas and terminology of an experiment before coming to the laboratory. The exercises in general involve measurements, graphical representation of the data, and calculation of a final result. The student should bear in mind that equipment can malfunction and final results may differ from expected values by what may seem to be large amounts. This does not mean that the exercise is a failure. The success of an experiment lies rather in the degree to which a student has: • mastered the physical principles involved, • understood the theory and operation of the instruments used, and • realized the significance of the final conclusions. The student should know well in advance which exercise is to be done during a specific laboratory period. The laboratory instructions and the relevant section of the text should be read before coming to the laboratory. All of the apparatus at a laboratory place is entrusted to the care of the student working at that place, and he or she is responsible for it. At the beginning of each laboratory period it is the duty of the student to check over the apparatus and be sure that all of the items listed in the instructions are present and in good condition. Any deficiencies should be reported to the instructor immediately. The procedure in each of these exercises has been planned so that it is possible for the prepared student to perform the experiment in the scheduled laboratory period. Data sheets should be initialed by your instructor or TA. Each student is required to submit a written report which presents the student’s own data, results and the discussion requested in the instructions. Questions that appear in the instructions should be thought about and answered at the corresponding position in the report. Answers should be written as complete sentences. If possible, reports should be handed in at the end of the laboratory period. However, if this is not possible, they must be submitted no later than the beginning of the next exercise OR the deadline set by your instructor. Reports will be graded, and when possible, discussed with the student. You may check with the TA about your grade two weeks after you have submitted it. 2 2. Combining Measurements Consider the simple function R = a b when a and b have uncertainties of Δa and Δb. Then ΔR = (a + Δa)(b + Δb)− a b = aΔb + b Δa + (Δb)(Δa) Since uncertainties are generally a few percent of the value of the variables, the last product is much less than the other two terms and can be dropped. Finally, we note that dividing by the original value of R separates the terms by the variables. ΔR R = Δa a + Δb b The RULE for combining uncertainties is given in terms of fractional uncertainties, Δx/x. It is simply that each factor contributes equally to the fractional uncertainty of the result. Example: To calculate the acceleration of an object travelling the distance d in time t, we use the relationship: a = 2 d t−2. Suppose d and t have uncertainties Δd and Δt, what is the resulting uncertainty in a, Δa? Note that t is raised to the second power, so that Δt/t counts twice. Note also that the numerical factor is the absolute value of the exponent. Being in the denominator counts the same as in the numerator. The result is that Δa a = Δd d + 2 Δt t Examination of the individual terms often indicates which measurements contribute the most to the uncertainty of the result. This shows us where more care or a more sensitive measuring instrument is needed. If d = 100 cm, Δd = 1 cm, t = 2.4 s and Δt = 0.2 s, then Δd/d = (1cm)/(100cm) = 0.01 = 1% and 2Δt/t = 2(0.2s)/(2.4s) = 0.17 = 17%. Clearly the second term controls the uncertainty of the result. Finally, Δa/a = 18%. (As you see, fractional uncertainties are most compactly expressed as percentages, and since they are estimates, we round them to one or two meaningful digits.) Calculating the value of a itself (2× 100/2.42), the calculator will display 34.7222222. However, it is clear that with Δa/a = 18% meaning Δa ≈ 6 cm s−2, most of those digits are meaningless. Our result should be rounded to 35 cm s−2 with an uncertainty of 6 cm s−2. In recording data and calculations we should have a sense of the uncertainty in our values and not write figures that are not significant. Writing an excessive number of digits is incorrect as it indicates an uncertainty only in the last decimal place written. 3. A General Rule for Significant Figures In multiplication and division we need to count significant figures. These are just the number of digits, starting with the first non-zero digit on the left. For instance: 0.023070 has five significant figures, since we start with the 2 and count the zero in the middle and at the right. The rule is: Round to the factor or divisor with the fewest significant figures. This can be done either before the multiplication or division, or after. 5 Example: 7.434× 0.26 = 1.93284 = 1.9 (2 significant figures in 0.26). 4. Reporting Uncertainties There are two methods for reporting a value V , and its uncertainty U . A. The technical form is ”(V ± U) units”. Example: A measurement of 7.35 cm with an uncertainty of 0.02 cm would be written as (7.35± 0.02) cm. Note the use of parentheses to apply the unit to both parts. B. Commonly, only the significant figures are reported, without an explicit uncertainty. This implies that the uncertainty is 1 in the last decimal place. Example: Reporting a result of 7.35 cm implies ±0.01 cm. Note that writing 7.352786 cm when the uncertainty is really 0.01 cm is wrong. C. A special case arises when we have a situation like 1500±100. Scientific notation allows use of a simplified form, reporting the result as 1.5×103. In the case of a much smaller uncertainty, 1500±1, we report the result as 1.500×103, showing that the zeros on the right are meaningful. 5. Additional Remarks A. In the technical literature, the uncertainty also called the error. B. When measured values are in disagreement with standard values, physicists generally look for mistakes (blunders), re-examining their equipment and procedures. Sometimes a single measurement is clearly very different from the others in a set, such as reading the wrong scale on a clock for a single timing. Those values can be ignored, but NOT erased. A note should be written next to any value that is ignored. Given the limited time we will have, it will not always be possible to find a specific cause for disagreement. However, it is useful to calculate at least a preliminary result while still in the laboratory, so that you have some chance to find mistakes. C. In adding the absolute values of the fractional uncertainties, we overestimate the total uncer- tainty since the uncertainties can be either positive or negative. The correct statistical rule is to add the fractional uncertainties in quadrature, i.e. ( Δy y )2 = ( Δa a )2 + ( Δb b )2 D. The professional method of measuring variation is to use the Standard-Deviation of many repeated measurements. This is the square root of the total squared deviations from the mean, divided by the square root of the number of repetitions. It is also called the Root- Mean-Square error. 6 E. Measurements and the quantities calculated from them usually have units. Where values are tabulated, the units may be written once as part of the label for that column The units used must appear in order to avoid confusion. There is a big difference between 15 mm, 15 cm and 15 m. Graphical Representation of Data Graphs are an important technique for presenting scientific data. Graphs can be used to suggest physical relationships, compare relationships with data, and determine parameters such as the slope of a straight line. There is a specific sequence of steps to follow in preparing a graph. (See Figure 1 ) 1. Arrange the data to be plotted in a table. 2. Decide which quantity is to be plotted on the x-axis (the abscissa), usually the independent variable, and which on the y-axis (the ordinate), usually the dependent variable. 3. Decide whether or not the origin is to appear on the graph. Some uses of graphs require the origin to appear, even though it is not actually part of the data, for example, if an intercept is to be determined. 4. Choose a scale for each axis, that is, how many units on each axis represent a convenient number of the units of the variable represented on that axis. (Example: 5 divisions = 25 cm) Scales should be chosen so that the data span almost all of the graph paper, and also make it easy to locate arbitrary quantities on the graph. (Example: 5 divisions = 23 cm is a poor choice.) Label the major divisions on each axis. 5. Write a label in the margin next to each axis which indicates the quantity being represented and its units. Write a label in the margin at the top of the graph that indicates the nature of the graph, and the date the data were collected. (Example: ”Air track: Acceleration vs. Number of blocks, 12/13/05”) 6. Plot each point. The recommended style is a dot surrounded by a small circle. A small cross or plus sign may also be used. 7. Draw a smooth curve that comes reasonably close to all of the points. Whenever possible we plot the data or simple functions of the data so that a straight line is expected. A transparent ruler or the edge of a clear plastic sheet can be used to ”eyeball” a reasonable fitting straight line, with equal numbers of points on each side of the line. Draw a single line all the way across the page. Do not simply connect the dots. 7 The Micrometer Caliper Also called a screw micrometer, this measuring device consists of a screw of pitch 0.5 mm and two scales, as shown in Fig. 3. A linear scale along the barrel is divided into half millimeters, and the other is along the curved edge of the thimble, with 50 divisions. Figure 3: Micrometer Caliper. The pointer for the linear scale is the edge of the thimble, while that for the curved scale is the solid line on the linear scale. The reading is the sum of the two parts in mm . The divisions on the linear scale are equal to the pitch, 0.5 mm. Since this corresponds to one revolution of the thimble, with its 50 divisions, then each division on the thimble corresponds to a linear shift of (0.50 mm)/50 = 0.01 mm . In Fig. 3, the value on the linear scale can be read as 4.5 mm , and the thimble reading is 44 × 0.01 mm = 0.44 mm. The reading of the micrometer is then (4.50 + 0.44) mm = 4.94 mm. Since a screw of this pitch can exert a considerable force on an object between the spindle and anvil, we use a ratchet at the end of the spindle to limit the force applied and thereby, the distortion of the object being measured. The micrometer zero reading should be checked by using the ratchet to close the spindle directly on the anvil. If it is not zero, then this value will have to be subtracted from all other readings. 10 Angle Scale Verniers This type of vernier appears on spectrometers, where a precise measure of angle is required. Angles arc measured in degrees (◦) and minutes (’), where 1 degree = 60 minutes. Fig. 4 shows an enlarged view of a typical spectrometer vernier, against a main scale which is divided in 0.5◦ = 30’. Figure 4: Angle Scale Vernier. The Vernier has 30 divisions, so that the sensitivity of the vernier is one minute. (There are also two extra divisions, one before 0 and the other after 30, to assist in checking for those values.) Each division on the vernier is by 1/30 smaller than the division of the main scale. When the index is beyond a main scale line by 1/30 of a division or 1’, line 1 on the vernier is lined up with the next main scale line. When that difference is 2/30 or 2’, line 2 on the vernier lines up with the next line on the main scale, and so on. Fig. 4 shows an example where degree and Vernier scale run from right to left. Again, reading the angle is a two step process. First we note the position of the index (zero line on the Vernier) on the main scale. In the figure it is just beyond 155.0◦. To read the vernier, we note that line 15 seems to be the best match between a vernier line and a main scale line. The reading is then 155.0◦ + 15′ = 155◦ 15′ = 155.25◦. The example shows one problem with working with angles, the common necessity of converting between decimal fraction and degree-minute-second (DMS) notation. We illustrate another place where this arises with the problem of determining the angle between the direction of light entering the spectrometer, and the telescope used to observe light of a particular wavelength. Example: The position of the telescope to observe the zeroth diffraction order is 121◦55′. Light of a certain wavelength is observed at 138◦48′. The steps in the subtraction are illustrated below, using DMS and decimal notation, respectively. Either method is correct. DMS decimal 138◦ 48′ 137◦ 108′ 138.80◦ − 121◦ 55′ − 121◦ 55′ − 121.92◦ ????? 16◦ 53′ 16.88◦ 11 Vectors - Equilibrium of a Particle APPARATUS 1. A force table equipped with a ring, pin, four pulleys, cords and pans 2. A set of 16 slotted masses: Set of known masses (slotted type) (4 × 100 g, 4 × 50 g, 2 × 20 g, 2 × 10 g, 1 × 5 g, 2 × 2 g, 1 × 1 g) 3. Protractor 4. Ruler INTRODUCTION Physical quantities that require both a magnitude and direction for their description are vector quantities. Vectors must be added by special rules that take both parts of the description into account. One method for adding vectors is graphical, constructing a diagram in which the vectors are represented by arrows drawn to scale and oriented with respect to a fixed direction. Using the graphic method we can rapidly solve problems involving the equilibrium of a particle, in which the vector sum of the forces acting on the particle must be equal to zero. While the graphical method has lower accuracy than analytical methods, it is a way of getting a feel for the relative magnitude and direction of the forces. It can also solve the problem of the ambiguity of the direction of a force where the analytical method uses an arctangent to determine direction, which gives angles in the range −90◦ to +90◦ .                                                           Figure 1: Graphical addition of two vectors The graphical addition of two vectors is illustrated in Figure 1. A suitable scale must be chosen, so that the diagram will be large enough to fill most of the work space. The scale is written in the work space in the form of an equation, as appears at the top of Figure 1. This is read as ”25 units of force are represented by 1 cm on the page”. 1 Questions (to be answered in your report): How close did the final values come to the values de- termined from the vector addition diagram? (You should have been able to get within 3 grams and 1.5 degrees.) Part II: Addition of Four Forces Producing Equilibrium                                        Figure 4 (a) Set up four pulleys and suspend unequal loads on the cords running over them. Change angles and loads until the system is in equilibrium (i.e. passes both tests). Be sure that the cords are still radial. Sketch the circle diagram to record loads (remember the weight of the pans) and angles. (b) Carefully draw the vector addition diagram (see figure 4). Note that your diagram may not close due to small errors. Questions (to be answered in your report): 1. How large is the resultant of the four force vectors in your diagram? 2. Why should we expect the vector addition diagram to close? (c) The discrepancy can arise from two sources, errors in the lengths of the lines in our diagram and errors in angles. We can estimate the uncertainty in each by noting the sensitivity of the ruler (∆L) and protractor (∆θ). (Sensitivity is the smallest quantity that can be read 4 or estimated from a scale.) Convert ∆L to a force by using your scale value. Convert ∆θ to radians and multiply by twice the largest force in the diagram. (180◦ = π Rad.) The sum of these two terms is an estimate of the uncertainty in the resultant. Questions (to be answered in your report): How does the size of your resultant in (b) compare with this uncertainty? Part III: Determination of X and Y Components of a Force                                            Figure 5 (a) Place a pulley on the 30◦ mark of the force table and apply a load over it. Note the total load and angle at the top of a data sheet. (b) If the resultant of the vector sum of two forces is the single force that is an exact equivalent of the two original forces, then we can reverse the process and find the two forces, in convenient directions, that is equivalent to any given single force. Draw a set of X-Y axes at the lower left of your data sheet. Choose a scale so that the vector representing your load will span most of the page. Draw the vector representing your load, assuming that the positive X axis is the 0◦ direction. Drop a perpendicular to the X axis from the head of your vector (this line makes an angle of 60◦ with the direction of the vector). Draw arrow heads on the two legs of the resulting right triangle, as if they were two vectors that added to the vector on the hypotenuse. 5 (c) The vector along the X axis and the vector parallel to the Y axis are called component vectors1. Convert to force values and sketch a circle diagram with the results. Set up these forces on your force table. Move the original load by 180◦, to 210◦. Test for equilibrium, and make any necessary adjustments to the load to balance. Questions (to be answered in your report): : How large an adjustment did you have to make? How does this compare with the uncertainty you found in II.(c)? (d) The sides of the right triangle can also be determined by trigonometry. Calculate the size of the component vectors trigonometrically. Show your calculation. How do they compare with the values you found graphically ? 1Components are scalars that have an accompanying indication of direction. 6 Part B: Determining the dissipated energy 1. With two spacers, measure the rebound distance D′. Use the average of two successive measurements that are within 5% of each other. 2. Evaluate the fraction of the original kinetic energy that is lost in the collision. 3. Explain the equations relating K and K ′ to D and D′ and then derive K ′/K = D′/D. ANALYSIS 1. If friction has been eliminated, what are the forces exerted on the glider? Draw the Free Body Diagram of the glider. Find the net force and apply Newton’s Second Law to determine the algebraic relation between the acceleration a, g and the angle θ of the track. Use the fact that sin θ = n H/L to express a in terms of n algebraically. What kind of a graph would you expect for a vs. n? 2. Review the material in the introduction to the lab manual on graphing, and using graphs. Select scales so that the graph of a vs. n takes up most of the page. Do include the origin. Plot your data points. Draw the single straight line that best fits your data points. Determine the slope of that line by using two widely separated points on the line that are not data points. What are the units of the slope? 3. Use the analysis of step 2 to relate your value of the slope to g. Determine g from your slope. 4. In order to compare your value of g with the standard value, a value for the experimental uncertainty is needed. Review the material in the introduction on ”Measurements and Un- certainty” and on using graphs. Determine the range of the six times measured for n = 4. Take this value as the uncertainty in time measurements, ∆t. Determine the uncertainty in the value of acceleration ∆a due to ∆t. The uncertainly in the slope measurement is just ∆a divided by the range of n. Finally, determine ∆g from the relation between g and the slope. 5. Compare your value with the standard value by calculating the Uncertainty Ratio: |g − gstandard| ∆g Values less than 1 indicate excellent agreement, greater than 4, disagreement and possible mistakes. Values between 1 and 4 are ambiguous, indicating fair or poor agreement. How well does your result agree with the standard value? 3 Air Track 1. Draw a free-body diagram showing the forces on the glider sitting on an airtrack at an angle θ. Derive the expression for a in terms of g and θ. Note sinθ=nH/L, then a = ____________________ (in terms of g, L, nH). (Where n are the number of blocks and H is their height. Sketch a graph showing how a will depend on n. 2. Check that the air track is level (horizontal). It should take the glider at least 10 seconds to move across the track, no matter at which end it is placed. If it is not sufficiently level, check with your instructor. 3. Measure D and L for the track D=______________ L=______________ H=1.27cm Do not forget units. 3. Releasing the glider and timing should be done by one person. Place one spacer under the single track support (n=1). With the air on, place the glider at the upper end with about one millimeter gap between the glider and the stationary spring. Take a few practice timings to get used to starting the clock at the time the glider is released, and stopping the clock when it hits the spring at the lower end. Be careful when reading the clock. The two dials have different divisions. When you are used to the process of release and timing, proceed. 4. What is the expression of acceleration a in terms of D and t? a=___________________ 5. Measure t and calculate a. With four blocks (n=4), take a total of 6 times. For n=3, first predict the time, then get the TA or professor to initial your prediction; then measure it. If your prediction is not close recalculate and do it again. n t1(s) t2(s) taverage(s) a(m/s2) 1 2 4 3 - predict 3 Atwood’s Machine APPARATUS 1. The apparatus consists of two composite masses connected by a flexible wire that runs over two ball-bearing pulleys. The make up of the composite masses at the beginning of the experiment is: Left Side Right Side ————– —————– 1× 1 g 2× 2 g 4× 5 g 1× 10 g 1× 250 g 1× 250 g 1× 500 g 1× 500 g 1× 965 g 1× 1000 g ———— ————– Total mass: 1750 g 1750 g 2. Stop clock 3. Pair of tweezers 4. Ruler INTRODUCTION This Atwood’s machine consists essentially of a wire passing over a pulley with a cylindrical mass attached to each end of the string, The cylinders are composed of three sections, the lower ones of 250 g, and the middle ones of 500 g. Note that the two top-most sections (the sections to which the wire is tied) , do not have the same mass. The one on the left has a mass of 965 g. The right hand one has a mass of 1000 g. Each of these sections has eight vertical holes drilled into its top. When all the small masses are in the left cylinder, the two cylinders have the same mass and the force (M1−M2)g = 0. In other words, there is no unbalanced force and the system remains at rest when the brake is released the 10 g mass is transferred from the left to the right hand cylinder, the difference between the two masses becomes (M1 −M2) = 20 g, while the sum (M1 + M2) remains unchanged. If now an additional 5 g mass is transferred, the mass difference becomes 30 g, while the sum is still unchanged, etc. With the small masses provided, it is possible to vary (M1 −M2) in 2 g steps from 0 to 70 g. A string, whose mass per unit length is approximately the as that of the wire, hangs from the two masses. It serves to keep the mass of the string plus wire on each side approximately constant as the system moves, therefore it keeps the accelerating force constant. 25 PRECAUTIONS In order to obtain satisfactory results in this experiment and in order to prevent damage to the apparatus, it is necessary to observe the following precautions: Release the brake only when the difference in mass on the two sides is less than 70 g. Preparatory to taking a run, raise the right hand mass until it just touches the bumper. Make sure that it does not raise the movable plate. Always release the brake when moving the masses. Keep your feet away from the descending mass. Before releasing the brake, make sure that the left hand mass is not swinging. Always stand clear of the suspended masses. The wire may break. PROCEDURE Start with a total load of 3500 g and all of the small masses on the left side. Move the left mass (M2) to its lowest posi- tion, (see Fig. 1) Measure the displacement s, which is the distance from the top of the left mass to the bumper. Transfer 10 g from the left to the right side (i.e. from M2 to M1, remember: if 10 g are transferred, then M1−M2 will be 20 g). Make two determinations of the time of rise of the left hand mass through the measured distance. If the two time determinations differ by more than 5%, repeat the measure- ment until you obtain agreement within 5%. Compute the average acceleration using the average of the values of the time. It will be well to practice the timing before recording any results. M1 M2 M2•g M1•g T1 T2 a a s Figure 1: Atwood’s machine Increase the mass difference (M1 −M2) by about 10 g noting the time required in each case and compute the corresponding accelerations. At least six different mass differences should be used. Setup the data table in the following way, do not forget to note M1 + M2: M1 −M2 time of rise [s] displacement acceleration [ g ] first second average s [m] a = 2s/t2 [m/s2] ANALYSIS Applying Newton’s second law to the descending mass (see Fig. 1) we have, M1 g − T1 = M1 a (1) and to the ascending mass, T2 −M2 g = M2 a (2) 26 where T1 is the tension in the wire above the descending mass, T2 the tension in the wire above the ascending mass, and g the acceleration due to gravity. T1 will be greater than T2 because there is friction and also because the wheels over which the wire runs are not without some mass, that means, a torque is required to accelerate them. Adding (1) and (2) and solving for the acceleration we get a = g M1 + M2 (M1 −M2) − T1 − T2 M1 + M2 (3) where (M1 +M2) is constant and (T1−T2) may also be considered as a constant if we assume that the friction remains constant as long as the total mass of the system does not change. If we plot the acceleration a versus the mass differences (M1 −M2), then equation (3) is represented by a straight line of slope g/(M1 + M2) and intercept (T1 − T2)/(M1 + M2). Consult the introduction to this manual for instructions concerning the graphing of data. Plot mass differences on the x-axis and corresponding accelerations the y-axis. Plot the data obtained on graph paper and draw the regression line which best ”fits” the points. From measurements of slope and intercept, calculate g and (T1 − T2). Your report should show your data table, graph, the method used to determine the slope and your calculations. Questions (to be answered in your report): 1. What is the advantage of transferring mass from one side to the other, instead of adding mass to one side? 2. How would your results be changed if you gave the system an initial velocity other than zero? 3. Solve for the tension in the wire above the descending mass for the case of the largest accel- eration. What would be the tension if a = g? 27 PROCEDURE Make any necessary adjustments of the three leveling screws so that the shaft is vertical. The detailed procedure for checking Eq. 1 experimentally will be left to the student. At least two values of r should be used, with two values of m for each r. A method for measuring r should be thought out, the diameter of the shaft is 1.27 cm. The value of f should be determined by timing 50 revolutions of the bob and then repeating the timing for an additional 50 revolutions. If the times for 50 revolutions disagree by more than one-half second either a blunder in counting revolutions has been made, or the point of the bob has not been maintained consistently in its proper circular path. In either case, the measurement should be repeated until a consistent set of values is obtained. It is suggested that you read the next section, on results and questions, before doing the experiment. RESULTS AND QUESTIONS 1. Exactly from where to where is r measured? Describe how you measured r. 2. Tabulate your experimental results. 3. Tabulate your calculated results for f , F from static tests, and F from Eq. 1 and the relative difference between the F ’s (in %), using the static F as standard. 4. Describe how to test whether the shaft is vertical without the use of a level. Why should it be exactly vertical? 5. What are the functions of the guard, the white background, and the counterweight on the crossarm? 6. Discuss your results. 30 Linear Momentum APPARATUS 1. The equipment shown in Figure 1. 2. Steel sphere. 3. Waxed paper. 4. A 30cm ruler. 5. Equal arm balance and known masses. 6. Meter stick. 7. Dowel rod to free stuck sphere from block. INTRODUCTION The principle of conservation of linear momentum is to be tested as follows: a steel sphere is allowed to slide down the track, and immediately after leaving the end of the track plunges into a hole in a wooden block and becomes stuck within the block. The block, which is suspended by four strings, is initially at rest, but swings as a pendulum because of the impact. The momentum of the sphere before the collision is compared to the momentum of the sphere and block just after the collision.                            Figure 1: Entire Apparatus. 31 NOMENCLATURE: m mass of the steel sphere, [g]. M mass of the wooden block, [g]. v velocity of the center of the sphere as it leaves the track, [cm/s] V common velocity of sphere and block immediately after impact, [cm/s] s, h, x, y, b, r various distances, indicated in Figures 2,3, and 4, all [cm] PROCEDURE Part I: Determination of the Velocity of the Sphere Before Impact Place the block on the platform, where it will he safely out of the way. Remove the slider guide and slider from the box (See Fig. 4) and clamp a strip of waxed paper to the floor of the box. Allow the steel sphere to roll down the track from its highest point. It will fall into the box and leave an imprint. The end of the track is horizontal. Determine the height b, through which the sphere falls; be aware that the track is a channel, and the lowest point of the sphere is below the upper edges of the channel. Make ten or more trials, and find the average value of the range r. From these data, calculate the time of flight, and the velocity of the center of the sphere as it leaves the track. hs Figure 2: Front view of the block.   Figure 4: Determination of the steel sphere’s velocity. y x h h h2 - x2 Figure 3: Side view of the block. 32 4. Determine the elongation produced by each load by subtracting the zero reading from each subsequent reading. 5. Show the dependence of the elongation upon the applied force by plotting the elongation as y-axis and the corresponding total force as x-axis. 6. Determine from the curve the range of forces used in which the elongation is proportional to the force. 7. Remove the load 100 g at a time, taking the scale reading in each case. Are these readings the same, for each load, as those found above? Part III: Force constant of the spring The force constant of the spring is the force ΔF required to produce an elongation Δl in the spring. In symbols, this may be expressed as k = ΔF/Δl in units of N/m. This is a constant only for the range of forces within which the proportionality of Part II exists. Determine an average value of the force constant of the spring from the curve plotted in Part II. Part IV: Dependence of the period in simple harmonic motion on the vibrating mass Consider a body, for which the distortion is proportional to the force producing it, held away from its normal position. There is now a restoring force in the body, which is proportional to the distortion. If the force applied to the body is removed, this restoring force returns the body to its normal position. However, its inertia carries it through that point producing a distortion or displacement on the other side. Now the action of the restoring force first brings the body to rest in a distorted position. The action is repeated and this simple harmonic motion continues until it is stopped by friction. As an example, consider as the body the spring with a load of 500 g suspended from it. Reference to the data of Part II will show that now, the spring is in a condition where any additional force will produce a proportional displacement. If the load is pulled down some distance x and released, a restoring force −k x acts on the body. As the body moves back to its equilibrium position, this restoring force diminishes. The minus sign indicates that the restoring force is opposite to the distortion. Since −kx is an unbalanced force, it produces an acceleration a. From Newton’s Law, F = M a, we get −k x = M a, where M is the mass of the system and the negative sign shows that x and a are oppositely directed. This yields − x a = M k (1) The period T in simple harmonic motion is given by T = 2π √ M k (2) 35 Here, M is the mass of the vibrating system consisting of the mass suspended from the spring (500 g in the example) plus a part of the mass of the spring. It can be shown that one third of the total mass of the spring is the part effective in determining the total M . 1. Determine the period of the simple harmonic motion occurring when the load on the spring is 500 g. Determine the average time required for at least fifty vibrations. 2. Repeat with loads of 600, 700 and 800 g. 3. Measure the mass of the spring. 4. Using equation (2), calculate the period to be expected in each case. Compare them with the experimental values of the periods. Obtain both, the difference and the percent difference. 5. What percentage error would be introduced in the calculated values of T for the 500 g load and the 800 g load, respectively, if the mass of the spring were neglected? 6. Plot two curves: T (M) and T 2(M) (the period and the square of the period of vibration vs. the mass supported by the spring) Part V: Dependence of the period on the amplitude The maximum value of the displacement in simple harmonic motion is the amplitude. Using a load of not over 600 g, try varying the initial amplitude of the vibration and note the effect on the period. Does the period depend upon the amplitude? 36 Buoyancy and Boyle’s Law Part A: Buoyancy and Archimedes’ Principle APPARATUS 1. Electronic balance with stand 2. Beaker 3. Metal object 4. Wooden block 5. Thread 6. Blue liquid INTRODUCTION The hydrostatic pressure P at a distance h below the surface of a fluid is given by P = P0 + ρ g h where P0 is the pressure at the surface of the fluid and ρ is the density of the fluid. The hydrostatic pressure exerts a normal force on all surfaces in contact with the fluid. As a result there is a net upward force, called the Buoyant Force FB, whose magnitude is equal to the weight of the fluid displaced, a relationship known as Archimedes’ Principle. FB = ρ g V where V is the volume of the object below the surface of the fluid. In this experiment, you will weigh objects in air and then measure the effect of submerging them in a fluid. A clearly labeled Free Body Diagram should be used to determine the forces on the submerged objects in order to relate your measurements to the density of the objects. If the fluid is water, assume the standard value for ρ of 1000 kg/m3. The electronic balance is turned on by pressing the button at the right. Pressing the button on the left quickly will change the units displayed. We will work with the gram scale. Note that this is the mass-equivalent of the force being measured, you do not have to actually multiply by the numerical value of g, leave it as a symbol and it will eventually cancel out. PROCEDURE Determine the density of a solid more dense than water. Weigh the metal object and then suspend it from the hook on the underside of the balance so that it is submerged in the beaker of water. This second weight, called the ”apparent weight” differs from the first due to the buoyant force. Draw the corresponding Free Body Diagram and use it to determine the forces involved, and to solve for the density of the submerged object. Calculate the buoyant force 37 5. Use the string to hang weights on the plunger. Use values of 500, 700, 1000, 1200 and 1500 g. Wait at least a minute after each weight is added, so that the gas can come back to room temperature. Increase the waiting time at the larger loads, so that the gas can return to room temperature after being compressed. Use the time to complete some of the calculations outlined below. Record the volume and complete the line in the table. Remember that padd = [M/(1 kg)]× p1, where p1 is the additional pressure due to a 1-kilogram weight. Trim ptotal of any digits that are not significant. ANALYSIS 1. We will use Boyle’s law in the form: V = k (1/p) (2) Calculate all the reciprocals and put them into the last column of the table. 2a. Plot the graph of V vs. 1/ptotal. The origin should be included, although it is not a data point. Be sure that your scales allow the graph to occupy most of the page. (We are following the usual practice of plotting the dependant variable on the y-axis.) 2b. Draw the single straight line that best represents the data. Use a transparent straight edge (like a plastic ruler) to help fit the line. The line should be drawn completely across the graph. 2c. Choose two points on the line (not data points) that are widely separated to use in calculating the slope. Note that the slope will have units. Record this value as kslope. 3. The graph of equation 2 would go through the origin. With experimental data, the straight line usually comes close to, but misses the origin. Determine the positive intercept with either axis. Assume that the uncertainties in the value of these intercepts are 0.2 cm3 and 0.04× 10−5Pa−1, respectively. We can see whether this intercept is consistent with 0 by calculating the uncertainty ratio, your intercept value divided by the appropriate uncertainty value. A small ratio (less than 2) indicates good agreement. Large values (greater than 5) means disagreement. Intermediate values can be described as ’fair’ or ’poor’ agreement and usually require further study. 4. The constant k can also be determined from the Ideal Gas Law, p V = n R T , where n is the number of moles of gas (ρV1/MW ), R is the gas constant (8.31 J/mol·K) and T is the absolute temperature in Kelvin. V1 is the volume measured at a load of 1 kg and ρ is the corresponding density. For air, 80% 14N2 and 20% 16O2, the molecular weight MW = 29 × 10−3 kg/mol, and ρ = 2.13 kg/m3 at the 1-kg load and room temperature. Calculate n R T from the data given, and compare with kslope, assuming an uncertainty in kslope of 0.2× 105 Pa·cm3. 40 Questions (to be answered in your report): 1. (a) What curve would equation 1 describe in a graph of p vs. V ? (b) How could we graph our data so as to obtain a straight line with slope k ? 2. How does your value of kslope agree with n R T? The uncertainty ratio here is |kslope − n R T | uncertainty in kslope 3. If the uncertainty in the slope is ΔV (from procedure 2b.) divided by the range of your (1/p) values, what is your uncertainty in kslope ? Density Table Metal ρ [g/cm3] Liquid ρ [g/cm3] Aluminum 2.7 Alcohol, Methyl 0.80 Brass (ordinary yellow) 8.40 Carbon Tetrachloride 1.60 Bronze - phosphor 8.80 Gasoline 0.68 Copper 8.90 Mercury (20◦C) 13.55 Gold 19.3 Water (0◦C) 0.999 Iron - wrought 7.85 Water (4◦C) 1.000 Iron - gray cast 7.1 Water (15◦C) 0.997 Lead 11.3 Water ( 100◦C) 0.958 Steel 7.8 Tungsten 19.3 Stone ρ [g/cm3] Zinc - wrought 7.2 Granite 2.7 Balsa wood (oven dry) 0.11 . . . 0.14 Limestone 2.7 Ebony 1.11 . . . 1.33 Marble 2.6 . . . 2.8 Oak 0.6 . . . 0.9 Mica schist 2.6 Pine - white (oven dry) 0.35 . . . 0.50 Sandstone 2.1 . . . 2.3 41