Oscillatory Motion Experiment 1: Oscillations of a spring As described in the text, a system of a simple spring exhibits simple harmonic motion and is a good introduction to a study of oscillations, which are widespread in nature. Procedure and data Part 1: Static Equilibrium We hang a spring from a stand, with a holder (m " ) attached. Record the bottom position of the holder (x " ). Then, add a weight m & (such as 100g mass) onto the holder so that m = m & + m " and allow the spring to stretch. When it is motionless, measure again the bottom position of the holder. From the change in the equilibrium position, x & x ", we can determine the spring constant k: When the spring is extended, the mass m exerts a force F = mg and at the new equilibrium position, F = m & g = k (x & x " ) (in magnitude). Thus, the spring constant for your spring is: k = m &g x & x ", (1) where m is the mass on the holder. Calculate k using above equation. Repeat this step for another mass, measuring a new equilibrium position. Calculate average k. Part 2: Dynamic oscillations While the spring is extended, measure the period of 10 oscillations T 23456738. The angular velocity is ω " and the predicted period of oscillation T is ω " = : k m 2π and T = = 2π@ m ω " k, (2) where k was measured above. Repeat for different masses. Be careful not to overstretch and deform your spring! Spring Part 1: A hanger and mass connected to a spring at one end For a hanging mass (mass m ) which is connected to a spring (force constant k ) as shown above we have: Restoring Force is given by: F = kx Thus: ma = kx = m 8C D 8D C The solution to the above differential equation is beyond the scope of this course, and is given by:
where: X is the amplitude of oscillation. x = Xcos(ωt + φ) φ is the phase (depends upon the starting position). The angular frequency of oscillation is : ω = @ K 2 The time period of oscillation T is given by: T = LM N = 2π@2 K Calculations and analysis From your measurement of x & x ", calculate T O738PQR38 and compare it to your T 23456738, the time for one full oscillation, for each of the masses used. In part 1, you measured the values of x & x ", for different masses m &, and calculated the intrinsic spring constant k. Use this value of k to calculate the period T for each mass m (which is the total mass m = m & + m " ). (a) x 1-x 0 (b) Figure 1: (a) spring with initial mass attached. (b) New mass m & attached to spring so that m = m & + m ". 2
For both mass 1 and mass2 perform the following steps: 1. Weigh the masses. 2. Pull the hanger such that the spring is extended by a few centimeters. 3. Decide on N, the number of oscillations for which you will measure the time. 4. Start the timer and release the glider at the same instant of time. 5. Record t, the time taken to complete a fixed number of predetermined oscillations N. 6. Calculate the time period T (=t/n) of the oscillations. 7. Calculate the spring constant of the spring. The notations and formulae used are as follows: m = Mass of the weight and hanger. t = Total duration for which N oscillations are observed [sec.] T = Time period for a single oscillation (T = t /N) [sec.] m N t T = t /N (kg) (s) (s) Mass 1 Mass 2 Discussion (spring data) Use the difference in the repeated measurements of the period to give an indication of the uncertainty in the result, and discuss the agreement between the predicted and measured periods. Briefly explain, what systematic uncertainties might have affected the results? Experiment 2: The simple pendulum The system we will study in this experiment is very straightforward: A mass M suspended by a light string from a fixed point. It provides an elegant example of simple harmonic motion, certainly one of the earliest to be noticed and studied. Indeed, the story is told that it was a swaying chandelier that first aroused Galileo's interest in mechanics and diverted him from the study of medicine. We can follow in Galileo's footsteps at least part way, exploring the factors that affect the motion of the pendulum. Clearly gravity is involved, and variables include the mass (M) of the suspended bob and the length (L) of the suspension string. We will consider the mass of the string itself to be negligible. Figure 2 shows the system, with the pendulum and string displaced by the angle θ from the vertical position. At this displacement, the pendulum will be subject to a restoring torque τ τ = (Mg sin θ)l, (3) 3
where g is the acceleration of gravity which we have measured before in Experiment 1. The negative sign indicates that the restoring force Mg sin θ is opposite in direction to the displacement θ. θ L Mg sin θ M θ Mg cos θ Figure 2: Pendulum forces. For small θ, sin θ θ, and in this approximation equation (3) can be written Mg τ = MgLθ. (4) In this approximation, the restoring torque is proportional to the displacement theta, so the condition for simple harmonic motion is satisfied. As shown in your textbook, the frequency (f) and the period (T) of the pendulum in this case are given by f = 1 2π @g L, T = 2π:L g. (5) This experiment can yield an excellent measurement of g. Notice that the mass M does not appear in this equation! In this experiment, we will check the dependence on L and the lack of dependence on M of the period, and also try to test the limits of validity of the simple harmonic motion approximation. We will then use equation (5) to determine g. Finally, we will study a pendulum operating in reduced gravity. Procedure and data Pendulum Part 1 The experiment consists of determining the frequency (period) of various pendulums of different M and L, for several initial angles of displacement. The equipment includes pendulum bobs of two quite different masses (steel and aluminum) and a string whose length you can vary. Start with the string approximately 100 cm. long. 4
Initiate oscillations by displacing the bob (leave no slack in the string) and determine the period by timing oscillations. Each oscillation involves a complete swing back to the starting position. The period T is determined by counting the number N of complete oscillations in a time t: T = t N. (6) There are two modes of doing this experiment: 1) and 2) are alternative procedures for taking data: 1. We can use the photogate apparatus in the pendulum mode. For this purpose the gate is mounted so the bob interrupts the light beam on each passage twice during each complete oscillation period. Set the Timer to the pendulum mode. Start the pendulum swinging but with a small amplitude. Press the RESET button on the timer and note the first time displayed. This is the period of the pendulum, the time for one complete oscillation. Repeat this measurement several times by pressing the RESET button and recording the first time measured. Take the average of these measurements to determine the period. This seems like an ideal experimental arrangement, BUT it requires careful alignment, and light reflected by the bob can lead to a false reading, so the data must be watched with care. You should do a few calculations with some data in the lab to make sure your data is accurate. 2. The second mode is much more mundane closer to what Galileo did, except we use a different kind of clock! This involves using a digital stopwatch as a timer, which we manually start and stop, while counting visually the number of oscillations. No photogate! No alignment problems to contend with! Just the need to correlate hands and eyes to start and stop the clock at an extreme of the oscillation, and count 50-100 complete oscillations visually for each period. It is important to do fairly long counts, in order that errors arising from problems of coordination will have a small percentage effect. Take data as follows: a. Choose one of the methods to use for your measurements. Determine N and t for two different bob masses (M) at each of three different pendulum lengths (L), approximately 100 cm, 50 cm and 25 cm. Use small oscillation amplitudes, (θ 10 ). From these data, you will determine the period T for each M and L combination. Do at least two trials for each mass and length combination (six combinations). If the two trials are reasonably consistent (within less than 5%), use the average. If not consistent, repeat until two consistent trials are obtained. b. For one M and L combination, determine the period T using the alternative method. c. Next, to check the limits of validity of the small θ approximation, measure T for larger oscillations, θ 30 45 using your standard method. Do these measurements with one value of M, at two different pendulum lengths. Record your estimate of θ for each case in a and b. 5
Part 2 In part 1, the pendulum is always swinging in a plane perpendicular to the surface of the earth. In a second type of apparatus, we can constrain the oscillation to lie in planes at other angles ψ to the vertical. That is, the displacement is still θ, and the motion is still simple harmonic, but the whole pendulum is leaning over at an angle ψ. In these cases, the whole force of gravity is not available to affect the pendulum motion only the component in the plane of motion, g cos ψ, is effective, and since cos ψ 1, the pendulum is operating in reduced gravity. In the second apparatus, the condition ψ = 0, cos ψ = 1 corresponds to our part 1 setup. For part 2, use the timer and visual counts to determine T at ψ = 0 and two other angles. Data and calculations Pendulum Part 1 The notations and formulae used are as follows: L = Length of the pendulum from the point of suspension to the center of the bob [cm.] t = Total duration for which N oscillations are observed [sec.] N = Number of complete oscillations in time t T = Time period for a single oscillation (T = t/n) [sec.] B O B g = 4πL L T L Small Oscillations (θ < 5 ) Large Oscillations (θ > 30 ) L t N T T L g t N T T L g cm sec sec s L cm/s L sec sec s L cm/s L Steel Alum Part 2 The notations and formulae used are as follows: ψ = Angle which plane of oscillation makes with the vertical plane. L = Length from beam to bob s center = cm. t = Total duration for which N oscillations are observed [sec.] T = Time period for a single oscillation (T = t/n) [sec.] 6
g 3mm = 4πL L T L ; ψ Q4oQ = cos p& q g 3mm r ; %Diff = ψ Q4oQ ψ g " ψ 100% ψ N t T T L g 3mm ψ Q4oQ %Diff (s) (s) (s) L (cm/s L ) O g " = g 3mm = g 3mm = g 3mm = Pendulum :Calculations and analysis Part 1 In this part of the experiment, you have experimentally determined values of T for six different combinations of M and L at small oscillation angles. Note that equation (5) can be written from which we obtain T = 2π xg L, (7) T L = 4πL L. (8) g Plot your values of T L vs L. According to equation (8), this should be a straight line. If all your points (at two different masses) fall on a single straight line, you have verified that the period does not depend on M, and that T L is a linear function of L. On the same graph, plot the values of T L vs L you obtained with large oscillations. Use a different symbol to distinguish these from the small oscillation data.) Are the two sets of data significantly different? You can now turn equation (8) around to solve for g: g = 4πL L T L. (9) Calculate g for each of your small oscillation measurements; determine the average and its Standard Deviation. Also calculate g o4z using your large angle oscillation data and equation (9). Do not include the g o4z values in your average of g. Separately, calculate g from the measurement made using the alternative method. Part 2 At ψ = 0, when the plane of oscillation is perpendicular to the earth, the effective acceleration of gravity, g 3mm, is equal to g. However, for other angles ψ, where the plane of oscillation is not perpendicular to 7
the earth, g 3mm is reduced. Equation (10) now holds, with g replaced by g 3mm. Using your measured values of T for different angles of ψ, calculate g 3mm from equation (10) for each value of ψ. We expect g 3mm = 4πL L T L. (10) g 3mm g = cos ψ. (11) Discussion (pendulum data) Compare your value of g ± Δg determined in part 1 from the small angle oscillation data to the accepted value of g. Does the experimental value agree with the accepted value, within the experimental error? What statistical errors may have been present and what would be their effect on the result? If you still have your writeup of Experiment 2, compare the values of g determined in free fall to those determined from the simple pendulum in part 1. How does the measurement made with the alternative method agree with your main result? What sources of error are different for the two methods? Which method would you expect to be more accurate? Discuss the sensitivity of your results to increasing the amplitude of the oscillation, in terms of the % difference between your measurements of g o4z and g in this experiment. For part 2, do your measurements show the effect of reduced gravity? How well do they agree with equation (11)? What sources of systematic error likely affected your result? How might you have reduced those errors? How does your measurement of g 3mm at ψ = 0 compare with your results from part? References Kestin and Tauc, University Physics, Chapter 12 Oscillations. 8