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6. Simple Harmonic Motion Goal: The purpose of this experiment is to investigate several properties of two important oscillating systems: the spring and the pendulum. Background: Oscillations are described in your text book. Very briefly, a simple harmonic oscillation results if the force F needed to pull an object away from its equilibrium position is proportional to its displacement x from equilibrium, F = kx (1) One such system consists of an object attached to a spring. The quantity k in Eq. (1) is called the spring constant and indicates how stiff the spring is. When set in motion, the spring oscillates with a period given by T = 27, k where M is the total mass hanging from the spring. -Veg Paper flag indicates Xog Weight holder 0.050 kg Another system that exhibits simple harmonic motion is the pendulum. When a mass suspended from a string swings through a small angle less than 30° about its equilibrium position, its period of oscillation is T = 27, (3) The situation is illustrated above Task 3 below. 2 ummy
Task 1: Determine the value of the spring constant using Eq. (1). • Record the equilibrium position of the unloaded spring above the table, which we denote by Vaa I measured 50 cm Attach the weight holder (0.05 kg) to the spring and measure the new height y. Then add up to 0.25 kg of mass, in increments of 0.05 kg. Note that the hanging mass Mincludes the mass of the holder itself. With the addition of each mass record the new height and calculate the displacements x=-y. Add a column to your data table with the corresponding weights Mg. I measured new height47 cm for total mass 0,05kg x=2.9 cm for a total mass of 0.1 kg I measured x = 5.9 cm for a total mass of 0.15 kg I measured x = 8.9 cm for a total mass of 0.20 kg I measured x= 12.2 cm for a total mass of 0.25 kg I measured x = 14.8 cm for a total mass of 0.3 kg i measured x = 18.1 cm • Graph the applied force Mg versus the displacement x. Find the best fit straight line of Graph I and determine the slope my and the y- intercept br. Using these two numbers (and units) write down how the force depends on the displacement, F=mx+b. According to Eq. (1), what does the slope of your graph represent? Task 2: Determine the value of the spring constant using Eq. (2). • Place 0.1 kg of mass on the holder, pull it downward a few centimeters, and release it. Determine the period of a single oscillation I defined as the time for one complete up and down motion. Determine I by measuring the time of twenty oscillations and dividing it by twenty. Repeat this measurement of I several times, pulling the spring downward to different initial displacements and recording the average period for each trial. Does the period depend on the displacement? I measured 20 oscillations, taking 13.2 seconds. I measured 20 oscillations, taking 13.1 seconds. I measured 20 oscillations, taking 12.9 seconds. The period is T=0.66 second The period is T = 0.66 second The period is T = 0.65 second • Repeat the period measurements for five different masses such that the largest attached mass should be about 0.25 kg. Record each of your five average periods and masses. For total mass of 0.05kg +0.05kg (holder), The period is T=0.56 second For total mass of 0.1kg +0.05kg (holder), The period is T=0.66 second For total mass of 0.15kg +0.05kg (holder), The period is T=0.75 second For total mass of 0.20kg +0.05kg (holder), The period is T=0.82 second For total mass of 0.25kg +0.05kg (holder), The period is T=0.89 second 1 3
• Plot Graph II of the square of the period, 7², versus the total suspended mass M. Find the best fit straight line of Graph II and determine the slope mu and the y-intercept kn. Using these two numbers (and units) write down how the square of the period depends on the suspended mass, T² = müM+b. According to Eq. (2), what does the slope of your graph represent? Does it agree with the value you obtained in Task 1 and what is the percent difference?The y-intercept of the graph bu might not be zero as we omitted the mass of the spring itself. Task 3: Investigate the period of a pendulum. This angle must be small. L Construct a pendulum by attaching a weight to a short string (say, L = 0.1 m) and suspend it from a stand. Determine the period of a single oscillation I by measuring the time of twenty oscillations and dividing it by twenty. Repeat this measurement of T several times, displacing the ball to different initial displacements (less than 20°) and recording the average period for each trial. Does the period depend on the displacement? No, the period seems not to depend on the displacement • Repeat the previous process for six different length L, reaching a maximum length of about 1.5 m. Record each of your average periods and lengths. Here is what I measured: L (m) T(sec) 0.296 1.1084 0.525 1.4708 0.478 1.3992 0.396 1.2866 1.355 2.3427 • Plot a Graph III of the square of the period, 7², versus the length L. Find the best fit straight line of Graph III and determine the slope mu and the y- intercept bm. Using these two numbers (and units) write down how the square of the period depends on the length, 7² = m+bm. According to Eq. (3), what does the slope of your graph represent? • According to Eq. (3), the slope of the best straight-line fit to your data points is 4²/g. What value of g do you deduce from the slope mmof Graph III? How well does it agree with the accepted value of g? Calculate the percent error. Task 4: Prepare a report containing your data sheet, graphs, and the answers requested in Tasks 1-3.