PASCO ME-8979 Simple Harmonic Oscillator Instructions

PASCO ME-8979 Simple Harmonic Oscillator Instructions

Purpose

The purpose is to measure the period of oscillation of a spring and mass system and compare it to the theoretical value.

Theory

For a mass attached to a spring, the theoretical period of oscillation is given by
where T is the time for one complete back-and-forth motion, m is the mass that is oscillating, and k is the spring constant.

According to Hooke’s Law, the force, F, exerted by the spring is proportional to the distance, x, by which the spring is compressed or stretched: F = kx, where k is the spring constant. Thus the spring constant can be experimentally determined by applying different forces to stretch the spring different distances. If force is plotted versus distance, the slope of the resulting straight line is equal to k.

Procedure

Measurements to Find the Spring Constant and Theoretical Period

1. Measure the mass of the cart and record it in Table 6.1.

2. Install the feet on the track and level it.

3. Install the end stops about 1 m apart.

4. Clamp the pulley at one end of the track.

5. Set the cart on the track and attach a spring to each end. Attach the other ends of the springs to the end stops (see Figure 6.1).

6. Tie a string to the lower attachment point of the cart. Run the string under one end stop and over the pulley as shown in Figure 6.1. Attach the mass hanger to the other end of the string. Adjust the pulley so that the string runs parallel to the track.

7. Let the mass hang freely and wait for the cart to come to rest. Record the mass of the hanger and the resting equilibrium position of the cart in Table 6.1.

8. Add mass to the hanger. Record the hanging mass and the new equilibrium position of the cart in Table 6.1. Repeat this for a total of 5 different masses, being careful not to over-stretch the springs.
   Measuring the Experimental Period

9. Displace the cart from equilibrium and let it go. Time 5 oscillations and record the time in Table 6.2.

10. Repeat this measurement at 5 times, using the same initial displacement.

11. Add a 250 g mass to the cart and repeat steps 9 and 10.

Calculations

Spring Constant and Theoretical Period

1. Calculate the forces applied by the hanging mass and write them in Table 6.1.

2. Using the data in Table 6.1, plot force versus position. Draw the best-fit straight line through the data points and determine the slope of the line. The slope is equal to the effective spring constant, k.
   k = __

3. Using the mass of the cart and the spring constant, calculate the theoretical period of the cart alone and with added mass. theoretical period of cart alone ____
   theoretical period of cart with added mass ____

Experimental Period

1. Using the data in Table 6.2, calculate the average time for 5 oscillations with and without the 250 g mass in the cart.
2. Calculate the period by dividing these average times by 5 and record the periods in Table 6.2.

Comparison

Calculate the percent difference between the measured and theoretical values:
cart alone percent difference = ____
cart with added mass percent difference = ____

Questions

1. Does the period of oscillation increase or decrease as the mass is increases? Does a more massive cart oscillate faster or slower?
2. If the initial displacement from equilibrium (the amplitude) is changed, does the period of oscillation change? Try it.

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