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In the first experiment we examined a situation involving periodic motion. Such situations are interesting in physics because to have periodic, or harmonic, motion generally requires that the accelerations involved are non-uniform. Thus, the simple equations of uniformly accelerated motion that are developed early in the class no longer apply, and the more general expressions that we call "first principles" must be used to develop useful equations.

Many forms of harmonic motion are commonly found in everyday situations, from the pendulums of grandfather clocks to the oscillations of a quartz crystal in a radio transmitter. In this experiment, you will examine the simple harmonic motion of a mass on a spring, one of the simplest forms of harmonic motion.

When a load is gradually applied to the free end of a spring suspended from a fixed support, the spring usually stretches until the tension in the spring just balances the weight of the load. Hooke’s law is the statement of this condition, in that the amount of stretch of the spring is directly proportional to the amount of load applied to it, within certain limits. Under these conditions, if the loaded spring is set into oscillation, it will undergo harmonic motion. This is the situation under study today.

When you have completed this experimental activity, you should be able to: (1) determine whether an elastic material obeys Hook’s law; (2) determine the force constant of a spring; (3) experimentally determine the period of vibration of a system in harmonic motion; (4) predict the period of a mass on a spring; and (5) understand the relation of frequency and period.

Suspend the spring from its hanger and place a weight hanger on the bottom of the spring. Adjust the height of the spring until the bottom of the weight hanger is at some convenient reference level. Begin to add weight to the weight hanger in 50 gram increments, noting the position of the hanger at each weight, until some 500 grams have been added to the hanger. In a similar manner, remove mass in 50 gram increments, noting the position of the hanger each time, until the hanger is empty again. Record each position accurately, and then take the average of the two positions as the actual position.

Make a Scatter plot your data as Load (in kg) versus Stretch of the spring (in m). The data points should lie along a reasonably straight line, indicating that the spring obeys Hooke’s law. Add a trendline of best fit to your plot, and determine the slope of the line. The force constant will be the slope of the line multiplied by 9.81 m/s², which converts it to units of N/m. Record your force constant.

Place a load of about 200 grams on the spring, and gently set the system into vertical oscillation. A reasonable amplitude is about 3 cm, as the system will oscillate for a long time even though the amplitude is small. Once the system is oscillating, time 20 complete oscillations of the mass. An online stopwatch is available. Record the value in your data table, and repeat the measurement two more times. The time for 20 oscillations should be consistent to within .10 seconds, else another measurement is called for.

Continue your observations with a total of 6 loads ranging from 200 grams to 600 grams, making at least 3 consistent measurements for each load. Record your data carefully in the data table, and also record the period of oscillation of each load. Remember, the period is the time for one complete oscillation.

For comparison, you should calculate the theoretical period of the system for each load. The period, T, is the reciprocal of frequency, f:

The frequency, f, is related to the angular frequency, w , that is determined by the force constant of the spring, k, and the effective mass of the system, M, by the relations:

Thus, the period should be:

It can be shown that for a spiral spring, the effective mass of the system is the total load of the spring plus one-third the mass of the spring, or:

where M_o is the total load on the spring (including the weight hanger) and m is the mass of the spring itself.

Use equation (4) to determine the theoretical period for each of the six loads, and compare the theoretical period with the experimental period for each using percent difference.

Report your value for the force constant of the spring. Did you and your partner get similar values? If you had a second spring with a much higher force constant, how would this change your results? Report your six values for period. How does the mass affect the period? What additional conclusions can you draw from the experiment?

1. What fractional error would be introduced into the theoretical period if the mass of the spring were neglected?
2. Would this error be significant in your experiment?
3. Does it appear from this experiment that the term is valid?

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