Simple Harmonic Motion



Simple Harmonic Motion©98

Experiment 7

Objective: To examine the significant parameters involved in the oscillatory motion of a body under the action of a spring and compare such motion with a simple pendulum.

Discussion:

Simple harmonic motion is the name given the most of all periodic motion, the period being defined as the observation of a complete cycle, i.e. back and forth or up and down. If a body hangs from a spring, stretching the spring because of its weight, a slight displacement of the body from its equilibrium position produces an oscillation. If an additional weight is added to this simple spring, the additional stretch of the spring to a new equilibrium position caused by the weight is proportional the weight added. That is, if Δx is the additional stretch of the spring and ΔW is the additional weight that caused this stretch, then

(1)

where k is the spring constant, a characteristic of how tightly the spring is wound and the flexibility of the material.

Except for the pull of gravity and a little friction from the air, only the spring acts on the mass suspended from it. The weight of the body, does not affect the oscillation of the body. However, the mass of the spring, since the spring takes part in oscillation, must be included in the total mass of the system.

(2)

Because not all parts of spring oscillate with the same amplitude, i.e. one end of the spring is fixed and the other end moves with the body, one third of mass of spring, or its effective mass, is included in the mass of the oscillating body. Thus the total mass is the sum of the mass of the object and the effective mass of the spring.

(3)

An even simpler example of harmonic motion is a point particle swinging at the end of a massless string. This is a simple pendulum and the period of this system is determined by:

(4)

Exercises For the Suspended Oscillator:

1. Determine the mass and period of several masses suspended from a spring.

a. Measure the spring constant of the large spring by suspending a number of known weights from it and using Equation 1.

b. Separately hang several bodies from the spring and measure their periods of oscillation.

c. Measure the mass of each body and then the spring.

2. Construct a graph showing the squared period of oscillation as a function of the total mass of the system. This curve should be a straight line. Is it?

3. Calculate the % deviation between the periods found for each mass from the graph and those found in part 1.

Exercises for the Simple Pendulum:

1. Data Acquisition.

a. Measure the length of the pendulum, from the point if pivot to the center of the mass.

b. Determine the period of oscillation of the pendulum using a small ( ................
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