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Simple Harmonic Motion: Simple Pendulum

A simple pendulum is an example of motion that is approximately a Simple Harmonic Oscillator (SHO). A SHO is used frequently by physicists as a model in describing systems which involve potential energy as a function of position. This model might represent an atom or other complex system. By using a simple model, difficult problems are often made clear.

Consider a particle subject to a linear restoring force. This could be a mass attached to a spring, or a simple pendulum. We will be using a simple pendulum in this lab experiment:


Figure 1: The Simple Pendulum

If we consider only motions of the pendulum in one vertical plane, the moment of inertia is:
 
I = ml2 (1)

and the torque can be written as:
 
t= I a = ml2 d2 q/d t2 = -mgl sin q (2)

The torque is taken as negative since it is acting in a direction such as to decrease the angle q . Then, the equation of motion can be written as:
 
d2 q/d t2 = - (g/l) sin q (3)

If we were using a spring we would have a differential equation in terms of the linear displacement x, but since this is a swinging system, must be used. Both types of systems are solved in the same manner. We make an approximation to be valid in the case of small angles (q<< p/2) so that sin q is approximately q (in radians). Note that theoretically, the frequency is independent of the amplitude (but remember that the amplitude must be somewhat small since we made this assumption to solve the problem to begin with). Since is measured in radians (angular measure per unit time) we can express the period, or time for oscillation as:
 
T = 2 p  Ö(l/g) (7)

Note that the period depends on the length of the pendulum and not the initial displacement of the bob. Because of this, pendulums have for ages been used to regulate time in many clocks.

OBJECTIVE:

To study the motion of a simple pendulum as an illustration of a simple harmonic oscillator and to make an experimental determination of the acceleration due to gravity.

APPARATUS:

PROCEDURE:

  1. Measure the length of the string. Make it about 100 cm. Be sure to measure it from the support to the center of mass of the object. This is the length that is needed to determine the period and other info.
  2. Start the pendulum swinging through a small arc, say about 5 degrees. Measure the time required to swing through about 50 cycles. Count these through the center of swing, since greater errors are possible at the top (either side) of the swing where it momentarily stops. Since it spends such a short time at the bottom, passing through, a more accurate determination of time can be made by measuring at this point. A cycle is from the middle, to the top, back through the middle, to the top and finally back to the middle again. That is one cycle.
  3. Record data for various lengths of the string, reducing from about 100 cm to 20 cm in increments of 20 cm; i.e. there should be five trials. Record Length, Time for 50 cycles, Period (time for one cycle) and period squared.
  4. For comparison, repeat this experiment with an initial angle of about 25 degrees. Do this for any convenient length.
  5. On one piece of graph paper plot the period versus length (for all five trials).
  6. On the same piece, plot the square of the period versus length, again for all five trials. Discuss the significance of the difference in shape of these curves.
  7. Compute the acceleration due to gravity from these curves. Explain in your data analysis section how and why you did what you did. The technique is up to you.
  8. Discuss the uncertainties. Why can you ignore the mass of the string? Or can you? How does that affect the length of the string and pendulum? What is the length? How do you determine it? Is the period independent of mass? Discuss these ideas and refer to your graphs and data to substantiate your comments and conclusions.
  9. Have fun!
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