Equipment Needed:

Statics Board

Thread

Stopwatch (ME-1234)

Utility Mount and Cord Clip

Mass and Hanger Set

### Theory:

Simple harmonic motion is not limited to masses on springs. In fact, it is one of the most common and important types of motion found in nature. From the vibrations of atoms to the vibrations of airplane wings, simple harmonic motion plays an important role in many physical phenomena.

A swinging pendulum, for example, shows behavior that is very similar to that of a mass on a spring. By making comparisons between these two phenomena, some predictions can be made about the period of oscillation for a pendulum.

The figure shows a simple pendulum with a string and a mass at an angle θ from the vertical position. Two forces act on the mass: the force of the string, T, and the force of gravity. The gravitational force, F = mg, can be resolved into two components. One component, F-radial, is along the string. The other component, F-tangential, is perpendicular to the string and tangent to the arc of the mass as it swings. The radial component of the weight, mg cosθ, equals the force, T, through the string. The tangential component of the weight, mg sinθ, is in the direction of motion and accelerates or decelerates the mass.

Using the congruent triangles in the figure, it can be seen that the displacement of the mass from the equilibrium position is an arc whose length, x, is approximately L tanθ. If the angle, θ, is relatively small (less than 20°), then it is very nearly true that sin θ = tan θ. Therefore, for small swings of the pendulum, it is apporximately true that F-tangential = mg tan θ = mg x/L. Since the tangential force is a restoring force, the equation should be F-tangential = -mg x/L. Comparing this equation to the equation for the restroing force of a mass on a spring, F = -kx, it can be seen that the quantity mg/L has the same mathematical role as k, the spring constant. On the basis of this similiarity, you can say that the period of oscillation for a pendulum is as follows:

where g is the acceleration due to gravity, and L is the length of the pendulum from the pivot point to the center of mass of the hanging mass.

In this part of the experiment we will investigate this equation for the period of the simple harmonic motion of a pendulum.

### Calculations:

- Calculate and record the Measured Period by dividing the total time by the number of oscillations.
- Calculate and record the Average Measured Period.
- Calculate the Theoretical Period.

### Questions:

- How well does the theoretical value for the period of oscillation for a simple pendulum compare to the measured period of oscillation?

=There is a range of 2.5%~3.5% percent error between the theoretical values for the period of oscillation for a simple pendulum and the measured period of oscillation, which is probably caused by lack of precision in using the stopwatch and operating the pendulum with our hands.

2. Does the equation for the period of a simple pendulum provide a good mathematical model for the physical reality? Why or why not?

=Yes, the equation for the period of a simple pendulum provides a good mathematical model for the physical reality because the percent errors between experimental and theoretical values are small –the equation is quite accurate.

3. How does increasing the mass of the simple pendulum affect the period of the pendulum?

=Increasing the mass of the simple pendulum doesn’t affect the period of the pendulum, according to the data from the experiment.

4. How does changing the length of the simple pendulum affect the period of the pendulum?

=The longer the length of string, the farther the pendulum falls; and therefore, the longer the period, or back and forth swing of the pendulum.