Lab 20 Physical Pendulum lab

Purpose: The purpose of this laboratory is to derive expressions for the period of various physical pendulums and verify the predicted periods by experiment.

The purpose of the apparatus is to measure the period of the pendulum so that the measured period could be compared to the period calculated theoretically.

Procedure:
In the first part of the lab we predicted the period for a metal ring. To do this we measured the diameter of the outer ring and the diameter of the inner ring. Then we used newton's second law to write an equation where the torque is equal to the moment of inertia of the ring times the angular acceleration. We rearranged the equation to get angular acceleration by itself and were able to find the angular velocity. We then used the angular velocity to calculate the period. Then we measured the actual period of the ring as a pendulum and we compared the two results.

For the second part of the experiment we cut out a half circle out and measured the radius of the half circle. Then we found the center of mass of the half circle so that we would know the distance of the force creating the torque as the pendulum swung. Then we derived an expression for the moment of inertia for the half circle because it was not one that we knew. Then we used newton's second law to and rearranged the equation to find the angular velocity. Using the angular velocity we found the period for the half circle as a pendulum. We did this for the pivot being at the top of the half circle and for the pivot being at the bottom of the half circle.

To find the angular velocity (omega) of a pendulum we wanted to use newton's second law equation and manipulate it so that it would look like this:

When we manipulate the equation to look like the one above, whatever is in the parentheses is omega^2. So we could then find omega and plug it into the following equation to find the period T:

The calculations for the period of the ring as a pendulum were as follows:

The equation for newton's second law was torque equaled moment of inertia times angular acceleration. We then solved for the angular acceleration to get it in the form that we wanted and the sintheta turned into just theta because theta was small. We added the negative sign because the force acting on the pendulum was always opposite its direction of motion. We then found omega and plugged it into the second equation to find the period which was .717 sec. We set up our ring on the stand and measured the period with logger pro and we got the measured period to be .719190 sec.

For the second part of the experiment we cut out a half circle and found the period if we let it swing as a pendulum in two positions.

The first thing we did was find the center of mass of the half circle, because the center of mass is where the force of the weight is moving the half circle as a pendulum. So to find the center of mass found an expression for a small piece of the mass and integrated it from 0 to R and the calculation was the following:

R being the radius of the half circle.

After we found the center of mass of the half circle we had to find the moment of inertia of the half circle and we did so by taking a small piece of the half circle and finding the moment of inertia of that piece and integrating it from 0 to R. The calculation went as follows:

After finding the moment of inertia we used newton's second law to find an equation that we could manipulate into the form alpha = -(    )theta. So that we could then find omega and use it to find the period. Our calculation to find the period for the half circle being hanged by the round side was:

 

The calculation to find the period for the pivot at this point:

the value that we measured on logger pro was:

Then we derived an equation to find the period if the pivot point were on the other side of the half circle:

 

The calculation to find the pivot point at this point:

The period value that we measured with logger pro for this point was.

Conclusion:
The period calculated theoretically for the half circle with the pivot on the round part was .7699 sec while the period measured was .6147 sec the values are pretty close so they are good for the equipment that we used. While the period calculated theoretically for the half circle with the pivot on the long part of the half circle was .3933 sec and the measured was .6189 sec. the values are more off for this one most likely because the calculation might be incorrect that is the only thing I can think of but I felt like it was the way it should have been done.