Lab 22 - Physical Pendulum Lab

Purpose To derive expressions for the period of differently shaped pendulums of varying mass and compare them to actual measured periods.

Apparatus: The apparatus was simple, a ring stand clamped to the lab table as shown below. The lower part of the ring stand had a photogate clamped onto it and was used to measure the period on logger pro. The first pendulum was a metal ring of mass 0.7519kg of outer radius .1465m and inner radius 0.1372m. However the ring was hung off an indentation that shifted the radius of the pendulum over by the average of the inner and outer radius, which turned out to be 0.14185m

Procedure: We first started by calculating the moment of inertia for every different shape. After we found the moments of inertia for the different pendulums, we then used torque equations to calculate the theoretical period for the oscillating pendulums. We used mgsin(theta)xRadius=I(alpha) and solved for alpha. After we solved for alpha, then we were able to solve for omega squared. When we divide 2Pi by omega, then we got the period.

 

Below are my calculations for the period of the hoop. Also included in the calculations are the %accuracy, which meant .07% error. This small amount of error in an experiment with limited technology is pretty amazing.

 The picture above shows the raw data on logger pro from the photogate for the hoop. The picture below was where we took the period information and took an average of it all.

 The next pendulum that we used was the semicircle supported  at the midpoint at the base.
I do not have the mass because I knew it would cancel out in the calculations, which I also told my lab partners we did not need. However, in retrospect it is very unprofessional of me to do so. I assume full responsibility for this; if my lab partners did not get the mass of the pendulum, I wish to be penalized for it alone because it was my fault.

Below are my calculations for the moment of inertia at the midpoint of the base, and the location of the center of mass.

 

The picture above is  of all the raw data put together to generate an average value.

 Avg= 0.83722056 sec

This picture below is of the raw data for the same pendulum, except we flipped it upside down. I completely forgot to take a picture of it before we took it down.

 This is the raw data put together to calculate an average value. Avg= 0.81718602 sec

 

Below are the calculations for the period of the semicircular pendulum with the pivot at the top.

 

Below are the % accuracy values for the semi-circle oriented two different ways.

 Below are the calculations for the moment of inertia around the center of mass for the semicircle, which were used for the parallel axis shifts. Also in this picture are the calculations for the moment around the top of the semi-circle.

 The two following pictures are my calculations for the triangular pendulum. Unfortunately, they are not very accurate.
:(

Conclusion: All in all I am actually pretty amazed at the accuracy of the period of the pendulums compared to the theoretical periods. All of the periods were within less than 1% error, which is unheard of for many lab experiments. The paper clips are located at the pivots for the semi-circular pendulum, and one on each side. The paper clips should not affect the moment of inertia because most of the mass is being supported by the hinge, and either way it would only add 1/3mr^2 for each of the clips. Compared to the rest of the pendulum, the mass and radius of the clips is very tiny. However, the tape has a larger effect on the moment of inertia because the piece used to mark the photogate is positioned at the furthest point on the pendulum. This would act as a point mass, which is mr^2, and this time the radius is actually much larger because it is at the outermost portion of the pendulum. I learned from this lab that pendulums are a much more accurate way to measure the value of gravity than a kinematics approach could ever be.

Lab Partners:
Haokun Zhang
Christian Rivera