28-Nov-2016: Lab 22 Physical Pendulum Lab

 1. Title: Lab 22 Physical Pendulum Lab
     Name: Qiwen Ye (Sherry)
     Partner: Matt
     Date: 28-Nov-2016

2. Purpose
The gold of the experiment was to determine the expression for the period of the ring and the semicircle disk pendulum and verified the predicted periods by experiment.

3. Introduction/Theory
Since we tried to find the period of the physical pendulum, we need to find the moment of inertia of an object first in order to set up the torque equation.
Part 1 - A ring of finite thickness and a little notch cut out at the top
Assumed the point of pivot was exactly half way between the inner and outer radii and that the notch did not impact how uniform the mass distribution was.

First, derived the expression for the moment of inertia of the ring by using parallel axis theorem, then set up the torque equation as shown:

The radius of the ring should be consider the outer and inner of the radius; therefore, the radius of the ring was:

So, the period of the ring in pendulum motion was:

Part 2 - A semicircular disk of radius R

At first,  calculated the moment of inertia of the semicircular disk that was rotating its midpoint of the diameter:

Then, calculated the center of mass of the semicircular disk by using integral method:

After we found the center mass of the semicircle, we used the parallel axis theorem to calculated new moment of inertia of the semicircular disk, including it was rotating about the midpoint of its base and rotating about a point on its edge, directly above the midpoints of the base.

1) Oscillation about the midpoint of its base 

2)oscillation about a point on its edge, above the midpoint of the base. 

The angular velocity and angular acceleration, rotating its midpoint of its base:

The angular velocity and angular acceleration, rotating a point on its edge, directly above the midpoint of the base.

4. Apparatus/Experimental Procedure

Part 1 - A ring of finite thickness and a little notch cut out at the top

Placed the ring as shown, and attached a thin piece of making tape to the base of the ring. Set up a photogate below the ring such that, for small oscillation, the marking tape passed through the photogatte. Used LoggerPro (opened Pendulum Timer.cmbl) to determine the period of oscillation of the ring.

Part 2 - A semicircle disk of radius R

Attached a thin piece of masking tape to the bottom of each object and attached a pin to the top of both side of the plate of each object as shown:

1) oscillating about the midpoint of its base

2) oscillating about a point on its edge, directly above the midpoint of the base

5. Measured Date

For the ring and the semicircle, measured the outer and inner of the radius of the ring and the radius of the semicircle:

6. Calculated Result/Graph

By using the photogate, we got the graph of period vs. time in LoggerPro. The Y-Intercept (b) was the experimental value of the period of the ring or the semicircle.

Part 1 - A ring

Part 2 - A semicircle

1) Oscillating about the midpoint of the base

2) Oscillating about a point on its edge, directly above the midpoint of the base.

7. Explanation

After we got the experimental value of the period in each part, we calculated the theoretical value of the period.

Part 1 - A ring

Part 2 - A semicircle

Compared the experimental and theoretical value of the period in each shape.

8. Conclusion

In this experiment, we calculated the period of an physical pendulum of the ring and the semicircle by using the moment of inertia, the center of mass and the parallel axis theorem of the object to set up equation of Newton's second law. Our range of error and uncertainty is between 0.19 to 1.24%. For the semicircle, we attached a pin to it as a pivot clip; however, we don't need to consider the mass of the pin when we calculate the moment of inertia because the radius of the pin to the pivot point and the mass of the pin are too small. The masking tape should be placed exactly in the center of the base of the semicircle because it would effect the value when we are measuring the moment of inertia. If the masking tape is not in the center of the base, the distance from the center of mass to the masking tape will become bigger, it means Md^2 become larger that the moment of inertia of the object will become larger.