04-Jun-2015: Physical Pendulum Lab

Purpose: To derive expressions for the period of various physical pendulums and verify the predicted periods through experiment.

Procedure: Our experiment consisted of three parts. The first part required us to determine the period of a physical pendulum composed of a solid ring, of mass M, outer radius R and inner radius r. The second part required us to determine the period of a semicircular plate of radius R, oscillating about the midpoint of its base. And the final part was to determine the period of the same semicircular plate of radius R, oscillating about a point on its edge, directly above the midpoint of the base.

Part 1: Solid Ring

Solid ring on ring stand with photo-gate

The apparatus consisted of two ring stands attached to each other to support an oscillating solid ring. A piece of tape is placed at the bottom of the ring so that it would pass through a photo-gate, consistently, every time the ring moved back and fourth. The photo-gate reads every other passing of the ring to measure the experimental period of the physical pendulum.


Finding the theoretical period of our physical pendulum required us to apply Newtons Second Law of Torque, the parallel axis theorem, and simple harmonic motion.

To begin, we took measurements of the ring's inner radius r, its outer radius R, and its mass M. Then we calculated the moment of inertia of the solid ring about its pivot. Because the moment of inertia was not known for our new pivot, we needed to use the parallel axis theorem solve for it. Below are the calculations.

Calculations for new moment of inertia

After we calculated the new moment of inertia, we applied Newton's Second Law of Torque and wanted to make our equation look like α = -(ω^2)ø. Because ø is very small, the sinø in our equation becomes ø. We can calculate the period by using the equation T = 2π/ω. Our theoretical value we calculated came out to T = 0.71767 s.

calculations for period of solid ring

Next we pulled the ring to the side and let go and began recording its period using logger pro and the photo-gate.

Graph of period for solid ring

Our experimental period came out to T = 0.718100 s. Comparing our theoretical value of T = 0.71767 s and our experimental value T = 0.718100 s, we concluded we had a percent error of 0.060 %, proving that our modeled equation α = -(ω^2)ø was very accurate.

Part 2: Semicircular plate with pivot about the midpoint of its base.

Semicircular plate on ring stand with photo-gate

The set up for part 2 remained the same as in part 1, except this time we wanted to measure the period a semicircular plate about the midpoint of its base. The concepts we used to calculate the object's period are also the same. We recorded the semicircle's radius R and had to solve for the object's center of mass and its moment of inertia with the pivot at the midpoint of its base. Then we had to apply Newton's Second Law of Torque and made the equation look like α = -(ω^2)ø so that we could use ω to solve for the period of the plate using the equation T = 2π/ω.

So first we were to derive the center of mass of our semicircular plate. Below is the derivation. The x center of mass = 0 and the y center of mass = 4R/3π

calculations for center of mass of semicircular plate (midpoint of its base)

Next, we were to find the moment of inertia about the midpoint of its base. The calculations are below, which came out to I = 1/2MR^2.

calculations for moment of inertia of semicircular plate (midpoint of its base)

Then we used Newton's Second Law of Torque again, and made the equation look like α = -(ω^2)ø. We calculated the period to be T = 0.7215 s. Below is the calculations for our theoretical value of T.

calculations for the period of semicircular plate (midpoint of its base)

We ran the the experiment and took the average of several points on the period vs time graph. We found the actual value to be T =  0.7126 s.

graph of period for semicircular plate (midpoint of its base)

Finally, comparing our two theoretical value T= 0.7215 s and experimental value T = 0.7126 s and found a percent error of 1.25 %. Once again our model worked at predicting the period of a physical pendulum.

Part 3: Semicircular plate with pivot from its edge, exactly above the midpoint of its base

We continued to step 3 with the same set up as in parts 1 and 2, except this time the plate was pivoted on its edge, directly above the midpoint of its base. And again we applied the same concepts in order to solve for the object's period. We had determine the moment of inertia, apply Newton's Second Law of Torque, and manipulate the equation to make it look like α = -(ω^2)ø.

Since we previously solved for the moment of inertia about the midpoint of its base, we were able to use this value to calculate the moment of inertia about the new axis by using the parallel axis theorem. Below is our calculations for the new moment of inertia about the pivot, which came out to I = 0.65117 MR^2.

calculations for moment of inertia of semicircular plate (directly above midpoint of base)

Applying Newton's Second Law of Torque and making the equation look like α = -(ω^2)ø, we were able to calculate the theoretical period to be T = 0.70707 s.

calculations for period of semicircular plate (directly above midpoint of base)

Then we pulled back the plate and let go to begin recording the period of the object about its new axis. Again we took the average value from the period vs time graph. We found the experimental value of the period to be T =  0.6946 s.

graph of period for semicircular plate (directly above midpoint of base)

Our theoretical value of T = 0.70707 s and experimental value T = 0.6946 s gave us a percent error of 1.80 %.

Conclusion/Uncertainty: Through experimenting with three particular physical pendulums, we were able to calculate the period of each object using its measured dimensions and get results which were verified by our experimental values. We solved for the theoretical values by finding the center of mass, the moment of inertia, the parallel axis theorem, Newton's Second Law of Torque, and simple harmonic motion. By manipulating our equations to look like α = -(ω^2)ø, we were able to find ω and solve for T. We also assumed that because the angle of oscillation was small enough, we let sinø=ø. There were not too many uncertainties during the course of our experiments other that the fact that the pivots were not perfectly frictionless. Sometimes the oscillations were not perfect because we had to pull back to side perfectly for the objects to oscillate evenly.