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.

20-May-2015: Conservation of Energy/Conservation of Angular Momentum

Purpose: Using our understanding of conservation of energy and angular momentum, we were to predict how high a pivoted stick/clay combination will rise after the stick inelastically collides with a stationary blob of clay.

Procedure: In order to set up our apparatus, we first grabbed a tall vertical rod and attached a smaller horizontal rod at the top so that we could hang a meter stick just above the table. Next, we pivot the meter stick near but not exactly at its end and wrap tape (inside out) around the bottom so it could swing and catch the stationary blob of clay at the bottom of the vertical. The clay blob was also wrapped with tap to ensure that they stuck when colliding.

lab  apparatus with video capture

Our goal was to calculate how high the stick/clay combination would rise after being released from a horizontal position by using our understanding of conservation of energy and conservation of angular momentum. Then, by setting up Logger Pro's video capture software along with a camera to record how high the combination rose in our experiment, we were to compare our theoretical result with our actual experimental result.

In order to derive our theoretical result, we had to first measure the the mass of the meter stick, the mass of the clay blob, and the length of the meter stick from the pivot to the bottom end. The idea is to break the problem down into three parts, from conservation of energy before the collision, to conservation of angular momentum during the collision, and back to conservation of energy to find the maximum height of the system.

Our measured values are below
Mass of stick - 142 g
Mass of clay - 32.0 g
Length of stick from pivot - 98.5 m

We know that the moment of inertia pivoted at the center is 1/12 ML^2, but the pivot is not exactly at the end of the stick. We had to apply the parallel axis theorem to derive our new moment of inertia for our calculations.

Part 1: Conservation of Energy

At the horizontal position, the meter stick has a potential energy and that energy transfers completely into kinetic energy at the bottom of the vertical. We want to use conservation of energy to solve for the angular velocity of the stick just before it collides with the clay blob. We set our gravitational potential energy to 0 at the top of the pivot so that it has negative potential energy at the vertical position.

Part 2: Conservation of Angular Momentum

Now that we have the angular velocity of the stick at the bottom of the rotation just before the collision, we can use conservation of angular moment to solve for the angular velocity of the stick/clay system after the collision. We also had to solve for the new moment of inertia of the system using the parallel axis theorem and adding the moment of inertia of the clay blob using its distance from the pivot.

Part 3: Conservation of Energy

Using the angular velocity of the system, we can solve for the maximum angle the system makes using conservation of energy once again. Then we can use that angle to solve for the maximum height of the system.

Below is our calculations for each part.

calculations for maximum height of the stick/clay system

Our theoretical max height of the system came out to 0.40 meters.

Because we have our theoretical value of the maximum height, we can now use our video capture to record the actual maximum height that the system reaches after swinging down from its horizontal position. We set the stick in the horizontal position, begin recording, and release the stick into motion.

video capture of maximum height of system

Above is the image of our maximum height of the system. We set a ruler at the bottom and marked a distance of .3 meters so that the video capture could scale the height of the stick and we set the origin of the axis at the point of collision. By plotting the point on the image, we were able to get the x and y position on a graph shown below.

position of maximum height of stick/clay

Results: Our experimental value came out to h = 0.3676 m. compared to our theoretical value of h = 0.4000 m, we have a percent error of 8.1 %.

Conclusion/Uncertainty: By using the conservation of energy and angular momentum theories, we were able to predict the maximum height of a swinging stick/clay system. We broke the problem down into a conservation of energy problem before the collision, a conservation of angular momentum problem during the collision, and again a conservation of energy problem after the collision. Then we used the video capture to record the actual maximum height of the system to find that our theoretical value to be 8.1% off of the experimental value. Although our percent error was slightly higher than we would have liked, we can say that our results were reasonably accurate. We can attribute some of the error to the fact that the pivot was not exactly frictionless and also the video capture method does not give us the exact value for the maximum height of the system because we can only stop the video at certain intervals and it is hard to perfectly pinpoint the center of the system at the height we decided was the max. Finally there is some propagated error in our measurements of the masses and lengths of stick and center of mass.