Moment of Inertia of a Triangle

Purpose: The purpose of the lab was to use the angular acceleration of a rotating disk system in order to find the moment of inertia of a uniform triangle on top of the system.

Procedure: In order to check our answer with the final one we get, we decided to start by symbolically calculating the moment of inertia of the uniform triangle. For the lab we had to calculate both when the triangle was up and when it was on its side, both around a center of where there was a hole in the middle of the triangle. We then would use a spinning uniform disk system with a hanging mass at one end in order to find the moment of inertia of the triangle by using the torque of the system. 

A representation of how the system is supposed to look like is shown below.

Data: The first graph we got was by spinning the triangle when it was on its side.

The second graph was when the triangle was standing up. Both gave us a different angular acceleration.

Calculations; Below is how we got the moment of inertia of the triangle by using the parallel axis theorem.

After getting the angular acceleration by using the spinning disk system and Logger Pro we used torque to solve for the moment of inertia of the uniform triangle.

Summary: The overall lab was a success because the calculated moment of inertia was close to the experimental moment of inertia. The error could have been because there might have been some friction between the disks and some air resistance.

Torque and Moments of Inertia (Disk)

Purpose: The purpose of the lab was to calculate the time at which a cart will reach the bottom of an inclined ramp by using the moment of inertia of the apparatus, which in this case was a solid uniform disk with a solid uniform cylinder in the center of the disk.

Procedure: Below is a picture of the disk and cylinder system. The moment of inertia was found by measuring the different parts of the systems like the radius of both the width as well as the diameter and mass.

Below is what the final set up would look like. By finding the moment of inertia of the system, a time can be found for when the cart reaches the bottom.

In order to get the time, the angular deceleration was needed so that the frictional torque of the system could be found. This was done by using Logger Pro with video capture and using a graph to find the angular deceleration of the system. Then we were given a problem where we had to find the time it took for a cart attached to the rotating system to reach the ground at a certain angle. The values were given to us and we solved it like regular problem. It served as a way to help us find the time of the system we had to physically set up.

Data: The data table below shows the information we got from Logger Pro when using video capture in order to find the deceleration of the system.

The graph below shows the angular velocity in the x and y direction which was used to acquire the angular deceleration of the system.

Using the equation Vt = sqrt(Vx^2 + Vy^2), the tangential velocity can be found and the graph below shows how the slope gives that tangential velocity.

Calculations: The calculations below are for how we solved the moment of inertia of the disk and cylinder system. The result was 1.92 x 10^-2 kg*m^2.

The calculations below show how the frictional torque, angular acceleration and the time it took the cart to drop 1 meter were acquired.

Summary: The lab was not a success because the time we calculated was off from what physically got. The ramp's angle in the calculated portion was 46 degrees and the time it took was 9.5 seconds. On the other hand, we physically set the ramp up to try and match that angle but we got about 12.5 seconds instead which is about 24% of error. We believe that the disk-cylinder system was tampered with or it was getting stuck somehow because after doing the trial multiple times the time kept increasing. Below is what the set up looked like at the end.

Angular Acceleration

Purpose: The purpose of this experiment was to see what factors affect the angular acceleration of a specific object which in this case was a solid disk.

Procedure: This picture shows the whole set up of how the lab looked. We had a platform with two spinning disks in which they were allowed to rotate without friction using the hose, with an air source, which was found within the platform. The air flowed in between the disks so that we could allow them not to touch which was meant to simulate no friction. The disks also had a hole in the middle so that air could be let out but since that was not necessary for the experiment, a pin to close up the hole was used. The whole system also had a string attached to the pin and at the end was a hanging mass which was used to help measure the angular acceleration of the disks.

The data was acquired using Logger Pro and the experiment was run multiple times and each of them with specific conditions to show the relationship between angular velocity and time and therefore acquire an angular acceleration. These conditions included changing the weight of the hanging mass, changing the weight of the disks and torque pulley.

Data: The graph below shows an example of one the test trials we did. The data table on the right gives us a set of angular velocity and time.

The graph below shows the actual first trial we did that went into our data.

This graph shows the result of the angular acceleration after doubling the hanging mass.

The final graph we documented was of when we tripled the hanging mass. As seen, the angular acceleration decreases the more the mass is increased.

The chart below shows were all the data was acquired from the graphs. We decided that no more graphs were necessary because all of them have the same picture but with different slopes because the angular acceleration is always different depending on the conditions. 

Calculations: There were no calculations because this lab was to see the relationship between angular velocity and time in order to get the angular acceleration of the whole system.

Summary: Overall, the lab was successful because the trends that are seen make sense. If the hanging mass is doubled then the angular acceleration is doubled and if the hanging mass is tripled then the angular acceleration is also tripled. To add on, if the radius is increased then the angular acceleration increases which makes sense since angular acc = radius x translational acceleration. One more trend was if the weight of the spinning disk is decreased then the angular acceleration is increased.