3) The moment of inertia is a quantity of measurement that express a body's tendency to resist in angular motion. In order to find the moment of inertia, we have to find where the axis is being rotated. If the axis of rotation is located at the edge of a rigid body, the moment of inertia will be a different quantity comparing the axis of rotation being on the center of mass. In this lab, we will find the moment of inertia where the axis of rotation will be at the center of mass. In order to find the center of mass of a right triangle, we use the approach of the density formula using calculus. Here is a step by step illustration of finding the center of mass of a right triangle in the x and y-coordinates.
Now that we found the center of mass of the right triangle, our next approach is finding the moment of inertia at the center. In order to find the moment of inertia, we use calculus of integration of the derivative of the moment of inertia which is the integral of dm multiplied by x squared. But since the integration between the points of the center of mass will get ugly to determine our answer, there's a simpler way to calculate the moment of inertia around a vertical end of the triangle. We can use the parallel axis theorem. The parallel axis theorem is the center of mass plus the moment of inertia of the entire object treated as a point mass at the center of mass. Here is a demonstration to prove the parallel axis theorem.
The center of mass in the horizontal axis is at 1/3 of the base, our variable "h" will be 1/3 because we moved our axis of rotation at the edge of the triangle to find the moment of inertia at the center of mass. Here is a step by step process on finding the moment of inertia at the center of mass.
Now that we found our quantity for the moment of inertia at the center of mass of axis of rotation, we will begin our measurement of our apparatus and compare our theoretical and experimental results are the same.
4) We use the same apparatus for the Rotational Acceleration lab. We do the same method to find the angular acceleration to find the moment of inertia doing the experimental method.
5) Here is a list of of the angular acceleration we tested in logger pro.
6) Here is a graph of our angular acceleration when the hanging mass accelerated our device in rotation.
7) The graph represents the angular acceleration when the hanging mass oscillates vertically since the force of gravity is the only force in the torque system. The angular acceleration when the hanging mass goes down does not equal when the hanging mass goes up. Also be aware that even though our experiment is "friction-less", there's still frictional torque in the system. the disk isn't truly frictoinless and there is some mass in the friction less pulley.
8) Here is a comparison of our theoretical and experimental results. Before we could show our theoretical results, we needed to measure the dimensions of our apparatus. Now that we measured the dimensions of the rigid bodies for moment of inertia, this is our theoretical results for the moment of inertia of a right triangle.
Before we find the moment of inertia of the right triangle in our experiment, our assumption is that the disk that is holding the right triangle during rotational motion is affected. Our assumption of the moment of inertia is the disk plus the triangle object. Therefore, the moment of inertia of the disk plus the triangle is equal to the moment of inertia of the disk plus the moment of inertia of the triangle. So in this free body diagram, the equation we derived is the sum of the moment of inertia of the disk in experiment one and moment of inertia of the disk plus the triangle in experiment two.
Using the average angular acceleration between the angular acceleration going up and down, here is how we found the moment of inertia of the triangle base side and height side.
What is peculiar is that the moment of inertia of the triangle base side is 25% error comparing the theoretical and experimental results, and the moment of inertia of the triangle height side is 3% error comparing the theoretical and experimental results. There could've been a frictional torque disturbance during the logger pro reading or a human error. But looking at the comparison of the moment of inertia of the height side is really close to each other; therefore, our theory proves the moment of inertia does affect the angular acceleration when the object goes in rotational motion.
Here is an extra activity of finding the frictional torque in the experiment.
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