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22-May-2017: LAB 18: A LAB PROBLEM--MOMENT OF INERTIA and FRICTIONAL TORQUE

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Lab18
Purpose:
Measure the total moment of inertia and the acceleration a cart, which is tied to a string connected to a disk, will experience. Derive the time of acceleration with the parameters above.

Introduction:
For a system, the total moment of inertia is the sum of every parts’ moment of inertia. So when we need to calculate the system moment of inertia, we just need to divide it into small parts. To get the acceleration of the cart, we can use the equation torque=Iα to calculate torque, then finally get the acceleration of the cart.

Setup:
Set equipment up like the plot below.


Procedure:
Measure the radius and height of each cylinder, which are the components of the whole system. It is hard to get the weight of each cylinder because they are welded together. Use the volume to calculate the percent of each cylinder. With the total weight, we can derive the weight of each part finally. The measurement and calculation is followed below.
r (mm)
h (mm)
Volume ()
Percent
Weight (g)
15.4
52.3
38966.6
7.23%
333.65
100.23
14.6
460784.8
85.54%
3947.67
15.4
52.3
38966.6
7.23%
333.65


So we get the system moment of inertia. Then use Logger Pro to analyze the rotation of the apparatus in the video. We will get points like below in this step.

From this figure, we can get the angular velocity from the angle verse time. Then find the angular acceleration from the angular velocity verse time. Here is the angular velocity verse time plot.

With the fitting result, we can derive the angular acceleration is the sloop of the fitting result, which is -0.1903 rad/s^2. The traveling angle of the wheel is about 40 degrees. Then use the net force equation and net torque equation to solve the acceleration.

The acceleration is 0.040227 m/s^2. The time is 7.051 s. To make sure the time is credible, we ran 5 trials. But among those trials, the average value is bigger than the time above. It claims that error occurs. The specific reason will be covered in the next part.

Conclusion:

We have some small error in this lab. One of the possible reasons is that the fitting error. The curve is oscillating during the time. So the linear fitting error might be large in this situation. Besides, the string is possibly strung on the cylinder, which is also a possible reason. But the method used in this part is right because the error is not too large to make the result incredible. From this lab, I learned the way to calculate the time of a specific cart’s acceleration and the way to get a whole system’s the moment of inertia. 

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