22-May-2017: Finding the moment of inertia of a uniform triangle about the center of mass

In the past, we have already learned how to calculate the moment of inertia of a triangle by the parallel theorem.

Then the inertia will be

Because the horizontal center is in 1/3 of the base from the left side, 

Today, we tried the formula in the real triangle to see whether it works.

Purpose

To determine the moment of inertia of a right triangular thin plate around its center of mass, for two perpendicular orientations of the triangle.

Plan

Set up the apparatus as lab on May 8th, which is

Because it can reduce the friction as much as possible by the flowing air.  

Rotate it first to get the inertia of the part without the triangle. 

Then use the same step as that lab to calculate the total inertia (which calculates the angular acceleration first, then use functions to calculate it; again, to reduce the error we use the average of the angular acceleration). 

And then put the small part component on the top to hold a standing triangle. 

Lie down the triangle and repeat the step above.

Compare the calculated value with the real value to see whether it fits. 

Set up & Data

Set up the apparatus as following, connect them to the laptop, and set up everything in Logger Pro. 

Rotate it without the triangle to get the inertia of the rest part. And we got

Weight the small mass and the triangle. We got the data. 

Rotate the triangle by release the small mass. And we got the data. 

After lie down the triangle, rotate it and collect the data again. 

Analyze

By the formula above we can first calculate the inertia of triangle.

Then use the other formula to calculate the actual inertia, then subtract them to get the triangle's inertia.

Compare them

Both of the difference are within 5%, so those are acceptable data, which implies that the parallel theorem works in these situations. 

Conclusion

In general, this lab confirmed the parallel theorem works in different situations by getting almost the same (since there are only 5% off). However, there still may be some problems on measures, which will give us some uncertainty. Moreover, we did not think about the deceleration on the apparatus. Even though this apparatus is very frictionless, we can find out every time the hanging mass goes up and down, the acceleration decreases a little bit, which implies that there is some frictional torque in it. It might slightly influence the result. Moreover, we only did the experiment once. There might be some unexpected factors (like someone accidentally hits the table).

To get a more precise result, we could consider about friction/resistance and repeat the experiment a few more times.