Help Deriving formula for moment of inertia lab

[HT3]

this lab is very similar to mine
basically i can only seem to derive a portion of the formula, I=mgb/al - mb^2
where b- radius, g-gravity, al- angular acceleration.
I do not get the right answer:
here is what i did do however,
Fnet=ma
T-mg=-ma (acceleration is downwards)
so T=(mg-ma)
so then

Tnet=I*al (where al is angular acceleration not acceleration)

so then
T*b=I*al
(b*mg-b*ma)/al = I

as u can see this is not the correct solution..but is a start
the solution i need to get to is
I=mgb/al - mb^2
can someone help me out - thanks :)

 

[HT3]

NEVERMIND I SOLVED IT i was on the right way i just realized that a=al*R and if i plug that in my answer is correct.
thanks to any1 who viewed it anyways

 

[Moetasim]

Hello,

I understand your frustration with not being able to fully derive the formula for moment of inertia. It can be a complex concept to understand and derive, especially in a lab setting. However, I would suggest going back to the basic definition of moment of inertia, which is the resistance of an object to changes in its rotational motion.

In this case, the moment of inertia is affected by the mass and the distribution of that mass around the axis of rotation. In your derivation, you have correctly identified the force equation (Fnet=ma) and the torque equation (T=I*al). However, you have not taken into account the distribution of mass in the object.

To fully derive the formula, you will need to use the parallel axis theorem, which states that the moment of inertia about a given axis is equal to the moment of inertia about a parallel axis through the center of mass plus the product of the mass and the square of the distance between the two axes.

In your case, the parallel axis theorem would be applied as follows:

I = Icm + mb^2

Where I is the moment of inertia, Icm is the moment of inertia about the center of mass, m is the mass of the object, and b is the distance between the axis of rotation and the center of mass.

By using this equation and substituting it into your torque equation, you should be able to derive the correct formula for moment of inertia:

T = I*al
T = (Icm + mb^2)*al
T = Icm*al + mb^2*al
T = mgb + mb^2*al

Then, rearranging for the moment of inertia:

I = mgb/al - mb^2

I hope this helps to clarify the derivation process for you. If you still have trouble, I suggest seeking assistance from your lab instructor or a fellow classmate. Keep in mind that understanding and deriving formulas is an important part of the scientific process, so don't get discouraged. Keep practicing and seeking help when needed. Good luck with your lab!