Why Can't I Use Torque = Ia for Calculating Pulley's Inertia?

[kp]

once again I have succeeded in confusing myself about basic angular motion...

If I have a pulley of with an unkown moment of inertia, and a bucket hanging from the pulley with a known mass, height, and acceleration in which it falls. Why can't i use:

Torque = Ia
rf = Ia
r * mg = Ia
r*mg/a = I

or can I, because when i use conservation of energy

mgh = 1/2mv^2 + 1/2Iw^2 and solve for I, i get different answers.

 

[Doc Al]

Torque = Ia
rf = Ia
r * mg = Ia
r*mg/a = I

This assumes that the force exerted on the pulley equals the weight of the hanging mass. Not true. The force does equal the tension in the cord. (If the force exerted on the pulley equalled the weight of the hanging mass, then by Newton's 3rd law the mass would be in equilibrium.)

 

[quicknote]

It is completely understandable to feel confused about basic angular motion. Inertia moments, also known as moments of inertia, refer to an object's resistance to rotational motion. This means that the moment of inertia of an object depends on its mass distribution and the axis of rotation.

In the example you have provided, you are trying to calculate the moment of inertia of a pulley using the equation Torque = Ia. While this equation is correct, it only applies in certain situations where the axis of rotation is fixed and the object is rotating about that axis. In your case, the pulley is not rotating about a fixed axis, but rather the bucket is falling due to gravity. Therefore, this equation cannot be used to calculate the moment of inertia of the pulley.

When using conservation of energy to solve for the moment of inertia, it is important to consider all forms of energy involved. In your equation, you have only accounted for the potential and kinetic energy of the bucket, but you have not included the rotational energy of the pulley. This is why you are getting different answers when solving for I.

To accurately calculate the moment of inertia of the pulley, you will need to consider the rotational energy of the pulley as well. This can be done by using the equation you mentioned, mgh = 1/2mv^2 + 1/2Iw^2, where w is the angular velocity of the pulley. By including this additional term, you should be able to obtain a more accurate value for the moment of inertia of the pulley.

I hope this helps clear up your confusion. Keep in mind that angular motion can be complex, and it is always important to consider all factors and forms of energy involved when solving for values such as moment of inertia.