A series of object are being rotated. The same net torque is being applied to ea

Question

A series of object are being rotated. The same net torque is being applied to each object. Rank the object in order of the resulting angular acceleration (1=lowest).
A solid disk spinning about its center. Total mass is 0.5M and the radius is 2R.
A hoop spinning about its center. Total mass is 2M and a radius of R.
A solid disk spinning about its center. Total mass is M and a radius of R.
An object consisting of two objects connected by a massless rod. Each object has a mass of 2M and is a distance R from the axis of rotation.

Explanation / Answer

We know that the torque is given by the relation

? = I?

For a given torque, the highest Moment of Inertia will result in the lowest acceleration and vice-versa.

1. For a Solid disk,

I = ½MR² =  ½(0.5*M)(2R)² = MR2
2. For a Hoop,

I = MR² = (2*M)R2 = 2MR2

3. For a solid disk,

I = (1/2)MR² = 0.5*M*R²
4. For an object consisting of two objects

I = 2(M)R² =2*2MR2 = 4MR²

thus the order with lowest accleration first is

4, 2 , 1 ,3

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