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 M and a radius of 2R.
A hoop spinning about its center. Total mass is M and the radius is 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.
A solid disk spinning about its center. Total mass is M and a radius of R.

Answer 1

a) 2 point masses of mass m each at distance r from the axis I = 2mr^2 ~ 4MR^2 This is only an approximation; you have not told us the shapes of the objects or anything at all about the mass or shape of the rod. Without this info we cannot calculate I for the system.

b) Uniform disc spinning about an axis normal to the plane of the disc and passing through the centre. I = mr^2/4 = MR^2. Note that I have made an assumption about the axis of rotation and the mass distribution; this might not be valid - the axis might be in the plane of the disc or at any angle to it.

c) Solid uniform sphere rotating about a diametrical axis I = 2mr^2/5 = (MR^2)/5. I have made an assumption about the mass distribution

d) Uniform disc spinning about an axis normal to the plane of the disc and passing through the centre. I = (mr^2)/4 = (MR^2)/4. Same qualification as in (b)

The ranking in increasing order of I and therefore decreasing order of accn is :-

1) 2 point masses: I = 4MR^2

2) solid uniform disc of radius: 2R I = MR^2

3) solid uniform disc of radius R: I= (MR^2)/4

4) Solid uniform sphere: I = (MR^2)/5