Question

Two uniform solid spheres have the same mass, but one has twice the radius of the other. The ratio of the larger sphere's moment of inertia about a central axis to that of the smaller sphere is

Answer 1

Answer:

The ratio is 4

Explanation:

The moment of inertia of a solid sphere about a central axis is:

$I=\frac{2mr^{2}}{5}$

With m the mass and r the radius of the sphere.

For the smaller sphere with mass M and radius R the moment of inertia is:

$I_{small} =\frac{2MR^{2}}{5}$ (1)

For the bigger sphere with mass M and radius 2R the moment of inertia is:

$I_{big}=\frac{2M(2R)^{2}}{5}$

$I_{big}=\frac{2^{2}*2M(R)^{2}}{5}$ (2)

The ratio between larger of the larger sphere's moment of inertia about a central axis to that of the smaller sphere is the ratio between (2) and (1):

$\frac{I_{big}}{I_{small}}=\frac{\frac{4*2M(R)^{2}}{5}}{\frac{2M(R)^{2}}{5}}$

The term $\frac{2M(R)^{2}}{5}$ cancels, so:

$\frac{I_{big}}{I_{small}}=4$