Moment of inertia for a hollow ball calculation

[AaronKnight]

Homework Statement

A uniform solid sphere of mass m and radius a has moment of inertia 2/5ma² about any
diameter. Material is removed from the sphere to make a concentric spherical cavity of
radius a/2. What is the mass of the resulting hollow ball ? Show that its moment of inertia
about a diameter is
(31/80)ma²
What does it mean that moments of inertia are additive and why are they so ?

Homework Equations

I have calculated the mass of the hollow ball to be (7/6)πρa³

The Attempt at a Solution

I've been attempting to solve this question by calculating the moment of inertia of the material removed from the the sphere and subtracting that from the moment of inertia given for a solid sphere, but this approach doesn't give me the correct coefficient of 31/80, I've checked my maths many times. I am just wondering if this is the correct way of doing this problem?

 

[fluidistic]

I wish I had time to actually help you in details for your problem... however may I point that you should give the mass of the resulting hollow ball in terms of m only, no rho the density.
Huge hint: The ratio of the volume of the sphere and hollow ball is equal to the ration of their mass.

 

[AaronKnight]

Thank you, I think I have managed to solve it now.

 

[SammyS]

Homework Statement

A uniform solid sphere of mass m and radius a has moment of inertia 2/5ma² about any
diameter. Material is removed from the sphere to make a concentric spherical cavity of
radius a/2. What is the mass of the resulting hollow ball ? Show that its moment of inertia
about a diameter is
(31/80)ma²
What does it mean that moments of inertia are additive and why are they so ?

Homework Equations

I have calculated the mass of the hollow ball to be (7/6)πρa³

The Attempt at a Solution

I've been attempting to solve this question by calculating the moment of inertia of the material removed from the the sphere and subtracting that from the moment of inertia given for a solid sphere, but this approach doesn't give me the correct coefficient of 31/80, I've checked my maths many times. I am just wondering if this is the correct way of doing this problem?

Notice that the result for the moment of inertia, (31/80)ma², is in terms of the mass, m, of the solid sphere, before it is hollowed out.