Inertia of a hollow cylinder and hollow sphere

[GayYoda]

Homework Statement

Consider a hollow cylinder of mass M with an outer radius R_out = 10 cm and an unknown inner radius R_in. If the hollow cylinder is to roll down an incline in the same time as a spherical shell of the same mass and the same outer radius, calculate R_in.

Homework Equations

I_cyl = MR^2/2 .
I_shell = 3/5(MR^2)

The Attempt at a Solution

I think the inertia of each are equal but I'm not sure

 

[PeroK]

You've given the moment of inertia of a solid cylinder. What you need is the inertia of a hollow cylinder of finite thickness.

 

[GayYoda]

I=M/2[(R_out)^2-(R_in)^2]?

 

[PeroK]

I=M/2[(R_out)^2-(R_in)^2]?

What happens in your formula if $R_{in}$ is close to $R_{out}$?

To calculate this inertia is not that easy. I suggest you consider the density of the cylinder.

 

[GayYoda]

i got it now its I=M[(R_out)^2+(R_in)^2]/2 as the density becomes M/[pi*h*[(R_out)^2-(R_in)^2] and when you sub it back into the intergral it becomes [(R_out)^4-(R_in)^4]/[(R_out)^2-(R_in)^2] = [(R_out)^2+(R_in)^2] because of difference of 2 squares