Which of the following objects has the greatest moment of inertia?

[fatcat39]

Homework Statement

Which of the following objects has the greatest moment of inertia?

Homework Equations

Hollow cylinder = 0.5M(R$$^{2}_{i}$$+R$$^{2}_{0}$$)
Solid cylinder = 0.5MR$$^{2}$$
Solid sphere = 0.4MR$$^{2}$$
Hollow sphere = (2/3)MR$$^{2}$$

The Attempt at a Solution

I know that the solid object has a lesser moment of inertia than the hollow one, but I'm not sure which is greater, the hollow cylinder or the hollow sphere. I think it's the hollow sphere, but can someone verify that?

Thanks in advance.

 

[fatcat39]

I read somewhere that the hollow cylinder's moment of inertia equation is actually 0.5MR^2, like what you find normally. Is this true? If so, then the hollow cylinder has a greater moment of inertia, right?

 

[Astronuc]

This is a useful reference.

http://hyperphysics.phy-astr.gsu.edu/hbase/mi.html

Keep in mind that the mass M will be related to the density of the material and its dimesions/geometry.

Having a hollow cylinder of mass M would mean the radii are greater than a solid cylinder of the same mass, assuming equal length. Mass has to occupy volume.

http://hyperphysics.phy-astr.gsu.edu/hbase/icyl.html#icyl2

http://hyperphysics.phy-astr.gsu.edu/hbase/icyl.html#icyl3

 

[Doc Al]

I read somewhere that the hollow cylinder's moment of inertia equation is actually 0.5MR^2, like what you find normally. Is this true?

No. 0.5MR^2 is for a solid cylinder. If the hollow cylinder is a thin cylindrical shell, then its moment of inertia is MR^2. (Use your equation for the thick hollow cylinder to see this.)

If so, then the hollow cylinder has a greater moment of inertia, right?

I presume all objects have uniform density and the same dimension R and mass M. If the cylinder is thin shelled enough, then its moment of inertia will be greatest. This should make sense, since it will have all of its mass as far from the axis as possible (for a given radius).