Moment of Inertia of a Hemisphere

[hp-p00nst3r]

Homework Statement

The moment of inertia of a uniform sphere of mass m and radius r about an axis passing through its center of mass is (2/5) mr^2. If half of the sphere is removed, what is the moment of inertia about the same axis?

Homework Equations

integral of r^2 dm

The Attempt at a Solution

http://img407.imageshack.us/img407/5250/mech221ps4moiim8.jpg
I think this is wrong since I've looked up the actual I for this and it gives me (2/5) mr^2 still. I also tried using the parallel axis theorem on this and it also gives me the (2/5) mr^2. What am I doing wrong?

 

[alphysicist]

Hi hp-p00nst3r,

Homework Statement

The moment of inertia of a uniform sphere of mass m and radius r about an axis passing through its center of mass is (2/5) mr^2. If half of the sphere is removed, what is the moment of inertia about the same axis?

Homework Equations

integral of r^2 dm

The Attempt at a Solution

http://img407.imageshack.us/img407/5250/mech221ps4moiim8.jpg
I think this is wrong since I've looked up the actual I for this and it gives me (2/5) mr^2 still. I also tried using the parallel axis theorem on this and it also gives me the (2/5) mr^2. What am I doing wrong?

About the third thing you have written down is this:

<br /> dI = \frac{1}{2} z^2 dm<br />

This should represent the moment of inertia of a representative disk, like the one you have in the diagram. However, the moment of inertia of a disk is (1/2) M R², where R is the radius of the disk.

From the figure, you can see that the radius of the disk is not z, so your expression for dI is not correct. The radius is actually y, so once you have used the expression for y in dI, I think you should get the right answer.