# How to calculate the moment of inertia of the rigid body?

1. Oct 28, 2004

### copperboy

I don't know how to calculate the following rigid bodies with different geometries, can anybody help me?

Thin spherical shell: I=(2/3)MR^2

Solid sphere: I=(2/5)MR^2

2. Oct 29, 2004

### Tide

Use the definition of moment of inertia:

$$I = \int r^2 dm$$

In the case of the shell the element of mass is $dm = M {dA} /{4 \pi R^2}$ where $dA = R^2 \sin \theta d\theta d\phi[/tex]. The distance to a point on the shell from the z-axis is [itex]R^2 \sin^2 \theta$ so

$$I = \frac {M}{4 \pi R^2} \int_{0}^{2 \pi} d\phi \int_{-\pi /2}^{\pi /2}R^4 \sin^3 \theta d\theta$$

from which the desired result follows.

In the case of the solid sphere you will work with a volume integral.

3. Oct 30, 2004

6. Nov 2, 2004

### ruffneck2

This is the first time i'm involved in physics studies.I'm having a lots of problems using these formulas and the text book i'm using does not make it easy. one from "James S. Walker"
Can someone please explain how and when to use these formulas.
Thank You.

7. Nov 2, 2004

### siddharth

This is one way to look at it

You must be familiar with the equation F=ma.

We use the Moment of Inertia to find the Torque.
Like how the force is directly propotional to acceleration, Torque is directly propotional to angular acceleration, the propotionality constant being the moment of inertia.

Also remember about what axis you are taking the moment of Inertia while solving problems. In most problems i have encounterd which ask you to find the angular acceleration, first find the torqure taking the vector product R X F. Next find the moment of inertia of the object about the required axis. Then the acceleration can be found

Also, for a solid sphere, perphaps deriving the moment of inertia by intergrating thin rings would be easier than a volume intergral?