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

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

Thanks in advance.

Tide

Use the definition of moment of inertia:

[tex]I = \int r^2 dm[/tex]

In the case of the shell the element of mass is [itex]dm = M {dA} /{4 \pi R^2}[/itex] where [itex]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[/itex] so

[tex]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[/tex]

from which the desired result follows.

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

copperboy

Could you please explaim why [itex]dA = R^2 \sin \theta d\theta d\phi[/tex] in detail?

arildno

Consider an area segment dA on a sphere with radius R.
We approximate this with a rectangle:
a)Two of the sides are arclengths along great circles; the length of each of these is [tex]Rd\theta[/tex]
b) The other two are arclengths in THE HORIZONTAL PLANE; the local radius there is [tex]R\sin\theta[/tex]
Hence, the arclenth is [tex]R\sin\theta{d\phi}[/tex]
c) Multiplying together, we get:
[tex]dA=R^{2}\sin\theta{d\theta}d\phi[/tex]

Tide

dA is a differential element of area on a spherical surface using spherical coordinates and it represents, to lowest order in differentials, the area of a rectangle [itex]R \sin \theta d\phi[/itex] high and [itex]R d\theta[/itex] units wide.

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.

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?