Rotational Mechanics - 2

Today let us deal with the calculation of moment of inertia.

In Rotational Mechanics - 1, we dealt with the question what is moment of inertia. Now lets deal with calculation of moment of Inertia (I).

I = ∫dm r²  .......   1

Ring (Mass M and radius R) (about an axis passing through centre of ring perpendicular to the plane of the ring):

For a ring of mass M and radius R, consider a small point object dm. All the point object dm are at a distance of r from the centre of axis. So

I = R² ∫ dm

I = M R²

Disc: (Mass M and radius R) (about an axis passing through centre of disc perpendicular to the plane of the disc):

Consider a small ring element of mass dm at a distance r from the centre. Then calculation of moment of inertia is as follows:

I = ∫dm r²

Calculation of dm is as follow:

dm = (M / πR²)(2πr dr)

Substitute dm in equation 1

I = ∫ 2 M r³ dr / R²

Click me

I =  ∫ 2 M R²/4

Note the limits of integration is from 0 to R in both cases