Question

What is Icm, the moment of inertia of the disk around its center of mass? You should know this formula well.

Answer 1

Answer:

$I_{cm}=\frac{MR^{2}}{2}$

Explanation:

First at all let's understand what is moment of inertia (I). The moment of inertia of a body is the rotational analog of mass in linear motion, this is, it determines the force we should apply to the body to acquire a specific angular acceleration. But in the rotational case we should specify about what point we are going to rotate an object so always the moment of inertia is defined respect to an arbitrary axis. It's usual to use the center of mass as an axis of rotation, because it's an unique point where we can assume all the mass of the object is concentrated.The moment of inertia respect of an axis that passes through the center of mass is denoted $I_{cm}$.

Now, if the disk you're talking about has uniform density the center of mass is exactly at the geometrical center of the disk, and the moment of inertia of a disk as that is:

$I_{cm}=\frac{MR^{2}}{2}$

Answer 2

$I_{cm} =mr^{2}$ is the moment of inertia of rigid body along its centre of mass.

The moment of inertia of a body is the rotational analog of mass in linear motion. If we consider a rigid body of mass m, assuming its mass concentrated at its center of mass (cm) which is at a distance of r from the axis of rotation of the body, then the moment of inertia of the rotating body is given by:  

                  $I_{cm} =mr^{2}$

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