Can mass of a body be taken concentrated at its centre of mass for the purpose of calculating its rotational inertia ?

Answer

To determine whether the mass of a body can be taken as concentrated at its center of mass for the purpose of calculating its rotational inertia, we can analyze the concept of rotational inertia (or moment of inertia) in detail.

### Step-by-Step Solution:

1. **Understanding Moment of Inertia**:
The moment of inertia (I) of a body is a measure of how difficult it is to change its rotational motion about an axis. It depends on the distribution of mass relative to the axis of rotation.

2. **Definition of Center of Mass**:
The center of mass of a body is the point at which the mass of the body can be considered to be concentrated for translational motion. However, for rotational motion, the distribution of mass around the axis of rotation is crucial.

3. **Example of Different Shapes**:
Consider two different shapes: a hollow sphere and a solid sphere. Both can have the same mass (m) and radius (r), and their centers of mass will coincide when they are positioned similarly.

4. **Calculating Moment of Inertia**:
- The moment of inertia of a hollow sphere about its center is given by:
\[ I_{hollow} = \frac{2}{3} m r^2 \]
- The moment of inertia of a solid sphere about its center is given by:
\[ I_{solid} = \frac{2}{5} m r^2 \]

5. **Comparison of Moments of Inertia**:
Although both spheres have the same mass and radius, their moments of inertia are different:
- \(I_{hollow} = \frac{2}{3} m r^2\)
- \(I_{solid} = \frac{2}{5} m r^2\)

6. **Conclusion**:
Since the moments of inertia are different despite having the same mass and center of mass, we conclude that the mass cannot simply be concentrated at the center of mass for the purpose of calculating rotational inertia. The distribution of mass relative to the axis of rotation is what determines the moment of inertia.

7. **Final Statement**:
Therefore, the statement that the mass of a body can be taken as concentrated at its center of mass for the purpose of calculating its rotational inertia is **false**.