# Find Minimum Moment of Inertia

This is a simple problem that aims to build a foundational understanding of JEE problems. JEE problems are known for involving more than one concept from same or different chapters. For solving this problem too, you should have an understanding of two fundamental concepts:

The moment of inertia of any object about an axis through its center of mass is the minimum moment of inertia for an axis in that direction in space.

The parallel axis theorem: It states that the moment of inertia about any axis parallel to an axis through the center of mass is given by

I(parallel axis) = I(CM) + Md² – (1)

Where I(parallel axis) is the moment of inertia of the body about an axis parallel to the axis through the center of mass

I(CM) is the moment of inertia of the body about an axis through the center of mass, M is the total mass of the body and d is the distance between the two parallel axes.

Putting it simply, the question wants you to calculate the moment of inertia of the given sphere about an axis through the center of mass of the sphere (which is the minimum moment of inertia of sphere). Since it is a uniform rigid sphere, we are talking about its central axis.

We also know that the CM of the sphere about its central axis is ? MR².

Using equation (1), we can write: ? MR² + Md² = I(CM)

putting values of I(CM), R and d in the equation we get:

(0.4+0.25) M = 6.5 or M = 10

Therefore, the minimum moment of inertia i.e the moment of inertia about the central axis = ? MR² = 4 kg m2

#### Practise This Question

Which of the following helps life sustane in water?