Question

State and prove parallel axis theorem.

Answer 1

Parallel axis theorem:

Parallel axis theorem states that the moment of inertia of a body about any axis is equal to the sum of its moment of inertia about a parallel axis through its center of mass and the product of the mass of the body and the square of the perpendicular distance between the two axes.

If I_C is the moment of inertia of the body of mass M about an axis passing through the center of mass, then the moment of inertia I about a parallel axis at a distance d from it is – given by the relation,
I = I_C + M d²
Let us consider a rigid body as shown in the figure. Its moment of inertia about an axis AB passing through the center of mass is I_C. DE is another axis parallel to AB at a perpendicular distance d from AB. The moment of inertia of the body about DE is I. We attempt to get an expression for I in terms of I_C. For this, let us consider a point mass m on the body at position x from its center of mass.

Parallel axis theorem

The moment of inertia of the point mass about the axis DE is, m (x + d)². The moment of inertia I of the whole body about DE is the summation of the above expression.

I = ∑ m (x + d)²

This equation could further be written as,

I = ∑ m(x² + d² + 2xd)

I = ∑ (mx² + md² + 2 dmx)

l = ∑ mx² + md² + 2d ∑ mx

Here, ∑ mx² is the moment of inertia of the body about the center of mass. Hence,I_C = ∑ mx²

The term, ∑ mx = 0 because x can take positive and negative values with respect to the axis AB. The summation (∑ mx) will be zero.

Thus, I = I_C + ∑ m d² = I_C + (∑m) d²

Here, ∑ m is the entire mass M of the object (∑ m = M).

I = I_C + Md²