Question

State and prove the theorem of parallel axes of moment of inertia.

Answer 1

Theorem of parallel axes: According to this theorem, "moment of inertia of a plane lamina about any axis in its plane is equal to its moment of inertia about a parallel axis passing through the centre of mass of the lamina plus the product of the mass of the lamina and square of the distance between the two axes."

Proof : Let AB be the axes about which the moment of inertia of the body is to be calculated and PQ be the axis passing the centre of gravity (G) of the body and is parallel to AB. Let h be the distance between these two axes.

Now, consider a particle P of mass 'm' whose distance from the PQ axis is 'x'.

Moment of inertia of the particle about

PQ = Mx²

Moment of inertia of the body about PQ is

IG = ∑mx²........(i)

Now moment of inertial of the particle about AB axis = m(x + h)².

Moment of inertia of the body about AB axis is,

(Total momentum of weight about centre of gravity is zero).