Theorems of Perpendicular and Parallel Axes

Theorems on Moments of Inertia:

Parallel Axis Theorem:

Allows to find MOI about an axis parallel to axis passing through centre of mass, provided MOI through the latter is known.

If d is the between these two parallel axes and M is the mass of the body then using this theorem:

Illustration:

Calculate the moment of inertia of a:

1. disc about an axis passing through its edge and perpendicular to the circular base of the disc
2. solid sphere about as axis touching the sphere at its surface.

Explanation:

Perpendicular Axis Theorem:

The moment of inertia of the body about Z- axis (axes X, Y, Z are mutually perpendicular) (passing through O and perpendicular to the plane of the body) is given by:

I_x = MOI about X – axis.

I_y = MOI about Y – axi

Illustration:

Calculate the moment of inertia of:

1. a ring of mass M and radius R about an axis coinciding with a diameter of the ring.
2. a thin disc about an axis coinciding with a diameter.

Explanation:

Let X & Y axis be along two perpendicular diameters of the ring.

By symmetry,

But we know that 

Similarly for at thin disc (i.e., a circular plate)

Moment of inertia about a diameter is