The root mean square distance of its constituent particles from the axis of rotation is called the radius of gyration of a body.

It is denoted by K.

Radius of gyration

K=r₁²+r₂²+…rₙ²n

The product of the mass of the body (M) and square of its radius gyration (K) gives the same moment of inertia of the body about rotational axis.

Therefore, moment of inertia I = MK² ⇒ K = √1/M

Parallel Axes Theorem

The moment of inertia of any object about any arbitrary axes is equal to the sum of moment of inertia about a parallel axis passing through the centre of mass and the product of mass of the body and the square of the perpendicular distance between the two axes.

Mathematically I = I_CM + Mr²

Where I is the moment of inertia about the arbitrary axis, I_CM is moment of inertia about the parallel axis through the centre of mass, M is the total mass of the object and r is the perpendicular distance between the axis.

Perpendicular Axes Theorem

The moment of inertia of any two dimensional body about an axis perpendicular to its plane (I_Z) is equal to the sum of moments of inertia of the body about two mutually perpendicular axes lying in its own plane and intersecting each other at a point, where the perpendicular axis passes through it.

Mathematically I_Z = I_X + I_Y

where I_X and I_Y are the moments of inertia of plane lamina about perpendicular axes X and Y respectively which lie in the plane lamina an intersect each other.

Theorem of parallel axes is applicable for any type of rigid body whether it is a two dimensional or three dimensional, while the theorem of perpendicular is applicable for laminar type or two I dimensional bodies only.

Moment of Inertia of Homogeneous Rigid Bodies

For a Thin Circular Ring

For a Circular Disc

For a Thin Rod

For a Solid Cylinder

For a Thin Spherical Shell

Equations of Rotational Motion

1. ω = ω₀ + αt
2. θ = ω₀t + 1/2 αt²
3. ω²= ω₀²+ 2αθ

Where θ is displacement in rotational motion, ω0 is initial velocity, omega; is final velocity and a is acceleration.