A rigid body rotates about a fixed axis. The rigid body consists of a large number of particles.
Let m1, m2, m3 etc., be the masses of the particles situated at distances r1, r2, r3 , … etc., from the fixed axis. All the particles rotate with the same angular velocity, but with different linear velocities depending on the values of ‘r’.
The angular momentum of a rigid body (L) = m₁ r₁²ω ₊  m₂ r₂²ω…
The angular momentum of a rigid body (L) = (m₁ r₁² + m₂ r₂²….) ω

The angular momentum of a rigid body (L) = Σ mr²ω …………………(1)

But, Σmr² = moment of inertia of the rigid body = I

Therefore,

Angular momentum of a Rigid body = I × ω……putting value of I in (1)

Angular momentum of a Rigid body = moment of inertia of the rigid body ( I) x × angular velocity (ω)

The S.I. unit for angular momentum is kgm²rad/s or kgm³/s

The moment of linear momentum is known as angular momentum. In other words the rotational analog of linear momentum is known as angular momentum
Consider a particle of mass m at a distance r from the axis of rotation. When a particle is in rotational motion about an axis, it has both linear velocity ‘v’ and angular velocity ‘ω’.

Angular momentum of the particle = linear momentum x perpendicular distance between
the particle and the axis of rotation or radius (r)

We know that momentum = mass × velocity
Substituting the value of momentum in the above equation we get,
Angular momentum = m v × r
we know that relation between Linear velocity and Angular velocity is defined as
Linear velocity (v) = Radius (r) x Angular velocity (ω)

Substituting the values we get
Angular momentum = m × (r. ω) × r
Therefore, the formula for Angular momentum is given as,
Angular momentum = mr²ω
The S.I. unit for angular momentum is kgm²s^-1

Torque is defined as the rate of change of angular momentum of an object. The definition of torque states that one or both of the angular velocity or the moment of inertia of an object are changing.
The formula for Torque acting on a body is given as,

Torque (τ) = moment of inertia (I) × angular acceleration (α)
→             →
τ     = I x α  

where τ = Torque
α = angular acceleration
I = moment of inertia
S.I. unit of torque is Newton metre (Nm)

The kinetic energy of the whole body is equal to the sum of the kinetic energy of all the particles present in the body. A rotating rigid body has kinetic energy because all atoms in the object are in motion.

The formula for kinetic energy of a rotating rigid body is given as,
Kinetic Energy (K.E.) = 1 /2 ( Moment of inertia (I) × (angular velocity)²)

where  ω = angular velocity

Radius of gyration is the distance between the given axis and the centre of mass of the body. The centre of mass of a body is point where the entire mass of the body is supposed to be concentrated. It is denoted by ‘K ’.

If M is mass of the body, the moment of inertia is given as,
Moment of inertia (I) = mass of the body (M) × (radius of gyration)²

So, the formula for radius of gyration (K) is given as,

where K = radius of gyration, I = Moment of inertia and M = mass of the body.