Rotational Kinetic Energy: Definition Formula Unit

Kinetic energy can be found in every moving material. We know how to calculate this for a body moving in one direction, but what about a rigid body rotating i.e. continuously changing its direction? Because each point on the rigid body has a different linear velocity, this may appear difficult. However, we can represent the kinetic energy of a rotating object using angular velocity (ω), which is the same for a completely rigid body.

The product of the moment of inertia and the square of the angular velocity is called rotational kinetic energy. Therefore by this definition, the rotational kinetic energy can be expressed in terms of the moment of inertia (I) and angular velocity (ω). The rotational K.E of an object is analogous to linear kinetic energy.

Formula of Rotational Kinetic Energy:

The formula for the rotational kinetic energy of an object is given by –

Where Erotational = Rotational Kinetic Energy
I = Moment of Inertia
ω = Angular Velocity.

Derivation of Rotational Kinetic Energy:

Let us consider a body PQR rotating about an axis AB with angular velocity ω [figure-1]. Due to this rotation, the body has some kinetic energy, which is rotational KE. Now the body can be considered as a sum of a large number of particles having masses m₁, m₂, m₃, …. etc. These particles are located at distances r₁, r₂, r₃, … etc. with respect to the axis of rotation AB. As the body is rigid, the angular velocities of all the particles remain the same i.e, angular velocity ω = constant. But the linear velocities (v) of all particles are not the same since the distances of the particles are different from the axis AB.

Now,
Particle of mass m₁ has velocity v₁. Thus, v₁ = ωr₁
Particle of mass m₂ has velocity v₂. Thus, v₂ = ωr₂
Particle of mass m₃ has velocity v₃. Thus, v₃ = ωr₃
The Kinetic Energy of particle 1 is = (1/2)m₁v₁² = (1/2)m₁ω²r₁²
The Kinetic Energy of particle 2 is = (1/2)m₂v₂² = (1/2)m₂ω²r₂²
The Kinetic Energy of particle 3 is = (1/2)m₃v₃² = (1/2)m₃ω²r₃²
…. and so on ….
Therefore, by summing up all the kinetic energies of the particles, we get the total Kinetic energy.
Total rotational kinetic energy = [(1/2)m₁ωr₁² + (1/2)m₂ωr₂² + (1/2)m₃ωr₃² + …. ]
= (1/2)ω[m₁r₁² + m₂r₂² + m₃r₃² + …. ]
= (1/2)ω² Σmr²
= (1/2)Iω²
Where I = Σmr² = Moment of Inertia about the axis.

Therefore, the rotational kinetic energy = Iω²/2

In translational motion the translational kinetic energy = (1/2)mv².

Unit and Dimension of Rotation K.E

The unit and dimension of rotational kinetic energy are given in the table below.

Derivation of SI Unit of Rotational K.E

Rotational Kinetic Energy = Iω²/2
Unit of Rotational K.E = [unit of Moment of Inertia] × [unit of Angular velocity]²
= [Kg ⋅ m²] × [s ^–1]²
= [Kg ⋅ m²s ^–2] = [Kg ⋅ m ⋅ s ^–2] × [m]
= [unit of force] × [unit of displacement]
= [unit of word done]
= [unit of energy]
= Joule

Derivation of Dimension of Rotational K.E

We know that, Rotational Kinetic Energy = (1/2) × [Moment of Inertia] × [Angular velocity]²
Dimension of Rotational K.E = [Dimension of Moment of Inertia] × [Dimension of Angular velocity]²
Dimension of rotational K.E = [ML²] × [T ^–1]²
= [ML²T ^–2]

Thus, the dimension of rotational kinetic energy is = [ML²T ^–2]

Rotational Power:

If an object rotates with angular velocity ω by the application of torque τ, then
Rotational Power = (work done)/time
= (torque × angular displacement) / time
= torque × angular velocity
= τω
Rotational Power P = τω

Rotational KE: Problem Solution

Rotational KE: Question Answers

Suggested Readings: