Rotational Motion – Stage 1 (Page 8)

Rotational Kinetic Energy & Work–Energy Theorem

Just as a moving body possesses kinetic energy due to translation, a rotating body possesses rotational kinetic energy. This concept is extremely useful in solving complex rotation problems.

1. Rotational Kinetic Energy

The kinetic energy of a rigid body rotating about a fixed axis is:

K_rot = ½ I ω²

where
I = moment of inertia about the axis
ω = angular velocity

2. Physical Meaning

✔ Larger moment of inertia → more energy required
✔ Faster rotation → much higher energy
✔ Energy depends strongly on ω²

This explains why high-speed rotating machines store large energy.

3. Total Kinetic Energy (Rolling Motion)

If a body is rolling without slipping, it has both translational and rotational kinetic energy.

K_total = ½ M v² + ½ I ω²

This formula is heavily used in incline and rolling problems.

4. Work Done by Torque

Work done by a torque when the body rotates through angle θ is:

W = τ θ (for constant torque)

θ must be in radians.

5. Work–Energy Theorem (Rotational)

Net work done by all torques acting on a rigid body is equal to the change in its rotational kinetic energy.

W = Δ(½ I ω²)

6. Linear vs Rotational Analogy

Force ↔ Torque
Mass ↔ Moment of Inertia
Velocity ↔ Angular velocity
½mv² ↔ ½Iω²

⚠ Common Student Mistakes

❌ Forgetting translational KE in rolling ❌ Using wrong moment of inertia ❌ Mixing degrees with radians

IIT-JEE Insight

IIT prefers energy methods over torque equations when forces are complicated or motion is constrained.

Next: Stage 1 – Page 9
Angular Acceleration, Dynamics & Equation τ = Iα

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