Physics

Introduction to Rotational Mechanics

Rotational mechanics deals with the motion of bodies that rotate about an axis. Just like linear motion, it has concepts of angular displacement, angular velocity, torque, and moment of inertia.

Key Concepts

Angular Displacement

Change in angular position of a rotating object.
θ (in radians)

Angular Displacement

Torque

The rotational equivalent of force.
τ = r × F

Torque

Moment of Inertia

Measure of an object's resistance to angular acceleration.
I = Σmr²

Moment of Inertia

Equations

Rotational Kinematics

θ = ω₀t + ½αt²

Rotational Dynamics

τ = Iα

Rotational KE

KE = ½Iω²

Angular Momentum

L = Iω

Law of Conservation of Angular Momentum

In the absence of external torque, the total angular momentum of a system remains constant.

Angular Momentum Conservation

Example Problems

1. A wheel of moment of inertia 2 kg·m² is rotating with angular speed 5 rad/s. Find rotational KE.

Solution: KE = ½Iω² = 0.5 × 2 × 25 = 25 J

2. Find torque when force of 10 N is applied at 0.5 m from axis.

Solution: τ = r × F = 0.5 × 10 = 5 N·m

3. A disc of I = 1.5 kg·m² has α = 2 rad/s². Find torque.

Solution: τ = Iα = 1.5 × 2 = 3 N·m

4. A gymnast reduces her moment of inertia to half. What happens to angular velocity?

Solution: L = Iω constant ⇒ ω doubles if I halves.