Rotational Motion & Moment of Inertia - Physics for JEE Main & Advanced

Rotational Motion

1. Rigid Body

Rigid body is a system of particles in which the distance between each pair of particles remains constant in time. In a rigid body the shape and size do not change during motion. Examples: fan, pen, table, stone.

Objects that deform (for example, two blocks joined by a spring, or a body that stretches) are not rigid bodies. For every pair of particles in a rigid body there is no relative approach or separation between them.

Consider two points A and B on a rigid body. In the figure shown the velocities of A and B with respect to the ground are

If the body is rigid, the components of velocities along the line joining the two points satisfy the geometric constraint

V_A cos θ₁ = V_B cos θ₂

Note : With respect to any given particle of a rigid body, the motion of any other particle of that rigid body is circular about that particle (instantaneously), since the distance between them is fixed.

V_BA denotes the relative velocity of B with respect to A.

1.1 Pure Translational Motion

A body is said to be in pure translational motion when every particle in the body undergoes the same displacement in a given time interval. In pure translation all particles have the same velocity and the same acceleration at any instant.

Example. A box pushed on a horizontal surface translates (if it does not rotate).

For a rigid body composed of many particles (m₁, m₂, m₃, ...), each particle has the same acceleration in pure translation. If the accelerations are a₁, a₂, a₃, ... then

But accelerations of all particles are equal, so

Therefore,

Where M is the total mass of the body and a is the acceleration of any particle or of the centre of mass of the body.

Similarly for velocities: if velocities of particles are v₁, v₂, v₃, ... and all are equal in pure translation,

Therefore the velocity of the centre of mass equals the common velocity of all particles.

The total kinetic energy of the body in pure translation is

1.2 Pure Rotational Motion

A body is said to be in pure rotational motion when every particle moves on a circular path about a fixed line (axis) and the perpendicular distance of each particle from the axis remains constant. Particles do not translate parallel to the axis. In pure rotation all particles have the same angular velocity and the same angular acceleration at an instant. Examples: rotating ceiling fan, hands of a clock.

In rigid-body pure rotation about a fixed axis, each particle describes a circle whose centre lies on the axis of rotation. If the angular displacement of the body is θ (in radians), the arc length s travelled by a particle at a perpendicular distance r from the axis is

s = r θ

The relation between linear and angular quantities is therefore

v = ω r

a_tan = α r

a_rad = ω² r

2. Kinematics of Rotational Motion

Angular displacement: θ (radian)

Angular velocity: ω = dθ/dt

Angular acceleration: α = dω/dt = d²θ/dt²

For constant angular acceleration, the kinematic equations (analogous to linear motion) are

• ω = ω₀ + α t
• θ = θ₀ + ω₀ t + ½ α t²
• ω² = ω₀² + 2 α (θ - θ₀)

Velocity of a particle at position vector r (measured from axis) in rigid body rotation is given by the vector relation

v = ω × r

Here ω is the angular velocity vector directed along the axis of rotation (right-hand rule). The magnitude of v is ω r sin φ where φ is the angle between ω and r; for planar rotation with r perpendicular to ω, |v| = ω r.

3. Dynamics of Rotational Motion

Torque (moment of a force): For a force F acting at position r (measured from chosen origin), the torque about that origin is

τ = r × F

The magnitude of torque about an axis is τ = r F sin θ, where θ is angle between r and F. Torque tends to produce rotational acceleration.

Rotational analogue of Newton's second law: For a rigid body rotating about a fixed axis, the net external torque about the axis equals the rotational inertia times angular acceleration:

∑τ = I α

Here I is the moment of inertia about the axis of rotation (see next section).

Work done by torque: If a torque τ acts and body rotates through angle dθ, the infinitesimal work is

dW = τ dθ

Rotational kinetic energy: For a rigid body rotating about a fixed axis with angular speed ω, the total kinetic energy is

K = ½ I ω²

Power supplied by torque is

P = τ ω

4. Moment of Inertia (Rotational Inertia)

Definition: The moment of inertia of a rigid body about a given axis is

I = ∫ r² dm

where r is the perpendicular distance of the mass element dm from the axis. The SI unit of moment of inertia is kg·m².

Properties:

• Moment of inertia depends on the mass distribution and the axis chosen.
• Moments of inertia add for composite bodies: the total I is the sum of I of constituent pieces about the same axis.

Parallel axis theorem (Steiner's theorem): If I_CM is the moment of inertia of a body about an axis through its centre of mass (parallel to the required axis), then the moment of inertia about an axis parallel to it and at distance d from it is

I = I_CM + M d²

Perpendicular axis theorem (for planar lamina): For a planar object lying in the xy-plane, moments about x and y axes and about z (perpendicular) axis through the same point satisfy

Rotational motion of rigid bodies is described by angular kinematics (θ, ω, α), rotational dynamics (torque τ and ∑τ = I α), and energy (K = ½ I ω²). The moment of inertia I is the rotational analogue of mass and depends on mass distribution relative to the axis. Useful theorems are the parallel-axis theorem and perpendicular-axis theorem. Combining translation and rotation is required for rolling motion and for many engineering applications.