Rotational Motion of Rigid Bodies – Complete Notes | Mechanics (UG Level)

Rotational Motion of Rigid Bodies

Rotational motion is one of the most important topics in classical mechanics. It explains how objects rotate about an axis and helps us understand the motion of wheels, gears, fans, planets, turbines, and even microscopic systems. In this part, we will build a strong conceptual foundation of rotational motion of rigid bodies using simple language, step-by-step explanations, and practical examples.

1. What is a Rigid Body?

A rigid body is an idealized object in which the distance between any two particles remains constant during motion. In reality, no object is perfectly rigid, but for most practical purposes (like wheels, discs, rods), deformation is very small and can be ignored.

Examples of rigid bodies:

A rotating wheel
A ceiling fan
A solid disc or ring
A door rotating about its hinges

When a rigid body rotates, all its particles move in circular paths around a common axis. This makes rotational motion different from straight-line (translational) motion.

Types of Motion of a Rigid Body

Pure Translational Motion: Every point of the body moves with the same velocity.
Pure Rotational Motion: Body rotates about a fixed axis.
General Motion: Combination of translation and rotation (e.g., rolling wheel).

Practice Questions

Define a rigid body and give two examples.
Why is a perfectly rigid body not possible in reality?

2. Rotational Motion and Axis of Rotation

In rotational motion, every particle of the rigid body moves in a circular path around a fixed line called the axis of rotation.

The axis of rotation can be:

Fixed (e.g., ceiling fan)
Passing through the body (e.g., rotating disc)
Outside the body (e.g., door rotating about hinges)

Each particle of the body has the same angular displacement, angular velocity, and angular acceleration, but their linear velocities are different.

3. Angular Variables (Rotational Kinematics)

Just like linear motion has displacement, velocity, and acceleration, rotational motion has corresponding angular quantities.

3.1 Angular Displacement (θ)

Angular displacement is the angle through which a body rotates about a fixed axis. It is measured in radians.

Relation between linear and angular displacement:

s = rθ

where:

s = arc length
r = radius
θ = angular displacement

3.2 Angular Velocity (ω)

Angular velocity is the rate of change of angular displacement with time.

ω = dθ / dt

Unit: rad/s

Relation with linear velocity:

v = rω

This shows that particles farther from the axis move faster.

3.3 Angular Acceleration (α)

Angular acceleration is the rate of change of angular velocity.

α = dω / dt

Relation with linear acceleration:

a_t = rα (tangential acceleration)

Example

A disc of radius 0.5 m rotates with angular velocity 4 rad/s. Find the linear velocity of a point on its rim.

Solution: v = rω = 0.5 × 4 = 2 m/s

Practice Questions

Define angular velocity and angular acceleration.
Why do particles farther from the axis move faster?

4. Equations of Rotational Motion

For constant angular acceleration, equations of rotational motion are similar to linear motion.

ω = ω₀ + αt
θ = ω₀t + (1/2)αt²
ω² = ω₀² + 2αθ

These equations are used to solve problems involving rotating wheels, discs, and pulleys.

Numerical Example

A wheel starts from rest and rotates with angular acceleration 2 rad/s². Find angular velocity after 5 seconds.

Solution: ω = 0 + 2 × 5 = 10 rad/s

5. Torque (Moment of Force)

In rotational motion, the turning effect of a force is called torque.

Torque (τ) = r × F

Magnitude:

τ = rF sinθ

where θ is the angle between force and position vector.

Torque depends on:

Magnitude of force
Distance from axis
Direction of force

Right-Hand Rule

Curl the fingers of your right hand in the direction of rotation. The thumb points in the direction of torque vector.

Real-Life Examples

Opening a door by pushing far from hinges
Using a long wrench to loosen a nut

Practice Questions

Define torque and write its SI unit.
Why is it easier to open a door by pushing at the edge?

6. Relation Between Torque and Angular Acceleration

Just as force causes linear acceleration, torque causes angular acceleration.

τ = Iα

This is the rotational form of Newton’s second law.

Here, I is the moment of inertia, which depends on mass distribution.

This equation shows:

More torque → more angular acceleration
Larger moment of inertia → smaller acceleration

Conceptual Question

Why does a heavy door require more effort to rotate than a light door?

7. Moment of Inertia

The moment of inertia (I) of a rigid body is the rotational analogue of mass in linear motion. It measures how difficult it is to rotate a body about a given axis.

Just as heavier objects are harder to accelerate linearly, objects with larger moment of inertia are harder to rotate.

For a system of particles:

I = Σ m_i r_i²

where:

m_i = mass of particle
r_i = perpendicular distance from axis of rotation

This formula clearly shows that particles farther from the axis contribute more to moment of inertia.

SI Unit of Moment of Inertia

The SI unit of moment of inertia is kg m².

8. Moment of Inertia of Common Rigid Bodies

For continuous bodies, moment of inertia is calculated using integration. However, for common shapes, standard results are used.

Thin rod about center: I = (1/12) ML²
Thin rod about one end: I = (1/3) ML²
Solid disc about center: I = (1/2) MR²
Ring about center: I = MR²
Solid sphere: I = (2/5) MR²

These formulas are extremely important for numerical problems and competitive exams.

Conceptual Question

Why does a ring have a larger moment of inertia than a disc of the same mass and radius?

9. Radius of Gyration

The radius of gyration (k) is the distance from the axis at which the entire mass of the body can be assumed to be concentrated without changing its moment of inertia.

Mathematically:

I = Mk²

k = √(I / M)

Radius of gyration gives an idea of how mass is distributed relative to the axis. A larger value of k means mass is distributed farther from the axis.

Practice Questions

Define radius of gyration.
What does a larger radius of gyration indicate?

10. Theorem of Parallel Axes

According to the parallel axis theorem, the moment of inertia of a rigid body about any axis is equal to the moment of inertia about a parallel axis through the center of mass plus Md².

I = I_cm + Md²

where:

I_cm = moment of inertia about center of mass
d = distance between the axes

This theorem is useful when the axis does not pass through the center of mass.

Example

Find the moment of inertia of a rod about one end if its moment of inertia about the center is (1/12)ML².

Solution:

I = (1/12)ML² + M(L/2)² = (1/3)ML²

11. Theorem of Perpendicular Axes

This theorem applies to planar bodies.

It states that the moment of inertia about an axis perpendicular to the plane is equal to the sum of moments of inertia about two mutually perpendicular axes in the plane.

I_z = I_x + I_y

This theorem is mainly used for discs, rings, and laminae.

12. Rotational Kinetic Energy

A rotating body possesses kinetic energy due to its rotation.

K = (1/2) Iω²

This formula is similar to translational kinetic energy (1/2 mv²).

For rolling motion, total kinetic energy is the sum of translational and rotational energy:

K = (1/2) mv² + (1/2) Iω²

Example

A disc rotates with angular velocity ω. If its moment of inertia is I, find its rotational kinetic energy.

Answer: (1/2)Iω²

13. Angular Momentum

The angular momentum of a particle about a point is defined as the moment of linear momentum.

L = r × p

For a rigid body rotating about a fixed axis:

L = Iω

Angular momentum is a vector quantity and its direction is given by the right-hand rule.

14. Conservation of Angular Momentum

If the net external torque acting on a system is zero, the total angular momentum of the system remains constant.

τ = 0 ⇒ L = constant

Real-Life Examples

Spinning ice skater pulling arms inward
Planetary motion
Divers rotating faster when they tuck their bodies

Conceptual Explanation

When a skater pulls arms inward, moment of inertia decreases. To conserve angular momentum, angular velocity increases.

15. Rolling Motion (Pure Rolling)

Rolling motion is a combination of translational and rotational motion.

For pure rolling:

v = rω

In pure rolling, there is no slipping between the body and the surface.

Practice Questions

What is pure rolling motion?
Why is friction necessary for rolling?

16. Important Exam-Oriented Questions

Derive expression for rotational kinetic energy.
Explain parallel axis theorem with example.
State and explain conservation of angular momentum.
Compare translational and rotational motion.

— End of Rotational Motion of Rigid Bodies (Complete Notes) —