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Chapter 6: System of Particles and Rotational Motion

Grade 11 Science | Chapter 6

System of Particles and Rotational Motion

Extended bodies can spin as well as move. This chapter introduces the centre of mass, torque, moment of inertia and angular momentum, with the law that conserves it.

Core Concepts

Key Principles

Worked Examples

Practice Sets

Contents

Introduction: Beyond the Point Particle

Centre of Mass

Torque, the Turning Effect

Moment of Inertia

Angular Momentum

Conservation of Angular Momentum

Key Reasoning (Principles)

Worked Examples (10)

Practice Sets A to D

10.

Summary and Exam Quick-Check

1. Introduction: Beyond the Point Particle

So far we treated objects as points. Real bodies have size and can rotate as well as move bodily. This chapter introduces the tools of rotational motion: the centre of mass that moves as if all the mass were there, the torque that causes turning, the moment of inertia that resists it, and angular momentum with its conservation law.

Core idea

Rotation mirrors straight-line motion: torque plays the role of force, moment of inertia the role of mass, and angular momentum the role of momentum. So τ = Iα mirrors F = ma.

2. Centre of Mass

The centre of mass is the point at which the whole mass of a body may be taken to act; the body moves as though all external forces acted there. For two masses on a line it lies closer to the heavier one. The motion of the centre of mass obeys Newton’s laws just as a single particle would, which is why we could treat objects as points earlier.

Diagram 1 – Centre of Mass

Two masses on a line with the centre of mass marked closer to the heavier one

Fig 1. For two bodies, the centre of mass lies on the line joining them, nearer the heavier body.

3. Torque, the Turning Effect

Torque is the turning effect of a force about an axis, equal to the force times its perpendicular distance from the axis, τ = r F sinθ. A larger force or a longer arm gives more torque, which is why a door handle is placed far from the hinge. Torque is the rotational equivalent of force, and a net torque produces angular acceleration.

Diagram 2 – Torque

A force applied at a distance from a pivot producing a turning effect, the torque

Fig 2. The torque of a force about the pivot is the force times its perpendicular distance from the pivot; it sets the body turning.

4. Moment of Inertia

The moment of inertia measures a body’s resistance to a change in its rotation, the rotational counterpart of mass. For a point mass it is I = m r², and for an extended body it depends on how the mass is spread about the axis: mass farther from the axis contributes more. Newton’s second law for rotation is τ = I α, torque equals moment of inertia times angular acceleration.

5. Angular Momentum

Angular momentum is the rotational analogue of momentum, L = I ω, where ω is the angular velocity. Just as a force changes momentum, a torque changes angular momentum, and the rate of change of angular momentum equals the applied torque. Angular momentum is a vector along the axis of rotation.

6. Conservation of Angular Momentum

When no net external torque acts, the angular momentum is conserved: I ω stays constant. So if a spinning skater pulls in their arms, reducing their moment of inertia, their angular velocity rises to keep I ω the same. This is the rotational version of the conservation of momentum.

Watch out

Conservation of angular momentum needs zero net external torque, not zero force. A skater’s internal muscle forces change I but exert no external torque, so L is conserved.

7. Key Reasoning (Principles)

Principle 1: Rotation mirrors translation

Every translational quantity has a rotational partner: force and torque, mass and moment of inertia, momentum and angular momentum, so τ = Iα mirrors F = ma.

Principle 2: Torque needs a lever arm

Torque is force times perpendicular distance, so the same force gives more turning effect when applied farther from the axis.

Principle 3: Angular momentum is conserved without external torque

With no net external torque, Iω stays constant, so reducing the moment of inertia speeds up the rotation.

8. Worked Examples

Example 1

Q: Find the torque of a 20 N force at a perpendicular distance of 0.5 m.

▶ Show Solution

τ = r F = 0.5 × 20.

= 10 N m.

Answer: 10 N m.

Example 2

Q: Find the moment of inertia of a 2 kg point mass at 3 m from the axis.

▶ Show Solution

I = m r² = 2 × 9.

= 18 kg m².

Answer: 18 kg m².

Example 3

Q: A torque of 12 N m acts on a body of moment of inertia 4 kg m². Find the angular acceleration.

▶ Show Solution

α = τ ÷ I = 12 ÷ 4.

= 3 rad/s².

Answer: 3 rad/s².

Example 4

Q: Find the angular momentum of a body with I = 5 kg m² spinning at 4 rad/s.

▶ Show Solution

L = I ω = 5 × 4.

= 20 kg m²/s.

Answer: 20 kg m²/s.

Example 5

Q: Where does the centre of mass of equal masses 6 m apart lie?

▶ Show Solution

For equal masses it is midway.

So 3 m from each.

Answer: At the midpoint, 3 m from each.

Example 6

Q: A 30 N force acts at 0.2 m from a pivot, at right angles. Find the torque.

▶ Show Solution

τ = 0.2 × 30.

= 6 N m.

Answer: 6 N m.

Example 7

Q: A skater with I = 4 kg m² at 2 rad/s pulls in to I = 2 kg m². Find the new angular velocity.

▶ Show Solution

Iω is conserved: 4 × 2 = 2 × ω’.

ω’ = 4 rad/s.

Answer: 4 rad/s.

Example 8

Q: Find the moment of inertia of a 3 kg mass at 2 m from the axis.

▶ Show Solution

I = m r² = 3 × 4.

= 12 kg m².

Answer: 12 kg m².

Example 9

Q: A wheel of I = 0.5 kg m² has angular acceleration 6 rad/s². Find the torque.

▶ Show Solution

τ = I α = 0.5 × 6.

= 3 N m.

Answer: 3 N m.

Example 10

Q: Why does a diver spin faster after tucking in?

▶ Show Solution

Tucking lowers the moment of inertia; with angular momentum conserved, the angular velocity rises.

Answer: Lower I means higher angular velocity (L conserved).

9. Practice Sets A to D

Set A – Multiple Choice (Basic)

1. Torque is force times: (a) mass (b) perpendicular distance (c) time (d) speed

2. The rotational analogue of mass is the: (a) torque (b) angular velocity (c) moment of inertia (d) force

3. Angular momentum is given by: (a) mv (b) Iω (c) Iα (d) rF

4. τ = Iα is the rotational form of: (a) F = ma (b) p = mv (c) W = Fs (d) v = u + at

5. Angular momentum is conserved when the net external torque is: (a) large (b) zero (c) constant (d) negative

▶ Reveal Answers

1. (b) perpendicular distance.

2. (c) moment of inertia.

3. (b) Iω.

4. (a) F = ma.

5. (b) zero.

Set B – Short Answer (Understanding)

1. Define the centre of mass.

2. Define torque and give its unit.

3. What does the moment of inertia measure?

4. Write the rotational form of Newton’s second law.

5. State the law of conservation of angular momentum.

▶ Reveal Answers

1. The point at which the whole mass of a body may be taken to act.

2. The turning effect of a force, τ = r F sinθ, measured in newton metres.

3. A body’s resistance to a change in its rotation, the rotational counterpart of mass.

4. τ = I α, torque equals moment of inertia times angular acceleration.

5. With no net external torque, the angular momentum Iω stays constant.

Set C – Application and Reasoning

1. Find the torque of a 50 N force at 0.4 m, at right angles.

2. Find the moment of inertia of a 5 kg point mass at 2 m.

3. A torque of 20 N m acts on I = 5 kg m². Find the angular acceleration.

4. Find the angular momentum of I = 3 kg m² at 6 rad/s.

5. Why is a door handle placed far from the hinge?

▶ Reveal Answers

1. τ = 0.4 × 50 = 20 N m.

2. I = 5 × 4 = 20 kg m².

3. α = 20 ÷ 5 = 4 rad/s².

4. L = 3 × 6 = 18 kg m²/s.

5. To give a larger lever arm, so the same force produces more torque to open the door.

Set D – Higher Order (Challenge)

1. A skater with I = 6 kg m² at 3 rad/s reduces I to 2 kg m². Find the new angular velocity.

2. Explain why τ = Iα is the rotational analogue of F = ma.

3. Two masses 2 kg and 4 kg are 9 m apart. Find the centre of mass from the 2 kg mass.

4. A flywheel of I = 0.4 kg m² is spun up to 10 rad/s by a torque of 2 N m. Find the time taken from rest.

5. Explain how a planet speeds up when nearer the Sun using angular momentum.

▶ Reveal Answers

1. Iω conserved: 6 × 3 = 2 × ω’, so ω’ = 9 rad/s.

2. Torque replaces force, moment of inertia replaces mass and angular acceleration replaces linear acceleration, giving the same form of law.

3. Centre of mass = (2 × 0 + 4 × 9) ÷ 6 = 6 m from the 2 kg mass.

4. α = τ ÷ I = 2 ÷ 0.4 = 5 rad/s²; time = ω ÷ α = 10 ÷ 5 = 2 s.

5. Nearer the Sun the planet’s distance, and so its moment of inertia, falls; with angular momentum conserved, its speed increases.

Chapter Summary

Centre of Mass

The point where the whole mass may be taken to act; moves by Newton’s laws.

Torque

Turning effect, τ = r F sinθ, the rotational force.

Moment of Inertia

Resistance to change in rotation, I = m r² for a point mass.

Rotational Second Law

τ = I α, mirroring F = ma.

Angular Momentum

L = I ω, the rotational momentum.

Conservation

With no external torque, I ω stays constant.

Quantity

Unit

Symbol

Torque

newton metre

N m

Moment of inertia

kilogram metre squared

kg m²

Angular momentum

kg m² per second

kg m²/s

8-Point Exam Quick-Check

The centre of mass moves as if all mass and external force acted there.

Torque = force times perpendicular distance, the turning effect.

Moment of inertia is the rotational counterpart of mass, I = m r squared for a point.

Rotational second law: torque = moment of inertia times angular acceleration.

Angular momentum L = I omega.

With no net external torque, angular momentum is conserved.

Reducing the moment of inertia raises the angular velocity.

Rotation mirrors translation throughout.

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Class 11 Physics Chapter 6: System of Particles and Rotational Motion, Complete Notes and Practice

This revision guide follows the current NCERT Class 11 Physics syllabus and develops rotational motion, covering the centre of mass, torque as the turning effect of a force, moment of inertia, the rotational second law torque equals I alpha, angular momentum and its conservation, with two diagrams, ten worked examples and graded practice. Visit SchoolRevise.com to revise, practise and excel.