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Physics · Class 11 Science

Ch 6System of Particles and Rotational Motion — Class 11 Physics, concept-first.

Every body studied so far in mechanics -- a block, a ball, a car -- was treated as though it were a single point, with all of its mass concentrated at one location.

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Key concepts

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Centre of Mass

The centre of mass (c.m.) is the single hypothetical point at which the entire mass of an extended, finite-sized object (or system of objects) can be treated as concentrated, allowing Newton's laws of motion -- which str…

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In previous exams

How often this chapter’s concepts have been examined — real appearance data, never estimated.

WBJEE

2 concepts · 4 appearances

2024, 2023, 2022, 2021

Chapter contents

The NCERT structure, section by section. Open a section to see its questions, then read the concept-first solution.

5.1

Introduction: Rigid Bodies and the Two Kinds of Motion

Every body studied so far in mechanics -- a block, a ball, a car -- was treated as though it were a single point, with all of its mass concentrated at one location.

5.2

Centre of Mass of a Two-Particle System

Consider two point masses and , located at positions and respectively. The centre of mass of this two-particle system is defined as the point whose position vector is the mass-weighted average of the…

5.3

Centre of Mass of a System of Particles and of a Rigid Body

The two-particle formula of Section 5.2 extends directly to any number of point masses at positions :

5.4

Motion of the Centre of Mass and Conservation of Linear Momentum

Differentiating the defining formula (Section 5.3) once with respect to time, and using as a constant (no mass enters or leaves the system),

5.5

Vector (Cross) Product and the Moment of a Force -- Torque

Multiplying two vectors together can be done in two genuinely different ways. The scalar (dot) product produces an ordinary number and measures how much one vector runs along the direction of the othe…

5.6

Angular Momentum of a Particle and of a System of Particles

The angular momentum of a single particle about a chosen reference point is defined, using the same vector (cross) product introduced in Section 5.5, as

5.7

Conservation of Angular Momentum, with Examples

Section 5.6 established that the total angular momentum of any system of particles, about a fixed point or a fixed axis, changes only under the action of a net EXTERNAL torque:

5.8

Equilibrium of a Rigid Body and the Principle of Moments

A single particle is in equilibrium when the net force on it is zero. An extended rigid body needs a SECOND, independent condition as well, because a body can have zero net force acting on it and stil…

5.9

Centre of Gravity

The centre of gravity of a body is defined as the single point at which the ENTIRE WEIGHT of the body -- the total downward gravitational force acting on every one of its constituent particles -- may…

5.10

Moment of Inertia

Moment of inertia is the rotational analogue of mass: just as a body's ordinary (inertial) mass measures its reluctance to change its state of straight-line motion under a force, moment of inertia mea…

5.11

Moment of Inertia of Simple Geometrical Bodies

Deriving the moment of inertia of even a simple uniform body from the defining integral (Section 5.10) needs calculus that is beyond WBCHSE's Unit 5 syllabus, which explicitly asks only for the final…

5.12

Radius of Gyration

The radius of gyration, denoted , of a rigid body about a given axis is defined as the particular distance from that axis at which, IF the body's ENTIRE mass were imagined concentrated as a single poi…

5.13

Theorems of Parallel and Perpendicular Axes

Two general theorems -- stated here without proof, as directed by the syllabus -- make it possible to find the moment of inertia of a rigid body about many different axes, starting from just ONE known…

5.14

Kinematics and Dynamics of Rotational Motion About a Fixed Axis

For a rigid body rotating about a FIXED axis with a CONSTANT angular acceleration , the angular displacement , angular velocity and angular acceleration obey three equations that are exact rotational…

5.15

Comparison of Linear (Translational) and Rotational Motion

Every physical quantity, and every governing equation, used across this entire chapter to describe the rotation of a rigid body about a fixed axis has an EXACT counterpart already met, earlier in the…

Summary

Centre of mass of a two-particle system, , extends to any system of particles or rigid body, , and need not lie inside the body's own material (e.g. a ring).

Sample & Board Papers

Sample papers and previous-year board questions for this subject.

+Show 2 questions2 questions
  1. Q1Define angular momentum (L⃗) and torque (τ⃗). Show that dL⃗/dt = τ⃗. **OR** A uniform solid sphere of mass M and radius R rolls down an incl…Preview
  2. Q2Define centre of mass. Write down its mathematical form. Two particles of masses 2g and 3g are situated at the positions (2 cm, 3 cm) and (4…Preview

More questions

26 Q
+Show 8 questions8 questions
  1. Example 1Two point masses $m_1 = 2\ \text{kg}$ and $m_2 = 3\ \text{kg}$ lie on the $x$-axis at $x_1 = 0$ and $x_2 = 5\ \text{m}$ respectively. Locate…Free
  2. Example 2Show, starting from the defining formula $\vec R = \dfrac{m_1\vec r_1 + m_2\vec r_2}{m_1+m_2}$ for a two-particle system, that the centre of…Free
  3. Example 3A thin uniform rod $AB$ of mass $3\ \text{kg}$ and length $1.2\ \text{m}$ has a small block of mass $1\ \text{kg}$ fixed rigidly at end $B$.…Free
  4. Example 4A shell of mass $10\ \text{kg}$ moving horizontally at $20\ \text{m/s}$ explodes in mid-air (an internal force only) into two fragments of m…Preview
  5. Example 5A force $\vec F = (3\hat i + 4\hat j)\ \text{N}$ acts at a point whose position vector, measured from the origin $O$, is $\vec r = (2\hat i…Preview
  6. Example 6A student sits on a rotating stool with arms outstretched, giving the student-plus-stool system a moment of inertia of $5\ \text{kg}\,\text{…Preview
  7. Example 7A light rigid rod $3\ \text{m}$ long is pivoted at its midpoint and can turn freely (like a see-saw). A weight of $20\ \text{N}$ hangs at a…Preview
  8. Example 8A uniform circular disc of mass $2\ \text{kg}$ and radius $0.5\ \text{m}$ has a moment of inertia $I = \tfrac12 MR^2$ about an axis through…Preview
+Show 10 questions10 questions
  1. Q9Define the centre of mass of a system of particles. Is the centre of mass of a rigid body always located within the material of the body its…Free
  2. Q10Distinguish between the centre of mass and the centre of gravity of a body. Under what condition on the surrounding gravitational field do t…Free
  3. Q11Two particles of masses $4\ \text{kg}$ and $6\ \text{kg}$ are $5\ \text{m}$ apart. Find the distance of their centre of mass from the $4\ \t…Free
  4. Q12In the Fosbury-flop technique, a high jumper arches the body over the bar so that the body's own centre of mass can actually pass just below…Preview
  5. Q13State the principle of conservation of angular momentum. Use it to explain why an ice skater who is spinning with arms outstretched spins no…Preview
  6. Q14Two forces of equal magnitude $F$, but opposite in direction, act along two separate parallel lines of action separated by a perpendicular d…Preview
  7. Q15Explain, with reference to a thin ring and a uniform disc of the same mass $M$ and the same radius $R$, spinning about an axis through the c…Preview
  8. Q16Define the radius of gyration of a rigid body about a given axis. A body of mass $8\ \text{kg}$ has a radius of gyration of $0.3\ \text{m}$…Preview
  9. Q17State the theorem of parallel axes. A uniform rod of mass $1.5\ \text{kg}$ and length $1.0\ \text{m}$ has a moment of inertia $I_{cm} = \tfr…Preview
  10. Q18State the theorem of perpendicular axes and the condition on the body under which it can be applied. For a uniform circular disc of mass $M$…Preview
+Show 8 questions8 questions
  1. Q19Three point masses of $1\ \text{kg}$, $2\ \text{kg}$ and $3\ \text{kg}$ are placed at the corners of an equilateral triangle of side $1\ \te…Free
  2. Q20A $60\ \text{kg}$ astronaut and a $15\ \text{kg}$ toolbox are initially at rest relative to each other, floating freely in space. The astron…Free
  3. Q21A force of $25\ \text{N}$ is applied at the end of a wrench of length $0.3\ \text{m}$ to turn a bolt. (a) Find the torque produced when the…Free
  4. Q22A flywheel of moment of inertia $4\ \text{kg}\,\text{m}^2$ rotates at $300$ revolutions per minute. Find (a) its angular velocity in $\text{…Preview
  5. Q23Calculate the moment of inertia of a solid sphere of mass $5\ \text{kg}$ and radius $0.2\ \text{m}$ about an axis through its centre. Also f…Preview
  6. Q24A uniform disc of mass $4\ \text{kg}$ and radius $0.25\ \text{m}$ spins at $10\ \text{rad/s}$ about a frictionless axle through its centre,…Preview
  7. Q25A wheel starts from rest and is given a constant angular acceleration of $2\ \text{rad/s}^2$ until it reaches an angular velocity of $20\ \t…Preview
  8. Q26A constant torque of $12\ \text{N}\,\text{m}$ is applied to a wheel of moment of inertia $3\ \text{kg}\,\text{m}^2$, initially at rest. Find…Preview