Class 11 Physics Chapter 6: System of Particles and Rotational Motion
Class 11 Physics Chapter 6: System of Particles and Rotational Motion
System of Particles and Rotational Motion is an important chapter of Class 11 Physics. It introduces the centre of mass, motion of a rigid body, torque, angular momentum, moment of inertia and rolling motion.
1. System of Particles
A system of particles is a collection of two or more particles considered together for studying their motion. The particles may interact with each other or may be acted upon by external forces.
Instead of studying every particle separately, we can often simplify the problem by considering the motion of the centre of mass.
2. Centre of Mass
The centre of mass of a system is the point where the entire mass of the system may be considered to be concentrated for describing its translational motion.
For two particles having masses m1 and m2 at positions x1 and x2:
For a system of many particles:
Similarly,
3. Motion of Centre of Mass
The velocity of the centre of mass of a system is related to the total momentum of the system.
where P is the total linear momentum and M is the total mass.
The motion of the centre of mass is determined by the net external force acting on the system.
Internal forces cannot change the motion of the centre of mass of an isolated system.
4. Rigid Body
A rigid body is an idealized body in which the distance between every pair of particles remains constant even when external forces act on it.
A rigid body can undergo:
- Translational motion
- Rotational motion
- Combined translational and rotational motion
5. Rotational Motion
In rotational motion, a body rotates about an axis and different particles of the body move along circular paths.
The basic angular quantities used to describe rotational motion are angular displacement, angular velocity and angular acceleration.
6. Angular Variables
Angular Displacement
Angular displacement is the change in angular position of a rotating body. It is represented by θ and measured in radians.
where s is the arc length and r is the radius.
Angular Velocity
Angular velocity is the rate of change of angular displacement.
For uniform rotation:
Angular Acceleration
Angular acceleration is the rate of change of angular velocity.
Relation Between Linear and Angular Quantities
Tangential acceleration is:
Centripetal acceleration is:
7. Equations of Rotational Motion
For constant angular acceleration, the equations of rotational motion are similar to the equations for linear motion.
These equations are useful in numerical problems involving uniformly accelerated rotational motion.
8. Torque
Torque is the turning effect of a force about a point or axis. It is also called the moment of force.
The magnitude of torque is:
where θ is the angle between the position vector and force.
The SI unit of torque is N m.
9. Angular Momentum
Angular momentum is the rotational analogue of linear momentum. For a particle:
The magnitude is:
For a rigid body rotating about a fixed axis:
where I is the moment of inertia and ω is angular velocity.
10. Relation Between Torque and Angular Momentum
The net external torque acting on a system is equal to the rate of change of angular momentum.
For a rigid body rotating about a fixed axis with constant moment of inertia:
This is the rotational form of Newton's second law.
11. Equilibrium of a Rigid Body
A rigid body is in mechanical equilibrium when there is no translational acceleration and no angular acceleration.
Therefore, the conditions are:
The first condition prevents translational acceleration, while the second prevents rotational acceleration.
12. Moment of Inertia
Moment of inertia is the rotational analogue of mass. It measures the resistance of a body to changes in its rotational motion.
For a system of particles:
The moment of inertia depends on:
- Mass of the body
- Distribution of mass
- Position of the axis of rotation
Its SI unit is kg m².
13. Radius of Gyration
The radius of gyration is the distance from the axis at which the entire mass of the body could be assumed to be concentrated to produce the same moment of inertia.
Therefore:
14. Parallel Axis Theorem
The parallel axis theorem relates the moment of inertia about an axis to the moment of inertia about a parallel axis passing through the centre of mass.
Here d is the perpendicular distance between the two parallel axes.
15. Perpendicular Axis Theorem
For a plane lamina, the moment of inertia about an axis perpendicular to its plane is equal to the sum of the moments of inertia about two mutually perpendicular axes in its plane passing through the same point.
16. Common Moments of Inertia
| Body | Axis | Moment of Inertia |
|---|---|---|
| Thin ring | Through centre, perpendicular to plane | I = MR² |
| Solid disc | Through centre, perpendicular to plane | I = ½MR² |
| Solid sphere | About diameter | I = ⅖MR² |
| Thin spherical shell | About diameter | I = ⅔MR² |
| Rod | Through centre, perpendicular to rod | I = 1/12 ML² |
| Rod | Through one end, perpendicular to rod | I = 1/3 ML² |
17. Rolling Motion
Rolling motion is a combination of translational and rotational motion. A wheel rolling on a surface has both types of motion.
For rolling without slipping:
The point of contact between the rolling body and the surface is instantaneously at rest relative to the surface.
Energy of a Rolling Body
The total kinetic energy of a rolling body is the sum of translational and rotational kinetic energies.
Quick Revision
| Concept | Formula |
|---|---|
| Centre of mass | xCM = Σmixi/Σmi |
| Centre of mass velocity | VCM = P/M |
| Angular velocity | ω = dθ/dt |
| Angular acceleration | α = dω/dt |
| Linear-angular relation | v = rω |
| Torque | τ = rF sinθ |
| Angular momentum | L = Iω |
| Rotational equation | τ = Iα |
| Moment of inertia | I = Σmr² |
| Radius of gyration | I = Mk² |
| Parallel axis theorem | I = ICM + Md² |
| Rolling without slipping | v = Rω |
Important Questions – System of Particles and Rotational Motion
Very Short Answer Questions
- What is a system of particles?
- Define centre of mass.
- Write the expression for the centre of mass of two particles.
- What is a rigid body?
- Define angular displacement.
- What is angular velocity?
- Define angular acceleration.
- What is torque?
- Define angular momentum.
- What is moment of inertia?
- Define radius of gyration.
- State the parallel axis theorem.
- State the perpendicular axis theorem.
- What is rolling without slipping?
Short Answer Questions
- Explain the concept of centre of mass with an example.
- How is the motion of the centre of mass affected by external forces?
- Differentiate between translational and rotational motion.
- Explain the relation between linear velocity and angular velocity.
- Why does a force produce torque about an axis?
- Explain the difference between torque and force.
- What factors affect the moment of inertia of a body?
- Why does the moment of inertia depend on the position of the axis?
- Explain the conservation of angular momentum with an example.
- Why does a rolling body possess both translational and rotational kinetic energy?
Long Answer Questions
- Derive the expression for the centre of mass of a system of particles.
- Derive the relation between torque and angular acceleration.
- Explain angular momentum and derive L = Iω for a rigid body rotating about a fixed axis.
- Explain the parallel axis theorem and write its mathematical expression.
- Explain the perpendicular axis theorem with suitable conditions.
- Derive the expression for kinetic energy of a rolling body.
- Explain the conditions required for the equilibrium of a rigid body.
Important MCQs
-
The centre of mass of a uniform circular disc lies at:
(a) circumference (b) centre (c) any point (d) outside the disc
Answer: (b) centre -
The SI unit of angular velocity is:
(a) m/s (b) rad/s (c) N m (d) kg m²
Answer: (b) rad/s -
Torque is given by:
(a) F/r (b) rF sinθ (c) r/F (d) mv
Answer: (b) rF sinθ -
The SI unit of moment of inertia is:
(a) kg m (b) kg m² (c) N m (d) kg/s
Answer: (b) kg m² -
The rotational analogue of mass is:
(a) torque (b) angular momentum (c) moment of inertia (d) angular velocity
Answer: (c) moment of inertia -
For rolling without slipping:
(a) v = R/ω
(b) v = Rω
(c) v = ω/R
(d) v = R²ω
Answer: (b) v = Rω -
For a rigid body in equilibrium, the net torque is:
(a) maximum (b) zero (c) infinite (d) negative
Answer: (b) zero -
The moment of inertia of a solid disc about its central axis is:
(a) MR² (b) ½MR² (c) ⅖MR² (d) ⅔MR²
Answer: (b) ½MR²
Numerical Problems
Numerical 1: Centre of Mass
Two particles of masses 2 kg and 3 kg are placed at positions 0 m and 5 m respectively on the x-axis. Find the position of their centre of mass.
= 15/5
= 3 m
Answer: 3 m from the origin.
Numerical 2: Angular Velocity
A wheel rotates through an angle of 20 rad in 5 seconds. Find its average angular velocity.
ω = 20/5
ω = 4 rad/s
Answer: 4 rad/s
Numerical 3: Torque
A force of 10 N acts perpendicular to a lever arm of length 0.5 m. Find the torque.
τ = 0.5 × 10
τ = 5 N m
Answer: 5 N m
Numerical 4: Moment of Inertia
Calculate the moment of inertia of a thin ring of mass 2 kg and radius 0.5 m about an axis through its centre and perpendicular to its plane.
I = 2 × (0.5)²
I = 0.5 kg m²
Answer: 0.5 kg m²
Numerical 5: Angular Momentum
A body has moment of inertia 4 kg m² and angular velocity 3 rad/s. Find its angular momentum.
L = 4 × 3
L = 12 kg m²/s
Answer: 12 kg m²/s
Numerical 6: Rolling Motion
A wheel of radius 0.2 m rolls without slipping with an angular velocity of 10 rad/s. Find the linear velocity of its centre.
v = 0.2 × 10
v = 2 m/s
Answer: 2 m/s
Assertion and Reason Questions
-
Assertion: The centre of mass of a uniform symmetrical body lies at its geometrical centre.
Reason: The mass distribution is symmetrical about the geometrical centre.
Answer: Both Assertion and Reason are true, and Reason correctly explains Assertion. -
Assertion: Torque is zero when the line of action of force passes through the axis of rotation.
Reason: The perpendicular distance between the axis and line of action of force is zero.
Answer: Both Assertion and Reason are true, and Reason correctly explains Assertion. -
Assertion: A body can have zero angular velocity but non-zero angular acceleration.
Reason: Angular acceleration represents the rate of change of angular velocity.
Answer: Both Assertion and Reason are true, and Reason correctly explains Assertion. -
Assertion: A rolling body has both translational and rotational kinetic energy.
Reason: The centre of mass translates while the body rotates about its centre of mass.
Answer: Both Assertion and Reason are true, and Reason correctly explains Assertion.
Exam Tips
- Learn the centre of mass formula and practice numerical problems.
- Remember the difference between angular displacement, velocity and acceleration.
- Understand torque using the perpendicular distance from the axis.
- Memorise standard moments of inertia of common bodies.
- Practice both parallel and perpendicular axis theorem problems.
- Remember that rotational equilibrium requires net torque to be zero.
- For rolling without slipping, always remember v = Rω.
- Do not confuse torque with work; both have the same dimensions but are different physical quantities.
Class 11 Physics Chapter 6 FAQs
1. What is Chapter 6 of Class 11 Physics?
Chapter 6 is System of Particles and Rotational Motion. It covers centre of mass, rigid body motion, torque, angular momentum, moment of inertia, rotational equilibrium and rolling motion.
2. What is the centre of mass?
The centre of mass is the point representing the average position of mass distribution of a system and is useful for describing its translational motion.
3. What is the formula for centre of mass?
For particles along the x-axis, the centre of mass is given by xCM = Σmixi/Σmi.
4. What is torque?
Torque is the turning effect of a force about an axis. Its magnitude is τ = rF sinθ.
5. What is angular momentum?
Angular momentum is the rotational analogue of linear momentum. For a rigid body rotating about a fixed axis, L = Iω.
6. What is moment of inertia?
Moment of inertia measures the resistance of a body to changes in its rotational motion. For a system of particles, I = Σmr².
7. What is the SI unit of moment of inertia?
The SI unit of moment of inertia is kg m².
8. What is the parallel axis theorem?
The parallel axis theorem states that I = ICM + Md², where d is the distance between the parallel axes.
9. What is rolling without slipping?
Rolling without slipping occurs when the point of contact of a rolling body with the surface is instantaneously at rest relative to the surface. The condition is v = Rω.
10. What are the conditions for equilibrium of a rigid body?
For complete mechanical equilibrium, the net external force and net external torque must both be zero: ΣF = 0 and Στ = 0.
Chapter 6 Formula Sheet
| Topic | Formula |
|---|---|
| Centre of mass | xCM = Σmixi/Σmi |
| CM velocity | VCM = P/M |
| CM equation | Fext = MaCM |
| Angular velocity | ω = dθ/dt |
| Angular acceleration | α = dω/dt |
| Linear velocity | v = rω |
| Tangential acceleration | at = rα |
| Centripetal acceleration | ac = rω² |
| Torque | τ = rF sinθ |
| Angular momentum | L = Iω |
| Rotational dynamics | τ = Iα |
| Moment of inertia | I = Σmr² |
| Radius of gyration | I = Mk² |
| Parallel axis theorem | I = ICM + Md² |
| Perpendicular axis theorem | Iz = Ix + Iy |
| Rolling condition | v = Rω |
| Rolling kinetic energy | K = ½Mv² + ½Iω² |
One-Minute Revision
- Centre of mass: Represents the mass distribution of a system.
- CM motion: Fext = MaCM.
- Angular velocity: ω = dθ/dt.
- Angular acceleration: α = dω/dt.
- Linear-angular relation: v = rω.
- Torque: τ = rF sinθ.
- Angular momentum: L = Iω.
- Rotational dynamics: τ = Iα.
- Moment of inertia: I = Σmr².
- Radius of gyration: I = Mk².
- Parallel axis: I = ICM + Md².
- Rolling: v = Rω.
Conclusion
Class 11 Physics Chapter 6 – System of Particles and Rotational Motion is a major foundation for understanding rotational mechanics. Students should give special attention to centre of mass, torque, angular momentum, moment of inertia, axis theorems and rolling motion.
For CBSE preparation, combine conceptual understanding with regular numerical practice because many questions in this chapter require correct identification of the axis, force, distance and rotational quantity.
Comments
Post a Comment
Thanks for your comments