Chapter 6: System of Particles and Rotational Motion – Class 11 Physics

Chapter 6 marks a major conceptual shift in mechanics. Earlier, motion was studied using the particle model. In this chapter, we extend mechanics to systems of particles and rigid bodies, laying the foundation for complete rotational dynamics.

This chapter can be divided into two major parts:

1. Mechanics of a System of Particles (Translational Analysis)
2. Rotational Motion of Rigid Bodies

It is one of the most important chapters for JEE Main, JEE Advanced, and NEET.

6.1 Introduction

Real objects possess finite size and internal structure. Therefore, their motion cannot always be described using the ideal particle model.

This section introduces:

• Difference between a particle and a rigid body
• Translational motion
• Rotational motion
• Pure translation
• Pure rotation
• Combined translation and rotation
• Fixed axis rotation

In pure translational motion, every particle of a body has the same velocity at a given instant.

In rotational motion about a fixed axis:

• Every particle moves in a circular path
• The plane of motion is perpendicular to the axis
• All particles share the same angular velocity

Rolling motion combines:

• Translation of centre of mass
• Rotation about centre of mass

6.2 Centre of Mass

The centre of mass (CM) simplifies the study of multi-particle systems by reducing translational motion to an equivalent single-particle description.

For a discrete system in one dimension:

In vector form:

where .

For continuous mass distribution:

Key Points:

• CM is the mass-weighted average position
• For symmetric homogeneous bodies, CM lies at geometric centre
• CM motion depends only on external forces

6.3 Motion of Centre of Mass

The fundamental equation governing CM motion is:

This implies:

The centre of mass moves as if the entire mass were concentrated at that point and all external forces acted there.

Important Results:

• Internal forces cancel due to Newton's Third Law
• Only external forces influence CM acceleration
• If , then is constant.

6.4 Linear Momentum of a System of Particles

Total linear momentum:

Relation with centre of mass velocity:

Conservation of Linear Momentum:

If ,

Applications include collisions, explosions, recoil, and fragmentation problems.

6.5 Vector Product of Two Vectors

The cross product is defined as:

Magnitude:

Properties:

This operation is essential in defining torque and angular momentum.

6.6 Angular Velocity and Its Relation with Linear Velocity

Angular velocity:

Linear velocity of a particle in a rotating body:

Angular acceleration:

6.7 Torque and Angular Momentum

Torque (Moment of Force):

Magnitude:

Angular Momentum:

Fundamental relation:

6.8 Equilibrium of a Rigid Body

Conditions for equilibrium:

Both translational and rotational equilibrium must be satisfied simultaneously.

6.9 Moment of Inertia

Moment of inertia is the rotational analogue of mass.

Definition:

Continuous form:

Parallel Axis Theorem:

Perpendicular Axis Theorem:

6.10 Kinematics of Rotational Motion About a Fixed Axis

Rotational analogues of linear motion equations:

Connections between linear and angular quantities:

6.11 Dynamics of Rotational Motion About a Fixed Axis

The fundamental equation of rotational dynamics is:

This is the rotational analogue of Newton's Second Law.

6.12 Angular Momentum in Case of Rotation About a Fixed Axis

For fixed axis rotation:

If external torque is zero:

This is the principle of conservation of angular momentum.

FAQs

Q1. What is the physical meaning of centre of mass?

The centre of mass is the effective point where the entire mass of a system can be assumed to be concentrated for analyzing translational motion.

Q2. Why do internal forces not affect centre of mass motion?

Internal forces cancel in equal and opposite pairs due to Newton's Third Law, so only external forces determine .

Q3. What is the difference between torque and force?

Force produces linear acceleration, whereas torque produces angular acceleration.

Q4. What is the rotational analogue of Newton's Second Law?

The rotational analogue is .

Q5. When is angular momentum conserved?

Angular momentum is conserved when external torque is zero, i.e., .

Conclusion

Chapter 6 establishes the complete bridge between translational and rotational mechanics. Beginning with the centre of mass concept, it develops conservation of linear momentum, introduces vector products, defines torque and angular momentum, and culminates in the fundamental rotational dynamics equation .

Mastery of this chapter is essential for solving rolling motion problems, angular momentum conservation problems, and advanced rigid body dynamics in competitive examinations like JEE Main and JEE Advanced.