CBSE Class 11 Physics Chapter 7 Notes: System of Particles and Rotational Motion

Chapter Introduction

System of Particles and Rotational Motion is the rotational analogue of Chapter 5 (Laws of Motion). Every linear concept — force, momentum, kinetic energy, Newton's laws — has a rotational counterpart — torque, angular momentum, rotational KE, and rotational laws. This chapter carries 8–10 marks in the board exam and is crucial for JEE/NEET. The moment of inertia formulas and the two axes theorems are guaranteed exam material.

What this guide covers: centre of mass, motion of COM, linear momentum of a system, torque, angular momentum, moment of inertia (formulas for all common shapes), parallel and perpendicular axes theorems, rolling motion (pure rolling, slipping), and conservation of angular momentum — all with worked numericals.

Key Definitions

• Centre of mass (COM): The point that moves as though all the mass of the system were concentrated there and all external forces acted on it. For two particles: X_cm = (m₁x₁ + m₂x₂) / (m₁ + m₂).
• Centre of gravity: The point where the total gravitational torque on the body is zero. Coincides with COM in a uniform gravitational field.
• Torque (τ): The rotational analogue of force. τ = r × F. Magnitude = rF sin θ. SI unit: N m. Vector, perpendicular to the plane of r and F (right-hand rule).
• Angular velocity (ω): Rate of change of angular position. ω = dθ/dt. Vector along the axis of rotation. SI unit: rad s^-1.
• Angular acceleration (α): Rate of change of angular velocity. α = dω/dt. SI unit: rad s^-2.
• Angular momentum (L): Rotational analogue of linear momentum. L = r × p = Iω. SI unit: kg m² s^-1.
• Moment of inertia (I): Rotational analogue of mass. I = Σm_ir_i² for discrete system; I = ∫r²dm for continuous body. SI unit: kg m². Depends on mass distribution and axis of rotation.
• Radius of gyration (k): k = √(I/M). The distance from the axis at which the entire mass could be concentrated to give the same moment of inertia.
• Rolling: Combined translational and rotational motion. Pure rolling means no slipping at the contact point.
• Couple: A pair of equal and opposite forces whose lines of action do not coincide. A couple produces pure rotation (net force = 0, net torque ≠ 0).

Detailed Concepts

1. Centre of Mass

Two-particle system: X_cm = (m₁x₁ + m₂x₂) / (m₁ + m₂) Y_cm = (m₁y₁ + m₂y₂) / (m₁ + m₂)

N-particle system: R_cm = (Σm_ir_i) / (Σm_i) = (Σm_ir_i) / M

Rigid body (continuous): R_cm = (1/M) ∫r dm

COM for common shapes (uniform density):

Motion of COM: F_ext = Ma_cm. The COM moves as if all external forces act on a single particle of mass M at the COM. Internal forces do not affect the COM.

Important: If no external force acts, the COM moves with constant velocity (or stays at rest). This is true even if individual parts of the system accelerate (e.g., an exploding firecracker — COM continues on the same parabolic path).

2. Linear Momentum of a System

Total linear momentum: P = Mv_cm

If F_ext = 0, then dP/dt = 0 → P = constant (conservation of linear momentum).

This means the COM moves with constant velocity when no external force acts.

3. Torque and Angular Momentum

Torque: τ = r × F. Turns the body about an axis.

Angular momentum of a particle: L = r × p, where p = mv.

Angular momentum of a rigid body rotating about a fixed axis: L = Iω

Newton's second law for rotation: τ_ext = dL/dt = Iα (when I is constant)

Conservation of angular momentum: If τ_ext = 0, then L = Iω = constant.

Application: A spinning skater pulls in arms → I decreases → ω increases (conservation of L). A diver in a tuck position spins faster than in an extended position.

Relation between linear and angular quantities:

Couple and torque: A couple is two equal and opposite forces separated by a distance d. Torque of a couple = F × d (independent of the reference point).

4. Moment of Inertia — Formulas

5. Parallel and Perpendicular Axes Theorems

Parallel axes theorem: I = I_cm + Md² where I_cm = moment of inertia about an axis through the centre of mass, d = perpendicular distance between the two parallel axes.

Example: I_end for a rod = I_cm + M(L/2)² = (1/12)ML² + (1/4)ML² = (1/3)ML²

Example: I_tangent for a disc = I_cm + MR² = ½MR² + MR² = (3/2)MR² (tangent ∥ to diameter → d = R → actually I_tangent = I_diameter + MR² = ¼MR² + MR² = 5/4 MR²)

Perpendicular axes theorem (planar bodies only): I_z = I_x + I_y where x, y are two perpendicular axes in the plane of the body, and z is perpendicular to the plane through the intersection point.

Example: For a disc, I_z = ½MR² and I_x = I_y (by symmetry) → I_x = I_y = ¼MR².

Important: The perpendicular axes theorem applies ONLY to planar (flat, 2D) bodies like discs, plates, and rings. It does NOT apply to spheres, cylinders, or 3D objects.

6. Rolling Motion

Pure rolling (no slipping): The point of contact is instantaneously at rest.

General formula for total KE in pure rolling: KE = ½Mv²(1 + I/(MR²))

Total KE for different shapes:

Rolling on an incline (no slipping, no friction loss):

Race down an incline: The solid sphere reaches the bottom first, then the disc, then the hollow sphere, and the ring last. This is because a smaller I/MR² means less energy goes into rotation and more into translation.

Rolling with slipping: When v ≠ ωr, kinetic friction acts to bring the body to pure rolling. Direction of friction:

• If v > ωr (sliding more than rolling): friction acts backward (opposes translation, aids rotation)
• If v < ωr (rolling more than sliding): friction acts forward (aids translation, opposes rotation)

Time to achieve pure rolling: t = (v − ωR) / (μg(1 + MR²/I)) — derived from simultaneous translation and rotation equations.

Important Numericals

Numerical 1: Centre of Mass

Two masses of 3 kg and 7 kg are placed at x = 2 m and x = 8 m respectively. Find the COM.

Numerical 2: Torque

A force F = 2î + 3ĵ N acts at point r = 1î + 2ĵ m. Find the torque about the origin.

Numerical 3: Moment of Inertia (Parallel Axes)

Find the moment of inertia of a solid sphere of mass 2 kg and radius 0.1 m about a tangent.

Numerical 4: Rolling on Incline

A disc rolls without slipping down an incline of 30°. Find the acceleration. (g = 10 m s^-2)

Numerical 5: Conservation of Angular Momentum

A disc of moment of inertia 0.2 kg m² rotating at 10 rad s^-1 has a ring of I = 0.1 kg m² dropped on it (no external torque). Find the new angular velocity.

Numerical 6: Total KE in Rolling

A solid sphere of mass 2 kg and radius 0.1 m rolls without slipping at 5 m s^-1. Find its total KE.

NCERT Exercise Reference

Common Mistakes

1. Confusing I = MR² with I = ½MR²: MR² is for a ring (all mass at distance R); ½MR² is for a disc (mass distributed from 0 to R). Always match the formula to the shape. 2. Applying perpendicular axes theorem to 3D bodies: It only works for planar (flat) bodies like discs, plates, and rings. Do not use it for spheres or cylinders. 3. Forgetting the +Md² term: When shifting axis from COM, I increases by Md², not decreases. I is always minimum about an axis through the COM. 4. Wrong rolling condition: v = ωr, not v = ω/r or v = r/ω. The point of contact is at rest (v_contact = v − ωR = 0). 5. Angular momentum conservation without checking torque: L is conserved only when τ_ext = 0, not just because the system is isolated from external forces (force and torque are different). 6. Not distinguishing between rolling with and without slipping: The formulas for acceleration on an incline (a = g sin θ / (1 + I/MR²)) apply only for pure rolling. 7. Torque direction: Use the right-hand rule for τ = r × F. Curl fingers from r to F; thumb gives the direction of τ. 8. I is always about an axis: Never say "the moment of inertia of a body" without specifying the axis. I changes with the axis.

Exam Tips

• Moment of inertia formulas for at least 5 shapes are essential — ring, disc, rod (centre and end), solid sphere, hollow sphere. These are 1-mark questions.
• Parallel axes theorem derivation and application is a guaranteed 3-mark question. Be able to derive I_end for a rod and I_tangent for a sphere from I_cm.
• Conservation of angular momentum numericals appear as 2–3 mark questions — remember the condition (τ_ext = 0).
• Rolling on incline: know the formula for acceleration a = g sin θ / (1 + I/MR²) and the result that solid sphere > disc > ring in order of reaching the bottom.
• For torque problems, always identify the pivot (axis) and compute τ = r × F carefully, using the perpendicular distance.
• Draw diagrams for rolling problems — show the point of contact, direction of friction, and the two types of KE.
• Perpendicular axes theorem: practise deriving I_diameter for a disc from I_⊥ = ½MR².
• For numericals involving I about tangent, always apply parallel axes theorem: I_tangent = I_cm + MR² (if tangent is at distance R from COM).

Quick Revision Notes

• COM: R_cm = Σm_ir_i / M. Moves as if all mass + all forces are at one point.
• Motion of COM: F_ext = Ma_cm. Internal forces do not affect COM.
• Torque: τ = r × F. Rotational analogue of force. SI unit: N m. Right-hand rule for direction.
• Angular momentum: L = Iω = r × p. Conserved when τ_ext = 0.
• Newton's 2nd law (rotation): τ = Iα = dL/dt.
• MOI formulas: Ring MR², Disc ½MR², Solid sphere (2/5)MR², Hollow sphere (2/3)MR², Rod (1/12)ML².
• Parallel axes: I = I_cm + Md². I is minimum about COM axis.
• Perpendicular axes: I_z = I_x + I_y (planar bodies only).
• Radius of gyration: k = √(I/M).
• Pure rolling: v = ωr. KE = ½Mv² + ½Iω² = ½Mv²(1 + I/MR²).
• Rolling on incline: a = g sin θ / (1 + I/MR²). Solid sphere wins the race.
• Rolling with slipping: friction acts to bring v = ωR; forward if v < ωR, backward if v > ωR.
• Couple: Two equal and opposite forces; torque = F × d, independent of pivot.

Important One-Mark Facts for Quick Recall

1. The moment of inertia is minimum about an axis through the centre of mass (I_cm). 2. When you shift an axis away from the COM, I always increases by Md² (parallel axes theorem). 3. The perpendicular axes theorem applies only to planar bodies (disc, plate, ring) — NOT to spheres or cylinders. 4. In pure rolling, the point of contact is instantaneously at rest (v_contact = v − ωR = 0). 5. A solid sphere rolls faster down an incline than a disc, which rolls faster than a ring — because less energy goes into rotation when I/MR² is small. 6. Angular momentum is conserved when τ_ext = 0, not when F_ext = 0 (force and torque are different). 7. A spinning skater spins faster when pulling arms in because I decreases and L = Iω is conserved. 8. Torque = r × F; its direction is given by the right-hand rule. 9. For a uniform circular disc: I_⊥ = ½MR², I_diameter = ¼MR² (from perpendicular axes theorem). 10. The total KE of a rolling body = ½Mv²(1 + I/MR²). For a ring this equals Mv²; for a disc, ¾Mv². 11. Rolling friction < kinetic friction < static friction: μ_r < μ_k < μ_s. 12. If a body rolls with slipping, friction acts to bring it to pure rolling: backward if v > ωR, forward if v < ωR. 13. The radius of gyration k = √(I/M). For a ring, k = R; for a disc (⊥ axis), k = R/√2. 14. The centre of mass of a uniform solid cone is at h/4 from the base on the axis. 15. A couple produces pure rotation (net force = 0, net torque ≠ 0).