Chapter 4 · Class 11-12 & JEE/NEET
Centre of Mass & Collisions
27 topics in 9 parts
Centre of Mass of Discrete Particles Centre of Mass of Continuous Bodies Centre of Mass of Uniform Rod Composite Body & Cavity Method Velocity of Centre of Mass Acceleration of Centre of Mass
Key ideas
4. Systematic Comparison & Parameter Classifications
| Body Shape | Axis of Rotation | Moment of Inertia (I) | Shape Factor (k² / R²) |
|---|---|---|---|
| Thin Ring / Hollow Cylinder | Central Axis Perpendicular to Plane | M R² | 1.0 (Slowest Down Incline) |
| Solid Disc / Solid Cylinder | Central Axis Perpendicular to Plane | (1/2) M R² | 0.50 |
| Solid Sphere | Diameter Axis | (2/5) M R² | 0.40 (Fastest Down Incline) |
| Spherical Shell (Hollow) | Diameter Axis | (2/3) M R² | 0.67 |
| Uniform Rod (Length L) | Through Center Perpendicular to Length | (1/12) M L² | L² / (12 R²) |
Exam Worked Example
Problem: A solid sphere and a solid cylinder of equal mass M and radius R roll down the same inclined plane without slipping from rest. Find the ratio of their accelerations down the incline.
● Step 1: Formula for linear acceleration in pure rolling: a = (g sin θ) / (1 + k²/R²).
● Step 2: For solid sphere, k²/R² = 2/5 = 0.4. Therefore, a_sphere = (g sin θ) / (1 + 2/5) = 5/7 g sin θ ≈ 0.714 g sin θ.
● Step 3: For solid cylinder, k²/R² = 1/2 = 0.5. Therefore, a_cyl = (g sin θ) / (1 + 1/2) = 2/3 g sin θ ≈ 0.667 g sin θ.
● Step 4: Ratio a_sphere / a_cyl = (5/7) / (2/3) = (5 * 3) / (7 * 2) = 15 / 14 ≈ 1.07. The solid sphere reaches the bottom first!
Checkpoint
1. When a ballet dancer pulls her arms inward during a spin on smooth ice, what happens to her kinetic energy?
All 27 topics in Centre of Mass & Collisions
The Centre of Mass
- System of N Discrete Particles
- Continuous Mass Distribution
- Uniform Rod
- Composite Bodies & Cavity Concept
Motion of the Centre of Mass
- Velocity & Acceleration of Centre of Mass
- Shift in Position of Centre of Mass
- Man-Plank & Boat System
Conservation of Linear Momentum
- Internal Forces & System Momentum
- Explosion of Bodies
- Centre of Mass Frame & Reduced Mass
- Work-Energy Theorem Relative to Centre of Mass
Impulse
- Instantaneous Force & Impact
- Momentum-Impulse Relation
- Force-Time Graphs
Collision in One Dimension (Head-on)
- Elastic Collision
- Inelastic & Perfectly Inelastic Collision
- Energy Loss in Collisions
Collision in Two Dimensions (Oblique)
- Oblique Impact of Smooth Spheres
- Deflection Angle & Kinetic Energy Transfer
Coefficient of Restitution
- Line of Impact
- Velocity of Approach & Separation
- Rebounds from Fixed Surface
Collisions with Springs, Wedges & Pendulums
- Ballistic Pendulum
- Collision with Wedge
- Collision with Spring-Mass System
Variable Mass System & Rocket Propulsion
- Rocket Propulsion
- Variable Mass Dynamics