Chapter 11 · Class 11-12 & JEE/NEET
Gravitation
56 topics in 14 parts
Escape velocity Newton's Law of Gravitation Vector Form of Gravitational Force Superposition of Gravity Gravitational Field Gravitational Potential
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 56 topics in Gravitation
Kepler's Laws
- Law of Orbits
- Law of Areas & Angular Momentum
- Law of Periods
- Elliptical Orbits
Newton's Laws Gravitation
- Definition of G
- Universal Law of Gravitation
- Inverse-Square Law
- Derivation from Kepler's Laws
- Experimental Evidence
Principle of Superposition of Gravitation
- Discrete Mass Systems
- Continuous Mass Distributions
Inertial & Gravitational Masses
- Inertial Mass
- Gravitational Mass
- Equivalence Principle
- Significance of the Equivalence Principle
Gravitational Field
- Field Due to Point Masses & Systems
- Neutral Points
- Field-Distance Graphs
Gravitation Due to Extended Bodies
- Spherical Shells
- Solid Spheres
- Rings & Rods
- Nonuniform Mass Distributions
Gravity
- Gravitation vs Gravity
- Mass & Density of Earth
- Surface Gravity
- Variation with Height, Depth & Latitude
- Weight on Planets
- Pendulums in Varying Gravity
Gravitational Potential
- Potential at a Point
- Potential Due to Earth
- Relation Between Field & Potential
- Potential of Spherical Shells & Solid Spheres
Gravitational Potential Energy
- Expression for Gravitational Potential Energy
- Gravitational Self-Energy
- Self-Energy of Particle Systems
- Work & Energy Conservation
Escape Velocity
- Escape Speed
- Escape from Planets & Heavenly Bodies
- Projection Speed & Maximum Height
- Speed at Infinity
Satellites
- Launching & Injection
- Orbital Velocity
- Orbital Period
- Orbital Acceleration
- Orbital Energy & Binding Energy
- Orbital Transfer
- Relative Motion
- Weightlessness
Geostationary & Geosynchronous Satellite
- Height & Time Period
- Equatorial Orbits & Communication Satellites
Tunnel SHM
- Diameter Tunnels
- Chord Tunnels
- Time Period & Oscillation Inside Earth
Binary Star Systems
- Motion About Centre of Mass
- Mutual Circular Orbits
- Three-Body & N-Body Orbits