Chapter 5 · Class 11-12 & JEE/NEET
Circular & Curvilinear Motion
33 topics in 7 parts
Angular Displacement Angular Velocity Angular Acceleration Constant Angular Acceleration Equations Radius of Curvature Linear & Angular Velocity Relation
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 33 topics in Circular & Curvilinear Motion
Kinematics of Circular Motion
- Angular Acceleration
- Angular Position
- Average Angular Velocity
- Important Terminology Related to Circular Motion
- Motion with Constant Angular Acceleration
- Radius of Curvature
- Relation Between Linear & Angular Quantities
- Relative Angular Velocity
- Uniform & Non-Uniform Circular Motion
- Centripetal & Tangential Acceleration
Dynamics of Circular Motion
- Centrifugal Force
- Centripetal Force
- Motion in Rotating Frame & Pseudo Force
Circular Motion in Horizontal Plane
- Motion on Flat Circular Track & Friction
- Conical Pendulum
- Motion on Rotating Turntable & Disc
- Rough Rotating Cone
- Motion Inside Cylindrical Drum & Hoop
Banking of Road
- Optimum Speed & Angle of Banking
- Maximum & Minimum Safe Speed with Friction
- Skidding & Overturning Thresholds
Vertical Circular Motion
- String & Light Rod Pendulum in Vertical Circle
- Critical Velocity & Minimum Speed at Lowest/Highest Point
- Tension & Normal Reaction in Vertical Circle
- Conditions for Oscillating, Leaving Circular Path & Completing Loop
Applications on Vertical Circular Motion
- Motion Inside Smooth Spherical Shell & Ring
- Roller Coaster Loop the Loop
- Motion on Spherical Surface & Hemisphere
- Motion on Inclined Plane
Work, Energy & Power in Circular Motion
- Work Done by Centripetal & Tangential Forces
- Power Delivered
- Conservation of Mechanical Energy
- Work-Energy Theorem on Curved Paths