Chapter 12 · Class 11-12 & JEE/NEET
Kinematics
44 topics in 16 parts
SUVAT — velocity SUVAT — displacement Projectile range Projectile max height Distance & Displacement Speed & Velocity
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 44 topics in Kinematics
Motion in a Straight Line
- Uniformly Accelerated Motion
- Equations of Motion
- Variable Acceleration
- Calculus-Based Kinematics
- Reaction Time & Braking
Displacement & Distance
- Distance vs Displacement
- Path Length
Velocity & Speed
- Average Velocity & Average Speed
- Instantaneous Speed
- Instantaneous Velocity
Acceleration
- Average Acceleration
- Instantaneous Acceleration
Graphs in Motion in One Dimension
- Acceleration-Displacement Graph
- Acceleration-Time Graph
- Graphical Representation of Motion
- Motion Diagrams
- Position-Time Graph
- Velocity-Displacement Graph
- Velocity-Time Graph
Motion Under Gravity
- Free Fall
- Vertical Motion
Motion Upon an Inclined Plane
- Motion Along Inclined Lines
Frame of Reference
- Non-Inertial Frames
- Inertial Frames
Motion in a Plane
- Two- & Three-Dimensional Motion
- 2D & 3D Kinematics
Position Vector, Velocity & Acceleration
- Position Vector
Trajectory
- Equation of Trajectory
Projectile Motion
- Collisions & Obstacles
- Projection on Inclined Plane
- Ground & Elevation Projection
Relative Motion
- Basic Relative Motion
- 1D Relative Motion
- Velocity of Approach & Separation
- River Crossing
- Rain & Wind
- Closest Approach
Constrained Motion
- String Constraints
- Rigid-Body Constraints
Vectors
- Vector Addition
- Dot Product
Kinematics of Circular Motion
- Angular Kinematics
- Constant Angular Acceleration
- Tangential Acceleration