Chapter 21 · Class 11-12 & JEE/NEET

Rotational Motion

61 topics in 13 parts

Full lessonClass 11 Physics · System of Particles and Rotational Motion Practice280 practice questions

Key formulas

Torque & Newton's Second Law for Rotation
$\vec{\tau} = \vec{r} \times \vec{F} = I \vec{\alpha}, \quad \tau = \frac{d\vec{L}}{dt}$
Angular Momentum & Conservation
$\vec{L} = \vec{r} \times \vec{p} = I \vec{\omega}, \quad \sum \vec{\tau}_{ext} = 0 \implies I_1 \omega_1 = I_2 \omega_2$
Parallel & Perpendicular Axis Theorems
$I = I_{cm} + M d^2, \quad I_z = I_x + I_y \quad (\text{for planar 2D lamina})$
Total Kinetic Energy in Pure Rolling
$K_{total} = \frac{1}{2} M v_{cm}^2 + \frac{1}{2} I_{cm} \omega^2 = \frac{1}{2} M v_{cm}^2 \left( 1 + \frac{k^2}{R^2} \right)$

Torque Angular Kinematics Linear-Angular Relations Torque Moment of Inertia Radius of Gyration

Key ideas

Rigid Body Rolling Down an Inclined Plane: Dynamics & Friction Direction

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 61 topics in Rotational Motion

Rigid Body Kinematics
- Basic Properties of Rigid Body
- General Rigid Body Motion
- Combined Rotation & Translation
- Velocity & Acceleration of Points
- Instantaneous Centre of Rotation
Angular Kinematics
- Angular Position, Velocity & Acceleration
- Constant Angular Acceleration
Moment of Inertia
- Standard Rigid Bodies
- Composite Bodies & Cutouts
- Triangular Lamina About Its Base
- Uniform Rectangular Lamina
- Parallel Axis Theorem
- Perpendicular Axis Theorem
Radius of Gyration
- Relation With Moment of Inertia
- Standard Bodies
- Composite Systems
Torque & Couple
- Moment of a Force
- Direction of Torque
- Couple
- Point of Application of Force
Rotational Dynamics
- General Rigid Body Dynamics
- Torque & Angular Acceleration
- Rotation About a Fixed Axis
- Falling & Hinged Rigid Bodies
- Physical Pendulum
Equilibrium of Rigid Bodies
- Translational & Rotational Equilibrium
- Beams, Rods & Ladders
- Friction in Equilibrium
- Buoyancy & Rotational Equilibrium
Toppling
- Tipping Condition
- Minimum Force for Toppling
Rotational Work, Energy & Power
- Rotational Kinetic Energy
- Kinetic Energy in Combined Rotation & Translation
- Work Done by Torque
- Rotational Power
- Work-Energy Theorem
- Conservation of Mechanical Energy
Angular Momentum & Angular Impulse
- About a Fixed Axis
- Free Rigid Body
- Particle
- Impulse
- Conservation
- Thermal Effects on Rotation
Rotational Collisions
- Particle-Rigid Body Collisions
- Bodies Coupling on a Common Axis
- Impulsive Rotation
- One-Dimensional Collision
Rolling Motion
- Pure Rolling
- Rolling With Slipping
- Rolling With Friction
- Direction of Friction
- Rolling on an Inclined Plane
- Rolling on a Moving Surface
- Energy in Rolling
- Kinematics of Points on a Rolling Body
- Rolling on a Loop
- Rolling Body Oscillations
- Compound Rolling Systems
Gear, Belt & Pulley Systems
- Gear Systems
- Belt & Pulley Systems
- String & Pulley Constraints