Chapter 19 · Class 11-12 & JEE/NEET
Oscillations (SHM)
38 topics in 11 parts
SHM period — spring Simple pendulum period Restoring force in SHM Acceleration-displacement relation SHM displacement SHM 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 38 topics in Oscillations (SHM)
Periodic Motion & Oscillations
- Functions & Condition for SHM
- Identification & Restoring Force
- Comparison with Uniform Circular Motion
Velocity & Acceleration in SHM
- Displacement, Velocity & Acceleration Equations
- Amplitude & Time Period Calculations
- Linear SHM
- Angular SHM
Phase & Phasor Diagrams in SHM
- Initial Phase & Epoch
- Phasor Diagram Analysis
- Relative Phase & Time Difference
- Same & Opposite Phase Conditions
Different Graphs in SHM
- Displacement-Time & Velocity-Time Graphs
- Velocity-Displacement & Acceleration-Displacement Curves
- Energy-Displacement Graphs
Energy in SHM
- Kinetic & Potential Energy Functions
- Total Mechanical Energy & Conservation
- Average Values of PE & KE
- Effective Force Constant from Potential Energy Relation
The Spring-Mass System
- Horizontal & Vertical Oscillations
- Series & Parallel Combination of Springs
- Cutting of Springs
- Two-Particle System & Two-Body Problem
- Collisions & Mass Detachment
Simple Pendulum
- Time Period & Frequency
- Accelerating Frames & Lifts
- Oscillations in Liquids & Buoyant Effects
- Large Length & Earth Radius Pendulum
Physical & Torsional Pendulums
- Compound Pendulum & Equivalent Simple Pendulum
- Torsional Oscillations
Superposition of SHMs
- Collinear Superposition & Parallel SHMs
- Mutually Perpendicular SHMs & Lissajous Figures
Special Cases of SHM
- Liquid Column in U-Tube
- Motion in Tunnel Through Earth
- Buoyancy & Floatation Oscillations
- Oscillations in Electric Field & Charged Systems
- Rolling Body Oscillations
Damped & Forced Oscillations
- Damping Constant & Damped Motion
- Forced Oscillations & Resonance