Chapter 27 · Class 11-12 & JEE/NEET
Waves & Sound
63 topics in 13 parts
Wave speed Progressive-wave equation Angular wave number Angular frequency Phase difference from path difference Phase difference from time delay
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 63 topics in Waves & Sound
Propagation of Disturbance
- Mechanical Waves
- Pulse & Wave
- Progressive Waves
- Types of Waves
- Wave Parameters
- Two-Dimensional Waves
- Seismic Waves
- Polarization of Mechanical Waves
Wave Function & Travelling Waves
- Wave Function
- Progressive Wave Equation
- General Expression for a Sinusoidal Wave
- Wave Graphs
- Phase Variation with Time
- Phase Variation with Distance
- Particle Velocity & Wave Slope
- Speed of Sinusoidal Waves
Wave Speed
- Waves on Strings
- Longitudinal Waves in Solids
- Sound in Gases
- Temperature Dependence
- Variable Tension & Density
Wave Energy & Intensity
- Energy of a Wave
- Power of a Wave
- Intensity of a Wave
- Sound Intensity Level
Reflection, Refraction & Transmission
- Reflection at Fixed & Free Ends
- Transmission at a Boundary
- Refraction of Mechanical Waves
- Echo & Depth
Superposition & Interference
- Resultant Amplitude & Phase
- Constructive & Destructive Interference
- Interference Conditions
- Mathematical Analysis of Interference
- Quinck's Tube
Beats
- Beat Frequency
- Tuning Fork Problems
Doppler Effect
- Moving Source & Moving Observer
- Moving Source & Stationary Observer
- Stationary Source & Moving Observer
- Reflected Sound
- Accelerated & Non-Collinear Motion
Standing Waves
- Formation & Equation
- Characteristics
- Nodes & Antinodes
- Phase & Energy Distribution
- Resonance
Vibrating Strings
- Both Ends Fixed
- One End Fixed & One End Free
- Fundamental & Overtones
- Melde's Experiment: Transverse Mode
- Melde's Experiment: Longitudinal Mode
- Sonometer
- Composite Strings
- Temperature Effects
Air Columns & Resonance Tubes
- Open Organ Pipes
- Closed Organ Pipes
- Resonance Tube
- End Correction
Standing Waves in Rods & Kundt's Tube
- Longitudinal Modes in Rods
- Kundt's Tube
Sound Waves & Their Properties
- Longitudinal Pressure Waves
- Displacement & Pressure Relation
- Quality of Sound