Chapter 17 · Class 11-12 & JEE/NEET
Mechanical Properties of Solids
44 topics in 12 parts
Pressure Mass density Relative density Hydrostatic gauge pressure Absolute pressure in a liquid Pressure difference between levels
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 Mechanical Properties of Solids
Elastic Behavior of Solids
- Cause of Elasticity
- Interatomic Forces & Potential Energy Curve
- Interatomic Force Constant
Stress
- Types of Stress
- Normal & Longitudinal Stress
- Shear & Tangential Stress
- Breaking Stress
- Oblique Planes
- Hoop Stress
- Accelerating Systems
- Rotating Systems
Strain
- Hooke's Law & Elastic Moduli
- Types of Strain
- Types of Modulus of Elasticity
- Relations Between Elastic Moduli
- Poisson's Ratio
Analogy of a Rod as a Spring
- Equivalent Spring Constant of a Rod
Bars of Composite Section
- Composite Rods Under Axial Load
- Wires in Series
- Wires in Parallel
- Wires in Series & Parallel
Elasticity & Plasticity
- Stress-Strain Curve & Elastic Regions
- Elastic Limit
- Elastic Aftereffect
- Elastic Fatigue
- Plastic Deformation & Ductility
Energy Stored in a Deformed Body
- Elastic Potential Energy in a Stretched Wire or Rod
- Elastic Energy Density
- Work Done in Stretching a Wire
- Sudden vs Gradual Loading
Young's Modulus
- Definition & Expression for Young's Modulus
- Experimental Determination of Young's Modulus
- Temperature Dependence of Young's Modulus
- Sagging of a Wire Under Load
Wire Elongation
- Elongation Under External Load
- Elongation Under Self-Weight
- Elongation of Tapered & Rotating Wires
Bulk Modulus
- Volumetric Strain & Compressibility
- Pressure-Volume Relationships in Liquids & Solids
Thermal Stress
- Fixed Rods Under Temperature Change
- Composite Rods Under Temperature Change
Shear Modulus
- Shear Stress & Strain
- Torsion & Angle of Twist
- Composite Shear Layers