Mechanical Properties of Solids
📐 Formula & Cheat Sheet (English)
Class 11 Physics Revision Notes & Formula Sheet
Chapter: Mechanical Properties of Solids (ठोसों के यांत्रिक गुण)
1. Basic Concepts (मूल अवधारणाएं)
- Deforming Force (विरूपक बल): An external force applied on a body which changes its shape or size or both.
- Restoring Force (प्रत्यानयन बल): The internal equal and opposite force developed inside a deformed body to regain its original shape/size.
- Elasticity (प्रत्यास्थता): The property of a material body by virtue of which it regains its original shape and size after the removal of the deforming force. (Example: Steel, Quartz)
- Plasticity (सुघट्यता): The property of a body by virtue of which it does not regain its original shape and size after removal of deforming forces. (Example: Mud, Putty)
2. Stress (प्रतिबल)
The internal restoring force per unit cross-sectional area of a deformed body.
$$\text{Stress} = \frac{\text{Restoring Force}}{\text{Area}} = \frac{F}{A}$$
- SI Unit: $\text{N/m}^2$ or Pascal ($\text{Pa}$)
- Dimensional Formula: $[M^1 L^{-1} T^{-2}]$
- Types of Stress:
- Longitudinal Stress (अनुदैर्घ्य प्रतिबल): Applied normal to the surface (Tensile or Compressive).
- Tangential / Shear Stress (अपरूपण प्रतिबल): Applied parallel to the surface.
- Hydraulic / Volumetric Stress (आयतन प्रतिबल): Applied uniformly from all sides by a fluid.
3. Strain (विकृति)
The ratio of change in dimension to the original dimension of the body.
$$\text{Strain} = \frac{\text{Change in Dimension}}{\text{Original Dimension}}$$
- Unit & Dimension: It is a dimensionless and unitless quantity.
- Types of Strain:
- Longitudinal Strain (अनुदैर्घ्य विकृति): $$\text{Longitudinal Strain} = \frac{\Delta L}{L}$$
- Shear Strain (अपरूपण विकृति): $$\theta = \frac{\Delta x}{L}$$
- Volume Strain (आयतन विकृति): $$ \text{Volume Strain} = \frac{\Delta V}{V}$$
4. Hooke's Law & Modulus of Elasticity (हुक का नियम)
Hooke's Law:
Within the elastic limit, stress is directly proportional to strain.
$$\text{Stress} \propto \text{Strain}$$ $$\text{Stress} = E \times \text{Strain}$$
Where $E$ is the Modulus of Elasticity (प्रत्यास्थता गुणांक) of the material.
- Unit of $E$: $\text{N/m}^2$ or $\text{Pa}$
5. Types of Elastic Moduli (प्रत्यास्थता गुणांक के प्रकार)
1. Young's Modulus of Elasticity ($Y$) (यंग प्रत्यास्थता गुणांक)
Defined for solids (wires/rods) only. Ratio of longitudinal stress to longitudinal strain.
$$Y = \frac{\text{Longitudinal Stress}}{\text{Longitudinal Strain}} = \frac{F / A}{\Delta L / L}$$
- Formula for a wire of radius $r$ suspended with mass $m$: $$Y = \frac{m \cdot g \cdot L}{\pi r^2 \cdot \Delta L}$$
2. Shear Modulus or Modulus of Rigidity ($\eta$) (दृढ़ता गुणांक)
Ratio of shear stress to shear strain. Defined for solids.
$$\eta = \frac{\text{Tangential Stress}}{\text{Shear Strain}} = \frac{F / A}{\theta} = \frac{F}{A \cdot \theta}$$
3. Bulk Modulus ($B$ or $K$) (आयतन प्रत्यास्थता गुणांक)
Ratio of hydraulic stress to volume strain. Defined for solids, liquids, and gases.
$$B = -\frac{\Delta P}{\Delta V / V} = -\frac{V \cdot \Delta P}{\Delta V}$$
(Negative sign indicates that volume decreases with an increase in pressure)
- Compressibility (संपीड्यता, $k$): The reciprocal of Bulk Modulus. $$k = \frac{1}{B} = -\frac{\Delta V}{V \cdot \Delta P}$$
6. Stress-Strain Curve (प्रतिबल-विकृति वक्र)
Key points on the curve for a metallic wire:
- Proportional Limit (समानुपाती सीमा - Point A): Hooke's law is strictly valid ($\text{Stress} \propto \text{Strain}$).
- Elastic Limit / Yield Point (प्रत्यास्थ सीमा / पराभव बिंदु - Point B): The maximum stress up to which the body regains its original shape completely after removing the load.
- Ultimate Tensile Strength (चरम सामर्थ्य बिंदु - Point D): The maximum stress a material can withstand without breaking.
- Fracture / Breaking Point (त्रोटन बिंदु - Point E): The point at which the material breaks.
- Ductile Materials (तन्य पदार्थ): Large plastic region between yield point and fracture point (e.g., Copper, Iron).
- Brittle Materials (भंगुर पदार्थ): Fracture point lies very close to the elastic limit (e.g., Glass, Cast Iron).
- Elastomers (प्रत्यास्थलक): Materials that can be stretched to large strains (e.g., Rubber, Tissue of aorta). They do not obey Hooke's law strictly.
7. Poisson's Ratio ($\sigma$) (पॉइसन अनुपात)
When a wire is stretched, its length increases while its diameter decreases.
$$\text{Lateral Strain (पाश्विक विकृति)} = -\frac{\Delta d}{d}$$ $$\text{Longitudinal Strain (अनुदैर्घ्य विकृति)} = \frac{\Delta L}{L}$$
$$\sigma = \frac{\text{Lateral Strain}}{\text{Longitudinal Strain}} = -\frac{\Delta d / d}{\Delta L / L}$$
- Theoretical Limits: $-1 \le \sigma \le 0.5$
- Practical Values: $0 \le \sigma \le 0.5$ (For most metals, $\sigma \approx 0.28 \text{ to } 0.33$)
8. Work Done in Stretching a Wire / Elastic Strain Energy
Energy stored in a stretched wire per unit volume:
-
Total Work Done (Potential Energy, $U$): $$U = \frac{1}{2} \times \text{Stretching Force} \times \text{Elongation}$$ $$U = \frac{1}{2} \cdot F \cdot \Delta L = \frac{1}{2} \frac{Y \cdot A \cdot (\Delta L)^2}{L}$$
-
Energy Density ($u$) [Energy per unit volume]: $$u = \frac{U}{\text{Volume}} = \frac{1}{2} \times \text{Stress} \times \text{Strain}$$ $$u = \frac{1}{2} \times Y \times (\text{Strain})^2$$
9. Thermal Stress (तापीय प्रतिबल)
When a rod clamped at both ends is heated or cooled:
- Thermal Strain: $$\text{Strain} = \alpha \cdot \Delta T$$
- Thermal Stress: $$\text{Stress} = Y \cdot \alpha \cdot \Delta T$$
- Thermal Force exerted on supports: $$F = Y \cdot A \cdot \alpha \cdot \Delta T$$
(Where $\alpha$ = Coefficient of linear expansion, $\Delta T$ = Change in temperature)
10. Summary Formula Table for Quick Revision
| Quantity | Formula | SI Unit |
|---|---|---|
| Stress | $\sigma = F / A$ | $\text{N/m}^2$ or $\text{Pa}$ |
| Longitudinal Strain | $\epsilon = \Delta L / L$ | Dimensionless |
| Young's Modulus | $Y = \frac{F \cdot L}{A \cdot \Delta L}$ | $\text{N/m}^2$ |
| Bulk Modulus | $B = -\frac{\Delta P \cdot V}{\Delta V}$ | $\text{N/m}^2$ |
| Compressibility | $k = 1 / B$ | $\text{m}^2/\text{N}$ or $\text{Pa}^{-1}$ |
| Shear Modulus | $\eta = \frac{F}{A \cdot \theta}$ | $\text{N/m}^2$ |
| Elastic Energy Density | $u = \frac{1}{2} \times \text{Stress} \times \text{Strain}$ | $\text{J/m}^3$ |
| Poisson's Ratio | $\sigma = -\frac{\Delta d / d}{\Delta L / L}$ | Dimensionless |
💡 MP Board Exam Important Tips:
- Steel vs. Rubber: Steel is more elastic than rubber because for a given strain, steel requires a greater force (Young's modulus of steel is higher than that of rubber: $Y_{\text{steel}} > Y_{\text{rubber}}$).
- Standard 2-mark derivations to practice:
- Derive the formula for work done in stretching a wire: $U = \frac{1}{2} F \Delta L$.
- Explain Stress-Strain graph with a labeled diagram.
- Be careful with units: Convert diameter to radius ($r = d/2$) and area $A = \pi r^2$ in numerical problems. Convert $\text{cm}$ or $\text{mm}$ to meters ($\text{m}$).