Hooke's Law — Revision Notes

⚡ 30-Second Revision

Hooke's Law (Springs): — $F = -kx$ (Restoring force, $k$: spring constant, $x$: displacement). \n- Hooke's Law (Solids): Stress $\propto$ Strain $\implies \text{Stress} = E \times \text{Strain}$ ($E$: Modulus of Elasticity). \n- Stress: $\sigma = F/A$ ($N/m^2$ or $Pa$). \n- Strain: $\epsilon = \Delta L/L$ (dimensionless). \n- Young's Modulus: $Y = \frac{\text{Tensile Stress}}{\text{Longitudinal Strain}} = \frac{F/A}{\Delta L/L}$. \n- Elastic Potential Energy (Spring): $U = \frac{1}{2}kx^2$. \n- Elastic Potential Energy (Wire/Volume): $U_{vol} = \frac{1}{2} \times \text{Stress} \times \text{Strain}$. \n- Springs in Series: $\frac{1}{k_{eq}} = \frac{1}{k_1} + \frac{1}{k_2} + \dots$. \n- Springs in Parallel: $k_{eq} = k_1 + k_2 + \dots$. \n- Elastic Limit: Max stress without permanent deformation. Hooke's Law holds within this limit.

2-Minute Revision

Hooke's Law is fundamental to understanding how materials deform elastically. For springs, it states that the restoring force is directly proportional to the displacement from equilibrium, $F = -kx$, where $k$ is the spring constant.

The negative sign indicates the force opposes displacement. For solid materials, this generalizes to stress being proportional to strain, with the constant of proportionality being the modulus of elasticity.

Young's Modulus ($Y$) is used for tensile/compressive deformation, defined as $Y = \frac{\text{Stress}}{\text{Strain}}$. Stress is force per unit area ($F/A$), and strain is the fractional change in length ($\Delta L/L$).

\n\nCrucially, Hooke's Law is valid only within the elastic limit, the point beyond which a material undergoes permanent deformation. Work done in deforming an elastic body is stored as elastic potential energy, given by $U = \frac{1}{2}kx^2$ for a spring.

For NEET, remember unit conversions, the definitions of stress and strain, and how to apply Young's Modulus. Also, be familiar with series and parallel combinations of springs and the interpretation of stress-strain curves, particularly the proportional and elastic limits.

5-Minute Revision

Hooke's Law is the bedrock of elasticity, stating that within the elastic limit, deformation is proportional to the applied force. For a spring, this is $F = -kx$, where $F$ is the restoring force, $k$ is the spring constant (stiffness), and $x$ is the displacement.

The negative sign signifies the restoring nature of the force. For bulk materials, this extends to 'stress is proportional to strain.' Stress ($\sigma = F/A$) is the internal restoring force per unit area, while strain ($\epsilon = \Delta L/L$) is the relative deformation.

The constant of proportionality is the modulus of elasticity, such as Young's Modulus ($Y = \frac{\text{Stress}}{\text{Strain}}$) for changes in length. \n\nKey Formulas: \n* Spring Force: $F = kx$ (magnitude) \n* Young's Modulus: $Y = \frac{F/A}{\Delta L/L}$ \n* Elastic Potential Energy (Spring): $U = \frac{1}{2}kx^2$ \n* Elastic Potential Energy (per unit volume for wire): $U_{vol} = \frac{1}{2} \times \text{Stress} \times \text{Strain}$ \n\nImportant Concepts: \n* Elastic Limit: The maximum stress a material can withstand without permanent deformation.

Hooke's Law applies below this. \n* Proportional Limit: The point up to which stress is strictly proportional to strain (linear region of stress-strain curve). \n* Springs in Series: $\frac{1}{k_{eq}} = \frac{1}{k_1} + \frac{1}{k_2}$.

\n* Springs in Parallel: $k_{eq} = k_1 + k_2$. \n\nWorked Example: A wire of length $1\,\text{m}$ and cross-sectional area $2 \times 10^{-6}\,\text{m}^2$ is stretched by $1\,\text{mm}$ by a force of $200\,\text{N}$.

Calculate Young's Modulus. \nSolution: \n1. Given: $L = 1\,\text{m}$, $A = 2 \times 10^{-6}\,\text{m}^2$, $\Delta L = 1\,\text{mm} = 1 \times 10^{-3}\,\text{m}$, $F = 200\,\text{N}$. \n2. Stress $= F/A = 200\,\text{N} / (2 \times 10^{-6}\,\text{m}^2) = 10^8\,\text{Pa}$.

\n3. Strain $= \Delta L/L = (1 \times 10^{-3}\,\text{m}) / 1\,\text{m} = 10^{-3}$. \n4. Young's Modulus $Y = \text{Stress} / \text{Strain} = 10^8\,\text{Pa} / 10^{-3} = 10^{11}\,\text{Pa}$. \n\nFocus on unit consistency, understanding the stress-strain curve, and applying the correct formula for different scenarios.

These are common traps in NEET.

Prelims Revision Notes

Hooke's Law is a critical topic for NEET, primarily tested through numerical problems and conceptual questions related to material properties. \n\n1. Hooke's Law for Springs: \n* Formula: $F = -kx$.

$F$ is restoring force, $k$ is spring constant (stiffness, $N/m$), $x$ is displacement from equilibrium. Negative sign indicates opposing direction. \n* Elastic Potential Energy: $U = \frac{1}{2}kx^2$.

Energy stored is proportional to $x^2$. \n* Spring Combinations: \n * Series: $\frac{1}{k_{eq}} = \frac{1}{k_1} + \frac{1}{k_2} + \dots$ (Total extension is sum of individual extensions). \n * Parallel: $k_{eq} = k_1 + k_2 + \dots$ (Total force is sum of individual forces).

\n* Cutting a Spring: If a spring of constant $k$ is cut into $n$ equal parts, each part has a spring constant $nk$. \n\n2. Hooke's Law for Solids (Stress and Strain): \n* **Stress ($\sigma$):** Restoring force per unit area.

$\sigma = F/A$. Unit: $N/m^2$ or $Pa$. \n* **Strain ($\epsilon$):** Fractional change in dimension. $\epsilon = \Delta L/L$ (longitudinal), $\epsilon_V = \Delta V/V$ (volumetric), $\epsilon_S = \phi$ (shearing).

Dimensionless. \n* **Modulus of Elasticity ($E$):** Constant of proportionality. $E = \text{Stress} / \text{Strain}$. \n * **Young's Modulus ($Y$):** For tensile/compressive stress. $Y = \frac{F/A}{\Delta L/L}$.

\n * **Bulk Modulus ($B$):** For volumetric stress. $B = \frac{-\Delta P}{\Delta V/V}$. \n * **Shear Modulus ($G$):** For shearing stress. $G = \frac{F/A}{\phi}$. \n\n3. Stress-Strain Curve: \n* Proportional Limit: Point up to which stress $\propto$ strain (linear region).

Hooke's Law strictly holds. \n* Elastic Limit: Max stress without permanent deformation. Material returns to original shape. \n* Yield Point: Stress at which plastic deformation begins. \n* Ultimate Tensile Strength: Max stress material can withstand before necking.

\n* Fracture Point: Point where material breaks. \n\n4. Key Points for NEET: \n* Always convert units to SI (e.g., cm to m, mm to m, $mm^2$ to $m^2$). \n* Be careful with diameter vs. radius for calculating area ($A = \pi r^2$).

\n* Understand the difference between proportional limit and elastic limit. \n* Practice problems involving energy calculations and combinations of springs. \n* Remember that $Y$ is an intrinsic material property, independent of geometry.

Vyyuha Quick Recall

Hooke's Law: For Stress, Strain Elasticity, Under Key Xtension. \n\n* Hooke's Law: The name itself. \n* For Stress, Strain Elasticity: Reminds you of the general form (Stress $\propto$ Strain) and the Modulus of Elasticity. \n* Under Key Xtension: Reminds you of the spring formula $F=-kX$ and the stored U energy $U = \frac{1}{2}kX^2$.