Hooke's Law and Young's Modulus

Understanding Hooke's Law

Hooke's Law describes how the force applied to an elastic material is related to the extension or compression it experiences. It is expressed as:

$$F = k \cdot x$$

• F is the force applied (in Newtons, N).
• k is the spring constant (in N/m), which measures the stiffness of the spring.
• x is the extension or compression (in meters, m).

Features of Hooke's Law

• Applies only within the elastic limit of the material.
• The relationship is linear, meaning the graph of force vs. extension is a straight line.
• Beyond the elastic limit, materials may not return to their original shape.

 

Young's Modulus

Young's Modulus is a measure of the stiffness of a solid material. It is defined as the ratio of stress (force per unit area) to strain (proportional deformation) in the linear elasticity regime of a uniaxial deformation:

$$E = \frac{\sigma}{\epsilon}$$

• E is the Young's Modulus (in Pascals, Pa).
• $\sigma$ is the stress (in Pa), calculated as force divided by area $\left( \sigma = \frac{F}{A} \right)$.
• $\epsilon$ is the strain, calculated as the change in length divided by the original length $\left( \epsilon = \frac{\Delta L}{L_0} \right)$.

 

Worked Example

 

Example

A metal wire of original length 2 m is stretched to 2.002 m by a force of 1000 N. Calculate the stress, strain, and Young's Modulus if the cross-sectional area of the wire is 0.0001 m².

• Stress: $\sigma = \frac{F}{A} = \frac{1000}{0.0001} = 10,000,000 \text{ Pa}$
• Strain: $\epsilon = \frac{\Delta L}{L_0} = \frac{0.002}{2} = 0.001$
• Young's Modulus: $E = \frac{\sigma}{\epsilon} = \frac{10,000,000}{0.001} = 10,000,000,000 \text{ Pa}$