15.1.1 Hooke's Law

Introduction

Hooke's law describes the relationship between the tension in a stretched elastic string or spring and its extension. It is a fundamental concept in mechanics and elasticity, used to model the behaviour of springs and elastic materials.

Hooke's law states:

$$T = \frac{\lambda x}{l}$$

where:

• $T$ is the tension in the string or spring ($\text{N}$)
• $\lambda$ is the modulus of elasticity ($\text{N}$)
• $x$ is the extension of the string or spring ($\text{m}$)
• $l$ is the natural length of the string or spring ($\text{m}$)

Natural Length ($l$)

• The natural length is the length of the string or spring when it is unstretched and not under tension.
• Its units are meters ($\text{m}$)

Modulus of Elasticity ($\lambda$)

• The modulus of elasticity is a measure of how "stretchy" the string or spring is.
• A larger $\lambda$ means the string is less stretchy, requiring greater force to achieve the same extension.
• Its units are newtons ($\text{N}$)

Extension ($x$)

• The extension is the additional length the string or spring gains when stretched.
• It is calculated as:

$$x = \text{stretched length} - \text{natural length}.$$

• Its units are meters ($\text{m}$)

Assumptions of Hooke's Law

• Hooke's law only applies within the elastic limit, where the string or spring returns to its original length when the tension is removed.
• For problems in this topic, assume Hooke's law is valid unless stated otherwise.

Worked Example

Note Summary