Hooke's Law

Hooke's Law Definition

Hooke's Law states that the force needed to extend or compress a spring is directly proportional to the extension or compression of the spring, provided the elastic limit is not exceeded.

Mathematically, this is written as:

Force (F) 27 Extension (e)

This can be expressed as an equation:

$F = k \times e$

where:

• F is the force applied to the spring (in newtons, N)
• e is the extension or compression of the spring from its original length (in metres, m)
• k is the spring constant (in newtons per metre, N/m), which measures the stiffness of the spring

The elastic limit is the maximum extension or compression a spring can undergo and still return to its original shape when the force is removed. Beyond this limit, the spring will be permanently deformed.

For example, a stiff spring has a large spring constant $k$, meaning it requires a large force to produce a small extension.

For instance, if a spring with a spring constant of 50 N/m is stretched by 0.02 m, the force applied is:

$F = 50 \times 0.02 = 1 \text{ N}$

Force-Extension Graphs

A force-extension graph shows how the force applied to a spring changes as the spring is stretched or compressed.

• Linear region: At first, the graph is a straight line passing through the origin, showing that force is proportional to extension. This linear region obeys Hooke's Law.
• Gradient: The gradient (steepness) of this straight line equals the spring constant $k$. A steeper gradient means a stiffer spring.
• Elastic limit: The point on the graph where the line stops being straight is the elastic limit. Beyond this, the spring no longer obeys Hooke's Law.
• Plastic deformation: Beyond the elastic limit, the spring stretches permanently and does not return to its original length when the force is removed.

If the force is removed while the spring is in the elastic region, the spring returns to its original length. If the force exceeds the elastic limit, the spring is permanently stretched.

Calculations Using Hooke's Law

The formula to calculate the force applied to a spring is:

$F = k \times e$

where:

• F is force in newtons (N)
• k is spring constant in newtons per metre (N/m)
• e is extension in metres (m)

You can rearrange this formula to find the spring constant or extension:

$k = \frac{F}{e}$

$e = \frac{F}{k}$

In practical experiments, the force applied is usually measured using a newtonmeter, and the extension is measured with a ruler or a metre stick. The extension is the change in length from the spring's original length.

For example, if a force of 3 N stretches a spring by 0.015 m, the spring constant is:

$k = \frac{3}{0.015} = 200 \text{ N/m}$

Applications and Limitations

Hooke's Law applies to many elastic materials and devices, such as springs, elastic bands, and some metals within their elastic limits.

Common applications include:

• Car suspension systems use springs to absorb shocks by stretching and compressing within the elastic limit.
• Scales and newtonmeters use springs to measure force by the amount they stretch.
• Trampolines use elastic materials that stretch and return to shape, following Hooke's Law within limits.

However, Hooke's Law has limitations:

• It only applies up to the elastic limit. Beyond this, materials undergo plastic deformation and do not return to their original shape.
• Some materials, like rubber bands, do not obey Hooke's Law perfectly even at small extensions.
• Repeated stretching beyond the elastic limit can permanently damage springs and elastic materials.

Understanding these limits is important in engineering and design to avoid failure of materials and ensure safety.