Work Done on a Spring

Work Done on a Spring

Work done is the energy transferred when a force moves an object through a distance. When you stretch or compress a spring, work is done on the spring by the applied force.

This work done is stored as elastic potential energy in the spring. The more you stretch or compress the spring, the more energy is stored.

The amount of work done on a spring (which equals the elastic potential energy stored) can be calculated using the formula:

Work done, $W$ = Elastic potential energy stored

$$W = \frac{1}{2} k x^2$$

• $W$ is work done (in joules, J)
• $k$ is the spring constant (in newtons per metre, N/m)
• $x$ is the extension or compression of the spring from its natural length (in metres, m)

This formula shows that the work done depends on both how stiff the spring is (spring constant) and how far it is stretched or compressed (extension).

For instance, if a spring with a spring constant of 200 N/m is stretched by 0.1 m, the work done on the spring is:

$$W = \frac{1}{2} \times 200 \times (0.1)^2 = 0.5 \times 200 \times 0.01 = 1 \text{ J}$$

Hooke's Law and Extension

Hooke's Law states that the force needed to stretch or compress a spring is directly proportional to the extension or compression, as long as the spring is not stretched beyond its elastic limit.

Mathematically:

$$F = kx$$

• $F$ is the force applied (in newtons, N)
• $k$ is the spring constant (in newtons per metre, N/m)
• $x$ is the extension or compression (in metres, m)

The spring constant $k$ measures the stiffness of the spring: a larger $k$ means a stiffer spring.

The limit of proportionality is the point beyond which Hooke's Law no longer applies. Beyond this limit, the spring will not return to its original length and may be permanently deformed.

A force-extension graph for a spring shows a straight line passing through the origin while Hooke's Law is obeyed. The gradient of this line is the spring constant $k$.

Elastic Potential Energy

Elastic potential energy is the energy stored in a spring (or any elastic object) when it is stretched or compressed.

The energy stored is equal to the work done to stretch or compress the spring.

On a force-extension graph, the elastic potential energy stored is represented by the area under the graph between zero extension and the current extension.

Since the force increases linearly with extension (Hooke's Law), the graph is a straight line and the area under it is a triangle. The area of this triangle is:

$$\text{Area} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times x \times F$$

Substituting $F = kx$, the energy stored is:

$$E = \frac{1}{2} k x^2$$

Energy is measured in joules (J).

For example, if a spring is stretched by 0.15 m and the force at this extension is 30 N, the elastic potential energy stored is:

$$E = \frac{1}{2} \times 0.15 \times 30 = 2.25 \text{ J}$$