Work Done by Spring Force

Setup

A spring of spring constant $k$ is attached to a wall. A body is attached to the free end. The natural (unstretched) length position is $x = 0$.

When the body is displaced by $x$ from the natural length (either stretched or compressed), the spring exerts a restoring force:

$F_{spring} = -kx$

The negative sign means the force always opposes the displacement — it pulls back when stretched, pushes back when compressed.

Derivation

Work done by the spring force as the body moves from $x = 0$ to $x = x_0$:

$W_{spring} = \int_0^{x_0} F_{spring} \, dx = \int_0^{x_0} (-kx) \, dx$

$W_{spring} = -k \cdot \frac{x^2}{2}\Big|_0^{x_0} = -\frac{1}{2}kx_0^2$

$\boxed{W_{spring} = -\frac{1}{2}kx^2}$

Why the work is negative

As the body moves away from equilibrium (stretching or compressing the spring), the spring force opposes this motion. Force and displacement are in opposite directions — negative work.

The spring takes energy from the body and stores it as elastic potential energy.

Work done by spring over a general displacement

If the body moves from $x_1$ to $x_2$:

$W_{spring} = \int_{x_1}^{x_2} (-kx) \, dx = -\frac{1}{2}kx_2^2 + \frac{1}{2}kx_1^2 = \frac{1}{2}k(x_1^2 - x_2^2)$

• If $|x_2| > |x_1|$: spring does negative work (stores energy)
• If $|x_2| < |x_1|$: spring does positive work (releases energy)

Work done when spring returns to natural length

If body is released from $x = x_0$ and returns to $x = 0$:

$W_{spring} = \frac{1}{2}k(x_0^2 - 0) = \frac{1}{2}kx_0^2$

Positive — the spring releases all the stored energy back to the body. This is why springs are useful as energy storage devices.

Connection to elastic potential energy

The elastic potential energy stored in a spring at extension $x$:

$U = \frac{1}{2}kx^2$

Work done by spring = decrease in elastic PE:

$W_{spring} = -\Delta U = -(U_f - U_i) = \frac{1}{2}kx_i^2 - \frac{1}{2}kx_f^2$

Consistent with the general formula above.