Conservative Work and Spring Energy

A spring exerts a conservative force. The work it does relates to potential energy change by $W_s = -\Delta U_s$. We examine all four cases: compression relaxing, compression increasing, stretch relaxing, and stretch increasing.

Case 1: Compression Relaxing

A spring compressed by $x_i$ relaxes to smaller compression $x_f < x_i$. The elastic potential energy decreases:

$$\Delta U_s = \frac{1}{2}k(x_f^2 - x_i^2) < 0$$

The spring does positive work:

$$W_s = -\Delta U_s = \frac{1}{2}k(x_i^2 - x_f^2) > 0$$

The spring pushes the block toward equilibrium.

Case 2: Compression Increasing

A spring compressed by $x_i$ is compressed further to $x_f > x_i$. The elastic potential energy increases:

$$\Delta U_s = \frac{1}{2}k(x_f^2 - x_i^2) > 0$$

The spring does negative work:

$$W_s = -\Delta U_s = \frac{1}{2}k(x_i^2 - x_f^2) < 0$$

The spring resists further compression. An external agent must do positive work against the spring force.

Case 3: Stretch Relaxing

A spring stretched by $x_i$ relaxes to smaller stretch $x_f < x_i$. The elastic potential energy decreases:

$$\Delta U_s = \frac{1}{2}k(x_f^2 - x_i^2) < 0$$

The spring does positive work:

$$W_s = -\Delta U_s = \frac{1}{2}k(x_i^2 - x_f^2) > 0$$

The spring pulls the block back toward equilibrium.

Case 4: Stretch Increasing

A spring stretched by $x_i$ is stretched further to $x_f > x_i$. The elastic potential energy increases:

$$\Delta U_s = \frac{1}{2}k(x_f^2 - x_i^2) > 0$$

The spring does negative work:

$$W_s = -\Delta U_s = \frac{1}{2}k(x_i^2 - x_f^2) < 0$$

The spring resists further stretching. An external agent must do positive work against the spring force.