Conservative Work and Gravitational Potential Energy

The Fundamental Relationship

When a conservative force acts on an object, the work it does is related to potential energy by:

$$W_c = -\Delta U = -(U_f - U_i) = U_i - U_f$$

The work done by a conservative force equals the negative of the change in potential energy. The negative sign captures a key insight: when potential energy decreases ($U_f < U_i$), the conservative force does positive work ($U_i-U_f > 0$). When potential energy increases ($U_f > U_i$), the conservative force does negative work ($U_i-U_f <0$).

Understanding the Sign Convention

Consider an object falling under gravity. As it descends from height $y_i$ to a lower height $y_f$, the gravitational potential energy decreases:

$$\Delta U_g = mg(y_f - y_i) < 0$$

since $y_f < y_i$. The work done by gravity is:

$$W_g = -\Delta U_g = mg(y_i - y_f) > 0$$

This is positive work. Gravity acts in the direction of motion and accelerates the object.

Conversely, when you lift an object upward against gravity, gravity does negative work because the displacement opposes the gravitational force. The potential energy increases by exactly the amount of negative work gravity performs.

Verification Using the Work Formula Explicitly

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