Work Done by Gravity

The result

When a body of mass $m$ moves through a vertical displacement $h$ (downward), gravity does work:

$W_{gravity} = mgh$

If the body moves upward by $h$, gravity does negative work: $W_{gravity} = -mgh$.

Crucially: the work done by gravity depends only on the vertical displacement, not on the path taken.

Derivation

Gravity is a constant force: $\vec{F}_g = mg$ downward.

For a body moving at angle $\theta$ to the vertical over displacement $d$:

$W = mgd\cos\theta$

The vertical component of displacement is $d\cos\theta = h$.

$W_{gravity} = mgh$

where $h$ is the vertical drop (positive downward).

Path independence — the key property

Consider three paths from point A (height $H$) to point B (height $0$):

1. Straight vertical drop
2. Inclined ramp
3. Curved path

For all three:

$W_{gravity} = mgH$

Only the vertical displacement matters. The horizontal component of displacement contributes nothing — gravity is vertical, so it has no component along any horizontal direction.

This path independence is the defining property of a conservative force. Gravity is conservative.

Sign convention

Taking upward as positive for displacement:

• Body moves down by $h$: displacement $= -h$, force $= -mg$ (downward in our convention means... )

Cleaner to state directly:

$W_{gravity} = mg \times (\text{vertical drop})$

• Moving down: $W_{gravity} > 0$ (gravity aids motion)
• Moving up: $W_{gravity} < 0$ (gravity opposes motion)
• Moving horizontally: $W_{gravity} = 0$ (gravity perpendicular to displacement)

Work done by gravity in a complete loop

If a body starts and ends at the same height (any closed path):

$W_{gravity} = mg \times 0 = 0$

Net work done by gravity over any closed path is zero. This is another way of stating that gravity is conservative.

Connection to gravitational potential energy

Work done by gravity = decrease in gravitational potential energy:

$W_{gravity} = -\Delta U = -(U_f - U_i) = mgh_i - mgh_f$

When the body falls (decreases height), PE decreases and gravity does positive work — the PE converts to KE.