Work is path-dependent only with friction

01
The tempting model
Assuming gravitational work depends on the route taken.
02
Why it feels right
A longer path feels like it should cost more energy.
03
Where it breaks · how to catch it
You integrated gravity along a winding path expecting a path-dependent answer.
04
The correct model
Conservative forces (gravity, spring) depend only on endpoints. Only non-conservative forces like friction make work path-dependent.
05
Repair · see it, then test it
worked example Work by gravity moving a mass up two different frictionless ramps to the same height?
Tempting move
Different — the ramps differ.
Correct move
Same: −mgh either way (endpoints only).
Test the repair EN-04 · transfer

A climber reaches the same ledge by a steep route and a winding route. What is true of the work done by gravity?
It is more negative on the winding route It is the same on both routes It is zero on both routes It depends on climbing speed

Make a fresh prediction before referring to the worked example.
Then practise it on the ladder
R17Endpoint work by gravityadds: Conservative work depends on endpoints, not the path.