Work is path-dependent only with friction

The mistake

Assuming gravitational work depends on the route taken.

Why it feels right

A longer path feels like it should cost more energy.

How to catch it

You integrated gravity along a winding path expecting a path-dependent answer.

The first principle

Conservative forces (gravity, spring) depend only on endpoints. Only non-conservative forces like friction make work path-dependent.

worked mini-example

Work by gravity moving a mass up two different frictionless ramps to the same height?

Common wrong move

Different — the ramps differ.

Correct

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.