Work by Variable Force

For a force varying with position, work is the integral W = ∫F·dx along the path.

Equivalently: area under F-vs-x curve from x_i to x_f.

Spring force F = −kx: W (compressing from 0 to x) = ½kx² (taken by the spring); −½kx² done BY the spring on the block.

Gravity near surface (W = mgh): constant force, simple. Gravity far from surface (W = −GMm·(1/r₁ − 1/r₂)): use integral.

Force ⊥ to motion (e.g., circular motion at constant speed): W = 0 instantaneously.

Work depends on path for non-conservative forces (friction, air drag).

For conservative forces, W = −ΔU. Total energy is conserved.

Used in: SHM analysis, escape velocity, pendulum, free-fall in real gravity.