Work by Variable Force
W = ∫F dx — see area under F(x) for linear, quadratic, or sinusoidal forces.
Key Notes
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.
Formulas
General formula
Area under F-vs-x curve.
Vector form (3D)
Line integral along path C.
Spring work
Work done BY spring is negative when stretched.
Universal gravity work
Path-independent for conservative force.
Important Points
Area under F-x graph = work done. Use this to read off problems graphically.
Conservative force: W depends only on endpoints (not on path).
Non-conservative: W depends on path (friction does more work over a longer path).
Spring stores PE: W_spring < 0 when stretching (you do positive work on spring).
When integrating, choose F as a function of x — then ∫F dx.
In 2D/3D: use line integral and dot product F · dr.
Work by Variable Force notes from sciphylab (also known as SciPhy, SciPhy Lab, SciPhy Labs, Physics Lab). Class 11 physics revision for JEE Mains, JEE Advanced, NEET UG, AP Physics 1/2/C, SAT, and CUET-UG.