Lesson 4 of 10 · 6 min
Work done by a variable force
NCERT § "Work Done by a Variable Force"
Anu cocks the dart gun at the game stall. The first centimetre of pull is easy, the last is hard: the spring pushes back harder the more she pulls. How much work does she do when the force keeps changing?
The lesson in notes
In short
When a force varies with position, split the displacement into small steps and add F(x)Δx; in the limit, W = ∫F(x) dx between the initial and final positions.
Graphically, the work done is the area under the force-displacement graph, taken with sign.
The work-energy theorem holds for variable forces as well; it is proved by integrating the second law over the displacement.
Work by a spring from x_i to x_f is −½k(x_f² − x_i²), which follows from integrating F = −kx.
Area below the displacement axis on an F-x graph represents negative work.