3.2 Work-Energy Theorem

Work-Energy Theorem

The work-energy theorem states that the net work done on an object equals its change in kinetic energy: W_net = ΔKE = KE_f - KE_i. This follows directly from Newton's second law integrated over displacement: W = ∫F·dx = ∫ma·dx = ΔKE. It applies to any net force, constant or variable, and lets you skip finding acceleration to relate force and speed.

Calculating Work with Variable Forces

Work is the integral of force over displacement: W = ∫F(x)dx, which equals the area under a Force vs. position graph. For a constant force, this simplifies to W = Fd·cos(θ). Since work is a scalar, only the component of force parallel to displacement contributes — perpendicular forces (like normal force in circular motion) do zero work.

Applying the Theorem

The work-energy theorem is powerful for finding final speed without solving for time or acceleration directly, especially when forces vary with position (like springs). If multiple forces act, sum all works (including negative work from friction or drag) to get W_net, then solve for ΔKE = (1/2)mv_f² - (1/2)mv_i².

Key terms

Work-Energy Theorem

W_net = ΔKE = KE_f - KE_i; the net work done on an object equals its change in kinetic energy.

Work formula (variable force)

W = ∫F(x)dx, the integral of force with respect to position — equal to the area under an F-x graph.

Work formula (constant force)

W = Fd·cos(θ), where θ is the angle between force and displacement.

Work done by a perpendicular force

Zero, since cos(90°) = 0; only the force component parallel to displacement does work.

Why use the work-energy theorem?

It relates force and speed directly without needing to solve for acceleration or time, especially useful for variable forces.

Try an AP-style question

A force F(x) = 3x² N acts on an object from x = 0 to x = 2 m. How much work does it do?

1. 8 J
2. 24 J
3. 12 J
4. 6 J