Inside This Unit: The Full Breakdown
This unit defines work as the integral of force over displacement, applies the work-energy theorem, uses energy conservation with gravitational and spring potential energy, and relates force to potential energy by F = −dU/dx.
Why it matters
Energy methods solve problems that would be hard with forces alone, and the calculus links (W = ∫F·dx, F = −dU/dx) are core Physics C skills.
Key concepts
- Work by a variable force is W = ∫F·dx; net work equals ΔKE.
- Mechanical energy is conserved when only conservative forces act.
- Spring energy is ½kx²; near Earth gravitational energy is mgh.
- The conservative force is F = −dU/dx, and equilibria are where dU/dx = 0.
Work and the Work-Energy Theorem
For a variable force, work is the integral W = ∫F·dx (the area under an F-x graph). The work-energy theorem, W_net = ΔKE, lets you find speed changes without computing acceleration. Power is the rate of doing work, P = dW/dt = F·v.
Potential Energy and Conservation
Conservative forces store potential energy; mechanical energy KE + U is conserved when only they act. Given U(x), the force is F = −dU/dx, so minima of U are stable equilibria. Friction removes mechanical energy, accounted for as nonconservative work.
AP exam tip
Use a U(x) graph to read off force (negative slope) and classify equilibria — a frequent FRQ task that rewards calculus reasoning over plugging in numbers.
Connections to other units
- Unit 2: Work is done by the forces from Newton’s laws.
- Unit 6: Rotational energy ½Iω² extends these energy methods.
