This AP Physics C lesson on work energy theorem notes gives the key ideas first, then worked examples and diagrams. After that, practise with the worksheet on work energy theorem and check yourself with the quiz.
Quick answer: Work done by a variable force is the integral W = ∫F dx, the area under the force–position graph. The net work equals the change in kinetic energy. For a conservative force, F = −dU/dx, so the force is the negative slope of the potential energy curve.
Level: AP Physics C (Mechanics and E&M, calculus-based) · Curriculum: AP PC:M Unit 3
Work energy theorem notes: the key ideas
How do you calculate work when the force changes?
For a constant force along the motion, W = Fd. When the force varies with position, split the path into tiny steps dx. On each step the force is almost constant, so the work is F dx, and adding them up is an integral.
Work is the energy transferred by a force acting through a displacement. For a force along the x-axis, W = ∫ F(x) dx from x₁ to x₂, which is the area under the F–x graph (positive where the force and displacement point the same way).
In general the work is the line integral of F · dr, so only the component of force along the displacement counts. For a spring that obeys F = −kx, the work you must do to stretch it from rest to x is ∫kx dx = ½kx².
How does work change kinetic energy?
Start from Newton’s second law and use the chain rule, dv/dt = v dv/dx. Then F dx = m v dv, and integrating gives the work–energy theorem.
Wnet = ΔK = ½mv₂² − ½mv₁². The net work done by all forces equals the change in kinetic energy; this holds for any force, conservative or not.
Aarav in Pune pushes a 2.0 kg cart with a force F = 3.0x² newtons (x in metres) from x = 1.0 m to 3.0 m. The cart starts at 1.0 m/s. W = ∫3.0x² dx = [x³] from 1.0 to 3.0 = 27 − 1 = 26 J. Kf = ½(2.0)(1.0)² + 26 = 27 J, so v = √(2 × 27/2.0) = √27 = 5.2 m/s.
How do you get force from a potential energy function?
A force is conservative if the work it does between two points does not depend on the path. Then a potential energy U(x) exists with ΔU = −W, and the force is minus the slope of the U graph.
F = −dU/dx and U(x) = −∫F dx. F is the force (N), U is potential energy (J), x is position (m). The force points toward lower U, down the slope of the graph.
| System | U(x) | F = −dU/dx |
|---|---|---|
| Spring | ½kx² | −kx (towards equilibrium) |
| Uniform gravity | mgy | −mg (downwards) |
| Planet’s gravity | −GMm/r | −GMm/r² (attractive) |
| Quartic well | ax⁴ − bx² | −4ax³ + 2bx |
What does an energy diagram tell you?
Because the total energy is E = K + U and K ≥ 0, the particle can only be where U ≤ E. Where U = E the kinetic energy is zero, a turning point. Equilibrium occurs where dU/dx = 0: a minimum (d²U/dx² > 0) is stable and a maximum is unstable.
Chiamaka in Lagos studies U = 2.0x⁴ − 8.0x² joules for a 0.50 kg particle. dU/dx = 8.0x³ − 16x = 0 gives x = 0 and x = ±√2 = ±1.41 m. d²U/dx² = 24x² − 16, which is −16 at x = 0 (unstable) and +32 at x = ±1.41 (stable). U(±1.41) = −8.0 J. If E = −4.0 J, then at x = 1.41 m, K = −4.0 − (−8.0) = 4.0 J and v = √(2 × 4.0/0.50) = 4.0 m/s.
Do not read the force from the height of the U graph. The force depends on the slope. Zero slope means zero force even if U is large, and the sign is the reverse of the slope: a positive slope means the force points in the negative direction.
Picture a ball rolling on the shape of the U graph. It speeds up going down, slows going up, and turns around where the ball would run out of height, which is where U meets the E line.
How is power connected?
Power is the rate of doing work, P = dW/dt = F · v. A force of 600 N acting on a motorbike at 20 m/s delivers 12 kW.
- W = ∫F dx; for a spring W = ½kx²
- Work–energy theorem: Wnet = ΔK
- Conservative force: F = −dU/dx
- Equilibrium: dU/dx = 0; stable if d²U/dx² > 0
- Power P = F · v; turning points occur where U = E
Work as an integral and potential energy curves: frequently asked questions
What is the work done by a variable force?
It is the integral of force over position, W = ∫F dx, which equals the area under the force–position graph. The sign is positive when the force has a component along the displacement.
What is the work–energy theorem?
The net work done on an object equals its change in kinetic energy, Wnet = ΔK. It follows directly from Newton’s second law, using F dx = mv dv.
How do you find force from potential energy?
Differentiate: F = −dU/dx. The minus sign means the force acts toward lower potential energy, down the slope of the graph.
How do you classify equilibrium on a U graph?
Equilibrium is at dU/dx = 0. If d²U/dx² is positive the point is a minimum and the equilibrium is stable; if it is negative the point is a maximum and it is unstable.
What is a turning point?
It is a position where U equals the total energy E, so the kinetic energy is zero. The particle momentarily stops and then moves back into the allowed region where U < E.
