Work and Energy with Variable Forces

In AP Physics 1, work is force times distance because the force is constant. In AP Physics C: Mechanics the force is allowed to change with position, and work becomes an integral. This is Unit 3, Work, Energy, and Power, which carries 15 to 25 percent of the exam.

Work as an integral

For a force that depends on position:

W = ∫ F(x) dx, taken from x₁ to x₂

• On a graph of F against x, the work is the area under the curve.
• Area below the x-axis counts as negative work.
• A constant force is the special case: the integral collapses to F · Δx.

Spring work

A spring pulls back with F = −kx. Integrating gives the work done by the spring from x₁ to x₂:

W_spring = −½k(x₂² − x₁²)

The energy stored in a spring stretched by x is U = ½kx². Work you do stretching it is the positive version of the same expression.

Force and potential energy

Force and potential energy are two views of the same thing, linked by a derivative:

F(x) = −dU/dx and U(x) = −∫ F dx

• Equilibrium is where dU/dx = 0, so the force is zero.
• Stable equilibrium is a minimum of U (d²U/dx² > 0): a small push brings the object back.
• Unstable equilibrium is a maximum of U.

Work energy theorem and power

The net work on an object equals its change in kinetic energy: W_net = ΔK. Power is the rate of doing work: P = dW/dt = F · v.

Worked example

A 2 kg object starts at rest at x = 0. A single force F(x) = 6x² + 4x (newtons, x in meters) pushes it to x = 2 m. How fast is it moving?

1. Work: W = ∫(6x² + 4x) dx from 0 to 2 = [2x³ + 2x²] from 0 to 2 = 16 + 8 = 24 J.
2. Work energy theorem: 24 J = ½(2)v², so v² = 24.
3. v ≈ 4.9 m/s.

A common mistake is to multiply the final force by the distance: F(2) = 32 N, and 32 N × 2 m = 64 J, almost three times the real work. The force was smaller for most of the trip, and only the integral accounts for that.

The free response approach

• Start from a definition. A derive question expects W = ∫F dx or F = −dU/dx written first, not a final formula pulled from memory.
• Write the limits on every integral and show the substitution.
• For graphs, say that the work is the area, then compute the area piece by piece.
• Keep units on the final answer. A correct number without units can lose the point.