Work-Energy with Variable Forces – AP Physics C | ShowMeClass

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Work-Energy with Variable Forces

AP Physics C · Unit 3: Work & Energy

Learning Objectives

Calculate work as W = ∫F·dx for variable forces

Apply the work-energy theorem: W_net = ΔKE

Interpret work as area under F-x curve

Analyze spring, gravity, and custom force functions

Key Equations

Graph Colors

F(x) — Force function

Shaded area = Work done

KE(x) — Kinetic energy

v(x) — Velocity

Key Insight

For constant F: W = Fd (rectangle).
For spring: W = ½kx² (triangle).
For any F(x): W = ∫F dx (integral).

The work-energy theorem connects force to motion via calculus.

WorkEnergyIntegralAP Physics C

x =—

F =—

W =—

KE =—

v =—

F(x)

KE(x)

v(x)

All

Force Model

Mass m (kg) 2.0

v₀ (m/s) 0

k / Coefficient 5

Amplitude / Offset 10

x range (m) 5

Speed 1×

Work-Energy Analysis

📝 Quick Quiz▼

1. Work done by a variable force F(x) from a to b equals:

F × (b−a) ∫ₐᵇ F(x) dx F(b) − F(a) dF/dx

2. Work done compressing a spring by x is:

kx ½kx² kx² ½kx

3. The work-energy theorem states W_net =:

mgh ΔKE ΔPE Fd cosθ