CalcRef

Applications of Integrals · Work & Force

Work (Spring)

W=∫abkx dx=12k(b2−a2)W = \int_a^b kx\, dx = \frac{1}{2}k(b^2 - a^2)

Work done compressing or stretching a spring from position a to b, where k is the spring constant (Hooke's law: F = kx).

Variables

SymbolNameUnit
kSpring constantN/m
aInitial displacementm
bFinal displacementm

Worked examples

A spring has k = 40 N/m. Find the work to stretch it from 0.1 m to 0.3 m.
  1. W = (1/2)(40)(0.3² - 0.1²) = 20(0.09 - 0.01) = 20(0.08) = 1.6 J

Answer: 1.6 J

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