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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).

  • 3 variables
  • 2 worked examples
  • 14 in Applications of Integrals

Variables

Variables used in the Work (Spring) formula
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

1 more worked examplePremium

A force of 20 N stretches a spring 0.5 m beyond its natural length. Find the work needed to stretch it 0.4 m from its natural length.

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Common questions

What is the Work (Spring)?

Work done compressing or stretching a spring from position a to b, where k is the spring constant (Hooke's law: F = kx). It is one of the applications of integrals formulas in the CalcRef reference.

What do the symbols in the Work (Spring) mean?

In the Work (Spring), k is the spring constant (N/m); a is the initial displacement (m); b is the final displacement (m).

How do you use the Work (Spring)?

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

Practice

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