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Work done by a spring
Intermediate
Stretching a spring stores energy proportional to x².
W = ½kx²Stretching a spring from 0 to x needs W = ½kx², the area of the triangle in the F–x graph. The spring does −½kx² on you in return.
k = 100 N/m, x = 0.2 m: W = 2 J.
Going from x₁ to x₂ the work is ½k(x₂² − x₁²), not ½k(x₂ − x₁)².
In more detail
A spring does work on an object when it is compressed or stretched, equal to minus one half k times x squared, where x is the displacement from equilibrium.
Real-world uses
- Springs in a clicking pen store and release energy.
- Spring-loaded door closers transfer stored energy to close a door.
- A catapult or bow stores energy in its deformed limbs, and the stored energy is found from the spring formula.
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Frequently asked questions
Why is the work negative?
Because the spring force opposes the displacement, so it removes energy from the object as it moves away from equilibrium.
How much energy is stored in a spring?
One half k x squared, which equals the negative of the work done by the spring.
Why is the energy proportional to x squared?
Because the force grows with extension, and the work needed to stretch it further increases with the stretch already present.
Related topics in Work, Energy & Power
Positive, negative and zero work · Work from a force–displacement graph · Work done by a variable force · Kinetic energy · Work–energy theorem · Potential energy