Spring Work Formula

Calculus 2 · Axiom Academy

From Hooke's law to the work integral — why stretching a spring takes a definite integral, and how that integral equals the area of a triangle. 1. Hooke's Law: The Linear Force A spring pushes back with a restoring force proportional to how far it is displaced from its natural length. Stretch or compress it by a distance x and the force you must supply is F = kx . Pull twice as far and the spring fights back twice as hard. x = displacement from natural length (m) k = spring constant, the stiffness (N/m) 2. Computing Work: The Integral Work is force times distance — but only when the force is constant. Here F changes at every position, so we add up infinitesimal bits of work from x = 0 to x = d . That sum is a definite integral. Evaluate the antiderivative at the endpoints: Each thin slice of stretch has width dx , over which the force is essentially constant. At position x the height of the slice is the force F = kx — taller as x grows. The total is the triangular area under the line F = kx : base d , height kd . The work you do compressing or stretching a spring does not vanish — it is stored as elastic potential energy , ready to be released back onto another object. The stored energy is exactly the work that went in: . Because the displacement is squared , the energy responds dramatically to how far the spring is moved: Doubling the displacement quadruples the stored energy ( 2^2 = 4 ). A stiffer spring (larger k ) stores more energy for the same displacement.

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