Work: Stretching a Spring — Calc II Explorer

Units

ft · lb m · N

The spring

natural length L = ft

stiffness k = lb/ft

The spring lies on a frictionless surface with its left end fixed to the wall and a block on its right end, so the only force in play is the spring's. Left alone the block rests at the spring's natural length L, and that resting spot is x = 0 — so x is how far the block has been moved, not how long the spring is. Moved to x, the spring pulls or pushes back with a force kx. The natural length only sets where the wall is drawn. Both fields accept expressions, so 3/2 works.

The process

from x = a = ft

to x = b = ft

Both are measured from the natural position, so 0 < a < b stretches the spring further and negative values squeeze it: a = −0.2 to b = −0.5 pushes the block in. The force is not constant over the process, so the work is not force times distance — that is what the second figure is for.

Where the spring is now

position x: —
force to hold it: —
you are: —

With x measured from the natural position the force is simply kx — a straight line through the origin of slope k, which is what the second figure draws. Over a sliver of the process the force barely changes, so the work done over that sliver is the area of the thin strip standing on it, and the shaded band is every such strip from a to b added up.

The spring

x =

The spring

Force at each x