Spring potential energy and Hooke's law review

The force required to stretch an elastic object such as a metal spring is directly proportional to the extension of the spring for small distances. The force exerted back by the spring is known as Hooke's law

Where $F_s$F, start subscript, s, end subscript is the force exerted by the spring, $x$x is the displacement relative to the unstretched length of the spring, and $k$k is the spring constant.

The spring force is called a restoring force because the force exerted by the spring is always in the opposite direction to the displacement. This is why there is a negative sign in the Hooke’s law equation. Pulling down on a spring stretches the spring downward, which results in the spring exerting an upward force.

The area under the force in the spring vs. displacement curve is the work done on the spring. Figure 1 shows a plot of force on the spring vs. displacement, where displacement is $0$0 when the spring is unstretched. The work done on a spring stores elastic potential energy $U_s$U, start subscript, s, end subscript in the spring until the spring goes back to its original length. Therefore, $U_s$U, start subscript, s, end subscript is equal to the work done and also to the area under the curve.

The area is a triangle with the following equation:

Note that the spring constant $k$k is the slope of the line since $k =\dfrac{\vert \vec F \vert }{ \vert \vec x \vert }$k, equals, start fraction, \vert, F, with, vector, on top, \vert, divided by, \vert, x, with, vector, on top, \vert, end fraction.

Although the spring force is a restoring force and has a negative sign, the elastic potential energy $U_s$U, start subscript, s, end subscript cannot be negative. As soon as the spring is stretched or compressed, there is positive potential energy stored in the spring.

For deeper explanations of elastic potential energy, see our video introducing springs and Hooke's law and the video on potential energy stored in a spring.

To check your understanding and work toward mastering these concepts, check out the exercise on calculating spring force and the exercise on calculating elastic potential energy.