Physics · Ch 4 — Work, Energy and Power

Elastic Potential Energy

4.2.6

Elastic Potential Energy

A stretched or compressed spring stores energy too, called elastic potential energy. Consider a mass on a smooth horizontal surface attached to a spring whose other end is fixed to a wall, with $x=0$ marking the spring's natural (equilibrium) length. Stretching the spring by a distance $x$ develops an internal restoring force obeying Hooke's law,

\vec F_s = -k\vec x

where $k$ is the spring's force constant and the negative sign shows the restoring force always opposes the displacement. The external applied force needed to produce this stretch is equal and opposite, $\vec F_a = +k\vec x$ -- and, unlike the constant force of gravity, this is a variable force, since it grows in proportion to $x$ itself.

The elastic potential energy stored is the work done by this applied force, integrated from the equilibrium position (the natural lower limit, since $U=0$ there) up to the final elongation $x$ :

U = \int_0^x F_a\,dx = \int_0^x kx\,dx = \tfrac{1}{2}kx^2

$\boxed{U = \tfrac{1}{2}kx^2}$ …

Figure 4.10Force-displacement graph for a spring

What this figure shows. Because the restoring spring force F = -kx is linearly proportional to the displacement x but opposite in sign, plotting F against x produces a straight line through the origin that passes only through the second and fourth quadrants. The elastic potential energy stored for a given elongation or compression is read off as the area of the shaded triangle bounded by this line, the displacement axis, and the vertical at that displacement, giving the same result, (1/2) k x s …

Figure 4.11Potential energy-displacement graph for a spring-mass system

What this figure shows. This graph plots the spring's potential energy as an upward-opening parabola in the displacement x, together with the complementary kinetic energy curve of the attached mass, on the same axes, showing that their sum -- the total mechanical energy -- stays constant (a horizontal line) at every displacement. As the mass oscillates back and forth, energy is repeatedly exchanged between the potential-energy parabola and the kinetic-energy curve, being purely potential at the extreme positions and purely kinetic as the mass crosses the equi …

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