The Work Done by a Spring and the Associated Potential Energy
A body attached to a spring which exerts a force on it described by
Hooke’s Law, Fx = -kx, moves along the x-axis from position x1 to
position x2. (Implicit in writing Hooke’s Law in this form is the
assumption that x = 0 corresponds to the equilibrium position of the
spring where Fx = 0.) The work done on it by that spring is
x2
x2
x1
x1
W spring   Fspring dx  
0
x1
x2
x
2
1
 1

 kx dx  - kx 2    k ( x 22  x12 ) .
2
 2
 x1
Now suppose that the spring provides the only force doing work on the body. This could be the
case if the supporting surface is frictionless, there is no air resistance and spring flexure
dissipates no energy. In such a case, by the work- kinetic energy theorem, Wspringv = Wnet = ΔK .
Now suppose that x2 > x1. If the body traverses a distance Δx = x2 – x1, then the spring does
1
negative work on it, W spring   k ( x 22  x12 ), and its kinetic energy decreases by this amount, i.e.,
2
1
K   k ( x 22  x12 ). However if the same body is pulled in the opposite direction through the
2
1
same path by the spring, then the spring does positive work on it, W spring  k ( x 22  x12 ), and its
2
1
kinetic energy increases by this same amount, i.e., K  k ( x 22  x12 ) . A body pulled from x1
2
to x2 then, by whatever means, acquires the potential to have the spring impart a kinetic energy,
1
K  k ( x 22  x12 ) , to it as it returns through this interval. Thus we call this quantity the
2
1
“spring potential energy,” i.e., U sp  k ( x 22  x12 ) , where we have let Usp symbolize the
2
1
spring’s potential energy. (Note that k ( x 22  x12 ) represents the potential energy difference
2
between body positions x2 and x1. It turns out that only potential energy differences matter.
Thus we can set Usp = 0 at any position we choose as long as our other designations of Usp in that
same problem are consistent with that arbitrary choice.)