How to Find the Velocity of an Object before Collision
Relating Impulse and Momentum through the Impulse-Momentum Theorem
How to Determine Whether a Collision Is Elastic or Inelastic

How to Calculate the Potential and Kinetic Energy in a Spring

In physics, you can examine how much potential and kinetic energy is stored in a spring when you compress or stretch it. The work you do compressing or stretching the spring must go into the energy stored in the spring. That energy is called elastic potential energy and is equal to the force, F, times the distance, s:

W = Fs

As you stretch or compress a spring, the force varies, but it varies in a linear way (because in Hooke’s law, force is proportional to the displacement).

image0.png

The distance (or displacement), s, is just the difference in position, xfxi, and the average force is (1/2)(Ff + Fi). Therefore, you can rewrite the equation as follows:

image1.png

Hooke’s law says that F = –kx. Therefore, you can substitute –kxf and –kxi for Ff and Fi:

image2.png

Distributing and simplifying the equation gives you the equation for work in terms of the spring constant and position:

image3.png

The work done on the spring changes the potential energy stored in the spring. Here’s how you give that potential energy, or the elastic potential energy:

image4.png

For example, suppose a spring is elastic and has a spring constant, k, of

image5.png

and you compress the spring by 10.0 centimeters. You store the following amount of energy in it:

image6.png

You can also note that when you let the spring go with a mass on the end of it, the mechanical energy (the sum of potential and kinetic energy) is conserved:

PE1 + KE1 = PE2 + KE2

When you compress the spring 10.0 centimeters, you know that you have 


image7.png

of energy stored up. When the moving mass reaches the equilibrium point and no force from the spring is acting on the mass, you have maximum velocity and therefore maximum kinetic energy — at that point, the kinetic energy is

image8.png

by the conservation of mechanical energy.

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