Spring Potential Energy

When you pull a spring you feel a force in the opposite direction from which you pull. Also the force increases with distance. This can be expressed as

in the x direction. Thus

which gives , i.e.  and . Thus we can simplify and write

which is our expression for spring potential energy

Example A spring with spring constant k has a mass of m on one end. The spring is stretched by a distance d. When released, how fast will the mass be moving when it returns to its original position? (Assume the motion occurs on a horizontal frictionless surface.)

Solution W_NC = 0 giving

Now x_f = 0, x_i = d m and v_i = 0. Thus

or

IMPORTANT NOTE:

The spring is an example of a variable force F = -kx which varies as distance. Thus the acceleration  is not constant and the constant acceleration equations cannot be used to solve the previous example. Also the variable force requires the integral definition of work as