3.5 Potential Energy

Potential energy is associated with the position of a body with respect to its surroundings. Near the Earth’s surface, gravitational potential energy is associated with height. As an object moves from one height above the earth’s surface to another, the work done in moving the object is equal to the object’s change in potential energy. Say that we begin with an object of mass m at a height h. Since we know both that work and energy and interchangable, and that energy is conserved, we know that the potential energy possessed by the object is equal to the work done when the object falls to earth.

Now, we shall let the object fall to earth so that all of its potential energy is converted. We then replace the s that represents the object’s displacement with h, the object’s original height. The object’s acceleration was due to the force of gravity, thus it was approximately 9.8 m/s². For this reason, we can replace the a in the above equation with g. Finally we get

where PE_g represents the gravitational potential energy of an object on earth. We could calculate the gravitational PE of the object on any other planet by letting g represent the planet’s acceleration of gravity.

Potential energy can also be elastic (resulting from tendancey of an object to change its length, volume, or shape in direct response to a force effecting such a change and to recove its original form upon the removel of the force). To hold a spring (or any other material that can be stretched/compressed and will exert a restoring force) either stretched or compressed an amount x from its normal length requires a force. This force is equal to the product of the spring’s displacement, x, and a constant. This constant, k, is particular to each spring, and is determined by the spring’s composition and history.

The spring exerts a restoring force in the opposite direction:

As the displacement of the spring increases, so does the magnitude of the displacing force (and the restoring force). The average applied force is

The spring will travel the distance x twice, once during compression/extension and once during restoration. Thus, elastic potential energy is defined as