11.8 : Potential Energy

A conservative force, such as a gravitational or elastic force, gives the body the capacity to do work. This capacity, measured as the potential energy, depends on the body's location or “position” relative to a fixed reference position or datum. The gravitational potential energy is considered zero at the reference point. Suppose a body is located at some vertical distance above a fixed horizontal reference or datum. In that case, the weight of the body has positive gravitational potential energy since it has the capacity to do positive work when the body is moved back down to the datum. Likewise, if the body is located at a vertical distance below the datum, the gravitational potential energy is negative since the weight does negative work when the body is moved back up to the datum.

If the height is taken as zero at the datum and the vertical upward distance is considered positive, then at some height above the datum plane, the gravitational potential energy is expressed as the product of the weight of the body and its height from the datum plane. Mathematically, this is expressed as follows:

If the body is at some distance below the datum plane, its gravitational potential energy becomes negative. Note that the datum plane for zero potential energy is arbitrary because it does not affect the change in the potential energy.

The work done on an elastic body is stored in the body in the form of elastic potential energy. This energy is available to do work on some other body while relieving its compression or extension. When a force compresses a spring, it causes a displacement x, which is directly proportional to the applied force. The elastic potential energy of the spring for a compression of x is the total work done on the spring and is expressed as follows:

When a spring is stretched, the force acts in the direction of the displacement, thus doing positive work on the spring and increasing its potential energy. The same is observed when a spring is compressed from an undeformed position.