Work and Power

Mechanical systems, an engine for example, are not limited by the amount of work they can do, but rather by the rate at which they can perform the work. This quantity, the rate at which work is done, is defined as power.

Equations for Power

From this very simple definition, we can come up with a simple equation for the average power of a system. If the system does an amount of work, W, over a period of time, T, then the average power is simply given by:

It is important to remember that this equation gives the average power over a given time, not the instantaneous power. Remember, because in the equation w increases with x, even if a constant force is exerted, the work done by the force increases with displacement, meaning the power is not constant. To find the instantaneous power, we must use calculus:

In the sense of this second equation for power, power is the rate of change of the work done by the system.

From this equation, we can derive another equation for instantaneous power that does not rely on calculus. Given a force that acts at an angle θ to the displacement of the particle,

Since = v ,

Though the calculus is not necessarily important to remember, the final equation is quite valuable. We now have two simple, numerical equations for both the average and instantaneous power of a system. Note, in analyzing this equation, we can see that if the force is parallel to the velocity of the particle, then the power delivered is simply P = Fv.

Units of Power

The unit of power is the joule per second, which is more commonly called a watt. Another unit commonly used to measure power, especially in everyday situations, is the horsepower, which is equivalent to about 746 Watts. The rate at which our automobiles do work is measured in horsepower.

Power, unlike work or energy, is not really a "building block" for further studies in physics. We do not derive other concepts from our understanding of power. It is far more applicable for practical use with machinery that delivers force. That said, power remains an important and useful concept in classical mechanics, and often comes up in physics courses.