Work, Energy and Power

The word work, $W$, in physics has a very precise meaning different from its colloquial usage. The work done on an object is defined as the force ($F$) acting on the object multiplied by the distance ($d$), in the direction of that force, that the force acts on the body. In symbols,

$$W = F \times d$$ (2.1)

The unit of work is therefore the "Newton-metre", which is also called the Joule, $J$, in honour of the physicist James Prescott Joule, a pioneer in the study of energy conservation. The work performed on a body can be used to increase its energy. How much work is required to lift an object of mass $m$, a height $h$? If $g$ is the acceleration due to gravity, then we know from Newton's Second Law that the force of gravity on the object is $mg$. Therefore we have to supply an equal but opposite force to move the object upwards a height $h$, doing an amount of work equal to $mgh$. This work, by definition, has been used to increase the potential energy of the object by $mgh$. That is,

$$U= m g h .$$ (2.2)

Energy due to position or the internal configuration of a material body is called potential energy, $V$. For example, a stone on a mountain top has gravitational potential energy due to its position, because when released it will roll down the mountain gaining speed, and hence kinetic energy, or energy due to motion. What is happening of course is that gravitational force does work on the stone, and this work causes the increase in kinetic energy of the stone. Conversely, to raise a stationary stone from the bottom of the mountain to the top requires us to perform work against the gravitational force. Then when the stone has been brought to the top, we say it has gained potential energy--the potential to increase its kinetic energy by rolling down the hill again. Energy due to motion in space is called kinetic energy, $T$. It is useful to note that the formula for the kinetic energy of an object of mass $m$ moving at speed $v$ is given by

$$T = {1 \over 2} m v^2 .$$ (2.3)

Notice that a massive object has more kinetic energy than a lighter one, even when they are both travelling at the same speed. Often one is interested at the rate at which work is performed, or the rate at which energy flows in some situation. Power is the rate of doing work,

$$P= {\Delta W \over \Delta t}$$ (2.4)

The metric unit of power is clearly $J/s$ and this is called a Watt, after James Watt, the person who invented the term "horsepower" to denote the power of his steam engines.