4.4 Power

4.4 Power

In the definition of work, it is not clear, whether the same amount of work is done in one second or in one hour. The rate, at which work is done, is often of interest in practical applications.

Power is the measure of the rate at which work is being done.

If work $\Delta W$ is done in a time interval $\Delta t$, then the average power $P_{\mathrm{av}}$ during the interval $\Delta t$ is defined as:

If work is expressed as a function of time, the instantaneous power $P$ at any instant is defined as:

where $\Delta W$ is the work done in short interval of time $\Delta t$.

Since $\Delta W = \mathbf{F}\cdot\Delta\mathbf{d}$

Hence $P = (\mathbf{F}\cdot\Delta\mathbf{d})/\Delta t = \mathbf{F}\cdot(\Delta\mathbf{d}/\Delta t)$

Since $\Delta\mathbf{d}/\Delta t = \mathbf{v}$

Hence

Thus, power may also be defined as the scalar product of $\mathbf{F}$ and $\mathbf{v}$.

The SI unit of power is watt, defined as one joule of work done in one second.

Sometimes, for example, in the electrical measurements, the unit of work is expressed as watt second. However, a commercial unit of electrical energy is kilowatt-hour.

One kilowatt-hour is the work done in one hour by an agency whose power is one kilowatt.

Therefore

or

For your information — Approximate Power

Example 4.2

A $70\,\mathrm{kg}$ man runs up a long flight of stairs in $4.0\,\mathrm{s}$. The vertical height of the stairs is $4.5\,\mathrm{m}$. Calculate his power output in watts.

Solution

Work done $= mgh$

Power $P = mgh/t$