Physics · Ch 5 — Work, Energy and Power

Power

5.10

Power

The Meaning of Power in Physics

In everyday language, "power" is often confused with "force" or "energy." In physics, power has a precise meaning: it is the rate at which work is done or the rate at which energy is transferred from one body to another. The key word is rate — power tells you how quickly work happens, not just how much work is done.

If a machine lifts a heavy load in 2 seconds and another machine lifts the same load in 10 seconds, both do the same amount of work. But the first machine has greater power because it does the work in less time. This distinction is crucial in everything from designing engines to understanding how your body uses energy during exercise.

Definition and Formula for Average Power

The average power $\bar{P}$ is defined as the total work done $\Delta W$ divided by the total time taken $\Delta t$ :

\bar{P} = \frac{\Delta W}{\Delta t}

Since work and energy are measured in joules (J) and time in seconds (s), the SI unit of power is the joule per second, which is given the special name watt (W):

1 \text{ W} = 1 \text{ J s}^{-1}

Note

The watt is named after James Watt, the Scottish engineer who improved the steam engine. You will also encounter the kilowatt ( $1 \text{ kW} = 10^3 \text{ W}$ ) and the megawatt ( $1 \text{ MW} = 10^6 \text{ W}$ ) in practical contexts.

Instantaneous Power

The formula $\bar{P} = \Delta W / \Delta t$ gives the average power over a time interval. To find the power at a specific instant, we take the limit as the time interval approaches zero — that is, we use the derivative:

P = \lim_{\Delta t \to 0} \frac{\Delta W}{\Delta t} = \frac{dW}{dt}

This is the instantaneous power. It tells you the rate at which work is being done at a particular moment.

Power in Terms of Force and Velocity

There is a very useful alternative expression for instantaneous power. Recall that work done by a constant force $\mathbf{F}$ over a small displacement $d\mathbf{r}$ is $dW = \mathbf{F} \cdot d\mathbf{r}$ . Substituting this into the definition of instantaneous power:

P = \frac{dW}{dt} = \frac{\mathbf{F} \cdot d\mathbf{r}}{dt} = \mathbf{F} \cdot \frac{d\mathbf{r}}{dt}

But $d\mathbf{r}/dt$ is the instantaneous velocity $\mathbf{v}$ . Therefore:

P = \mathbf{F} \cdot \mathbf{v}

P = \mathbf{F} \cdot \mathbf{v}

This is a powerful result. It tells you that the instantaneous power delivered by a force is the dot product of the force vector and the velocity vector of the point of application.

Watch out

A common mistake is to think that a large force always means large power. If the velocity is zero (e.g., you are pushing against a wall that does not move), the power is zero — no work is being done, no matter how hard you push. Power depends on both force and how fast the point of application moves.

If the force is constant and the motion is in the same direction, the formula simplifies to $P = F v$ , where $F$ and $v$ are the magnitudes of force and velocity.

The Kilowatt-Hour: A Unit of Energy, Not Power …

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