# Power

The quantity work has to do with a force causing a displacement. Work has nothing to do with the amount of time that this force acts to cause the displacement. Sometimes, the work is done very quickly and other times the work is done rather slowly. For example, a rock climber takes an abnormally long time to elevate her body up a few meters along the side of a cliff. On the other hand, a trail hiker (who selects the easier path up the mountain) might elevate her body a few meters in a short amount of time. The two people might do the same amount of work, yet the hiker does the work in considerably less time than the rock climber. The quantity that has to do with the rate at which a certain amount of work is done is known as the power. The hiker has a greater power rating than the rock climber.

Power is the rate at which work is done. It is the work/time ratio. Mathematically, it is computed using the following equation.

Power = Work / time

or

P = W / t

The standard metric unit of power is the Watt. As is implied by the equation for power, a unit of power is equivalent to a unit of work divided by a unit of time. Thus, a Watt is equivalent to a Joule/second

The expression for power is work/time. And since the expression for work is force * displacement, the expression for power can be rewritten as (force * displacement) / time. Since the expression for velocity is displacement / time, the expression for power can be rewritten once more as force * velocity. This is shown below.

Power = Work/Time = (Force * Displacement)/Time

Power = Force * (Displacement/Time)

Power = Force * Velocity

This new equation for power reveals that a powerful machine is both strong (big force) and fast (big velocity).

Example 1: An elevator can carry a maximum load of 1800 kg (elevator + passengers) is moving up with a constant speed of 2 ms–1. The frictional force opposing the motion is 4000 N. Determine the minimum power delivered by the motor to the elevator in watts as well as in horse power.

Solution: The downward force on the elevator is

F = mg + Ff = (1800 × 10) + 4000 = 22000 N

The motor must supply enough power to balance this force. Hence,

P = F.v = 22000 × 2 = 44000 W = 59 hp