# Power

## Power is the rate at which work is done or energy converted

### Work done

Power is a measure of the rate at which work is done - work done per unit time - and can be expressed as

P = Work / time

= W / dt (1)

where

P = power (W, ft lb/s)

W = work done (J, ft lb)

dt = time taken (s)

Work can be expressed as

* W = F s (1b)*

* where *

* F = force (N, lb)*

* s = distance (m, ft) *

Since work is the product of the applied force and the distance, power *(1)* can be modified to

P = F v (1c)

where

F = force (N, ft)

v = velocity (m/s, ft/s)

### Energy converted

Power is also a measure of the rate at which energy is converted from one form to another and can be expressed as

P = E / dt (2)

where

P = power (W, ft lb/s)

E = energy converted (J, ft lb)

dt = time taken (s)

### Example - Power needed to lift a Mass

A mass of *1000 kg* is lifted *10 m* in *10 seconds.*

The work done can be calculated as

W = (1000 kg) (9.81 m/s^{2}) (10 m)

= 98100 J (Nm)

The power required is work done per unit time

* P = (98100 J) / (10 s)*

* = 9810 W (J/s) *

* = 9.8 kW*

### Example - Work done by an Electric Motor

The work done by a *1 kW* electric motor in *1 hour* can be calculated by modifying *(1)* to

W = P dt

= (1 kW) (1000 W/kW) (1 h) (3600 s/h)

= 3600000 J

= 3600 kJ

### Example - Electric Winch lifting a Mass

An electric winch is lifting a mass of *100 kg 10m* above the ground. The electric motor in the winch is *500 W*. The force (weight) acting on the mass due to the acceleration of gravity can be calculated as

*F = m g *

* = (100 kg) (9.81 m/s ^{2})*

* = 981 N*

* = 0.98 kN*

The work done by the winch can be calculated with *(1b)* as

*W = F s*

* = (981 N) (10 m)*

* = 9810 Nm (J)*

The time required for the winch with the actual motor to lift the mass can be calculated by modifying *(1)*

*dt = W / P*

* = (9810 J) / (500 W) *

* = 19.6 s*

### Power - kW versus Horsepower

Power can also be expressed in *horsepower (hp)*:

*1 kW = 1.341 hp*

*1 hp = 0.7457 kW*

* *

*1 kW = 1000 Watts = 3412 Btu/h = 737.6/550 British hp = 1.341 British hp = 10*^{3}/9.80665 kg_{f}m/s = 737.6 ft lb_{f}/s = 10^{3}/(9.80665 75) metric hp*1 hp (English horse power) = 745.7 W = 0.746 kW = 550 ft lb/s = 2545 Btu/h = 33000 ft lb/m = 1.0139 metric horse power ~= 1.0 KVA*

### Comparative Power Outputs

Comparative power outputs are indicated below:

- human sound - normal speech : 10
^{-8}kW - human output: 0.1 kW (daily average), 0.2 kW (walking), 1 kW (running), 1.7 kW (sprinting)
- small scooter : 4 kW
- family car : 40 kW
- light aircraft : 150 kW
- sports car : 240 kW
- helicopter : 400 kW
- formula 1 car : 600 kW
- locomotive : 1500 kW
- cargo vessel : 6000 kW
- military aircraft : 30000 kW
- warship : 60000 kW
- airliner : 80000 kW
- launch rocket : 400000 kW