# Torque - Work done and Power Transmitted

## The work done and power transmitted by a constant torque.

### Work done

Work done is *the force multiplied with the distance moved by the force* - and can be expressed as

W = F s (1)

where

W = work done (J, Nm)

F = force (N)

s = distance moved by force (s)

For an angular motion

the work done can be expressed as

W = F θ r

= T θ (2)

where

W = work (Joules)

θ = angle (radians)

r = radius (m)

T = torque or moment (Nm)

### Power transmitted

Power is the ratio between the work done and the time taken and can be expressed as

P = W / dt

= T θ / dt

= T ω

= 2 π n T

= 2π (n_{rpm }/ 60) T

= 0.105(3)n_{rpm}T

where

P = power (Watts)

dt = time taken (s)

ω = θ / dt = 2 π n = angular velocity (rad/s)

n = speed (rev/s)

n_{rpm}= speed (rev/min, rpm)

**Note!** - a machine must rotate to produce power! A machine with no rotation can deliver torque - like an electric motor - but since no distance is moved by force - no power is produced. As soon as the machine starts to rotate power is produced.

### Example - required Torque to produce Power

A machine rotates with speed *3000 rev/min (rpm)* and consumes *5 kW*. The torque at the shaft can be calculated by modifying (3) to

T = P / 2 π n

= (5 kW) (1000 W/kW) / 2 π (3000 rev/min) / (60 sec/min)

= 15.9 Nm