# Elevators - Force and Power

## Required force and power to lift an elevator.

### Work done by Lifting the Elevator

The work done by lifting an elevator from one level to an other can be expressed as

* W = m a _{g} (h_{1} - h_{0}) (1)*

*where *

*W = work done (J, ft lb _{f})*

*m = mass of elevator and passengers (kg, lb _{m})*

*a _{g} = acceleration of gravity (9.81 m/s^{2}, 32.17 ft/s^{2})*

*h _{1} = final elevation (m, ft)*

*h _{0} = initial elevation (m, ft)*

The generic equation for work done by a force can be expressed as

*W = F _{c} s (2)*

*where *

*W = work done by force (J, ft lb _{f})*

*F _{c} = force acting on the elevator at constant speed (N, lb_{f})*

*s = distance moved by elevator (m, ft)*

### Forces acting on the Elevator

Since works done in *(1)* and *(2)* are equal - the equations can be combined to

*F _{c}* s = m a

_{g}(h

_{1}- h

_{0}) (3)

#### Force at constant Speed

Since the difference in elevation and the distance moved by the force are equal - *(3)* can be modified to express the force required to move the elevator at constant speed to

*F _{c}* = m a

_{g}(4)

#### Force at start/stop

When the elevator starts or stops - the acceleration or deceleration force in addition to the constant speed force can be expressed as

*F _{a} = m (v_{1} - v_{0}) / t_{a} (5)*

*where *

*F _{a} = acceleration force (N, lb_{f})*

*v _{1} = final velocity (m/s, ft/s)*

*v _{0} = initial velocity (m/s, ft/s) *

*t _{a} = start or stop (acceleration) time (s)*

### Power required to move the Elevator

The power required to move the elevator can be calculated as

*P = W / t *

* = m a_{g} (h_{1} - h_{0}) / t (6)*

*where *

*P = power (W, ft lb _{f})*

*t = time to move the elevator between levels (s)*

### Example - Force and Power to Lift an Elevator

An elevator with mass *2000 kg* including passengers are moved from level *0 m* to level *15 m*. The force required to move the elevator at constant speed can be calculated as

*F _{c}* = (2000 kg) (9.81 m/s

^{2})

* = 19820 N*

The power required to move the elevator between the levels in *20 s* can be calculated as

*P = (2000 kg) (9.81 m/s^{2}) ((15 m) - (0 m)) / (20 s)*

* = 14865 W *

* = 14.9 kW*