# Work done by Force

When a body is moved as a result of a force being applied to it - ** work is done ** .

### Work done by a Constant Force

The amount of work done by a constant force can be expressed as

W_{ F }= F s (1)

where

W_{ F }= work done (J, ft lb_{ f })

F = constant force acting on object (N, lb_{ f })

s = distance object is moved in direction of force (m, ft)

The unit of work in SI units is * joule (J) * which is defined as the amount of work done when a force of * 1 Newton * acts for distance of * 1 m * in the direction of the force.

*1 J (Joule) = 0.1020 kpm = 2.778x10*^{ -7 }kWh = 2.389x10^{ -4 }kcal = 0.7376 ft lb_{ f }= 1 (kg m^{ 2 })/s^{ 2 }= 1 watt second = 1 Nm = 1 ft lb = 9.478x10^{ -4 }Btu*1 ft lb*_{ f }(foot pound force) = 1.3558 J = 0.1383 kp m = 3.766x10^{ -7 }kWh = 3.238x10^{ -4 }kcal = 1.285x10^{ -3 }Btu

This is the same unit as energy .

The work done by a constant force is visualized in the chart above. The work is the product * force x distance * and represented by the area as indicated in the chart.

#### Example - Constant Force and Work

A constant force of * 20 N * is acting a distance of * 30 m * . The work done can be calculated as

W_{ F }= (20 N) (30 m)

= 600 (J, Nm)

** Example - Work done when lifting a Brick of mass 2 kg a height of 20 m above ground **

The force acting on the brick is the weight and the work can be calculated as

* W _{ F } = F s *

* = m a _{ g } s (2) *

* = (2 kg) (9.81 m/s ^{ 2 } ) (20 m) *

* = 392 (J, Nm) *

** Example - Work when Climbing Stair - Imperial units **

The work made by a person of * 150 lb * climbing a stair of * 100 ft * can be calculated as

* W _{ F } = (150 lb) (100 ft) *

* = * 15000 * ft lb *

### Work done by a Spring Force

The force exerted by springs varies with the extension or compression of the spring and can be expressed with Hooke's Law as

* F _{ spring } = - k s (3) *

* where *

* F _{ spring } = spring force (N, lb _{ f } ) *

* k = spring constant *

The work done by a spring force is visualized in the chart above. The force is zero with no extension or compression and the work is the half the product force x distance and represented by the area as indicated. The work done when a spring is compressed or stretched can be expressed as

* W _{ spring } = 1/2 F _{ spring_max } s *

* = 1/2 k s ^{ 2 } (4) *

* where *

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

* F _{ spring_max } = maximum spring force (N, lb _{ f } ) *

#### Example - Spring Force and Work

A spring is extended * 1 m * . The spring force is variable - from * 0 N * to * 1 N * as indicated in the figure above - and the work done can be calculated as

* W _{ spring } = 1/2 (1 N/m) (1 m) ^{ 2 } *

* = 0.5 (J, Nm) *

The spring constant can be calculated by modifying eq. 4 to

* k = 2 (0.5 J)/ (1 m) ^{ 2 } *

* = 1 N/m *

### Work done by Moment and Rotational Displacement

Rotational work can be calculated as

* W _{ M } = T θ (5) *

* where *

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

* T = torque or moment (Nm, ft lb) *

* θ = displacement angle ( radians ) *

#### Example - Rotational Work

A machine shaft acts with moment * 300 Nm * . The work done * per revolution (2 π radians ) * can be calculated as

* W _{ M } = (300 Nm) ( 2 π ) *

* = 1884 J *

### Representations of Work

Force can be exerted by weight or pressure:

* W = ∫ F ds *

* = ∫ m a _{ g } dh *

* =∫ p A ds *

* =∫ p dV (6) *

* where *

* W = work (J, Nm) *

* F = force (N) *

* ds = distance moved for acting force, or acting pressure (m) *

* m = mass (kg) *

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

* dh = elevation for acting gravity (m) *

* p = pressure on a surface A, or in a volume (Pa, N/m ^{ 2 } ) *

* A = surface for acting pressure (m ^{ 2 } ) *

* dV = change in volume for acting pressure p (m ^{ 3 } ) *

### Power vs. Work

Power is the ratio of * work done to used time * - or * work done per unit time. *

## Related Topics

### • Dynamics

Motion - velocity and acceleration, forces and torque.

### • Mechanics

Forces, acceleration, displacement, vectors, motion, momentum, energy of objects and more.

### • Thermodynamics

Work, heat and energy systems.

## Related Documents

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### Energy

Energy is the capacity to do work.

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### Torque - Work done and Power Transmitted

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