# 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 of bodies and the action of forces in producing or changing their motion - velocity and acceleration, forces and torque.

### • Mechanics

The relationships between forces, acceleration, displacement, vectors, motion, momentum, energy of objects and more.

### • Thermodynamics

Work, heat and energy systems.

## Related Documents

### 1st Law of Thermodynamics

The First Law of Thermodynamics simply states that energy can be neither created nor destroyed (conservation of energy). Thus power generation processes and energy sources actually involve conversion of energy from one form to another, rather than creation of energy from nothing.

### Bodies Moving on Inclined Planes - Acting Forces

Required forces to move bodies up inclined planes.

### Energy

Energy is the capacity to do work.

### Force

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### Heat, Work and Energy

Heat vs. work vs. energy.

### Impact Force

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### Potential Energy - Hydropower

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### Power

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

### Pump Power Calculator

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### Torque - Work done and Power Transmitted

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