# Work Done by Force

## Work done by a force acting on an object

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= work done (J, ft lb_{F}_{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= (20 N) (30 m)_{F}

= 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.*