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# 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

WF = F s                                         (1)

where

WF = work done (J, ft lbf)

F = constant force acting on object (N, lbf)

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 lbf = 1 (kg m2)/s2 = 1 watt second = 1 Nm = 1 ft lb = 9.478x10-4 Btu
• 1 ft lbf (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

WF = (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

WF = F s

= m ag s                               (2)

= (2 kg) (9.81 m/s2) (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

WF = (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

Fspring = - k s                        (3)

where

Fspring = spring force (N, lbf)

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

Wspring = 1/2 Fspring_max s

= 1/2 k s2                             (4)

where

Wspring = work done (J, ft lbf)

Fspring_max = maximum spring force (N, lbf)

#### 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

Wspring = 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

WM = T θ                        (5)

where

WM = rotational work done (J, ft lb)

T = torque or moment (Nm, ft lb)

#### Example - Rotational Work

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

WM = (300 Nm) (2 π)

= 1884 J

### Representations of Work

Force can be exerted by weight or pressure:

W = ∫ F ds

= ∫ m ag 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)

ag = acceleration of gravity (m/s2)

dh = elevation for acting gravity (m)

p = pressure on a surface A, or in a volume (Pa, N/m2)

A = surface for acting pressure (m2)

dV = change in volume for acting pressure p (m3)

### 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 torques
• Mechanics - Forces, acceleration, displacement, vectors, motion, momentum, energy of objects and more
• Thermodynamics - Effects of work, heat and energy on systems

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