Work done is the product of applied force and distance

Work of a Force

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

The amount of work is the product of applied force and distance and can be expressed as

WF = F s                                         (1)


WF = work done (J, ft lb)

F = force acting on the object (N, lb)

s = distance object is moved in the direction of the 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 and a spring force can be visualized as the area under the graph in distance force diagrams like

Work - constant force and distance diagram - spring force and distance diagram

Example - Constant Force and Work

A constant force 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 - 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

WF = 1/2 k s2                                               (2)


k  = spring constant (N/m)

The spring constant can be calculated with Hooke's Law as

k = - F / x                                             (3)

  = - (1 N) / (- 1 m)

  = 1 N/m

Using (2)

WF = 1/2 (1 N/m) (1 m)2 

   = 0.5 (J, Nm)

Example - Work 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 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 of a Couple Moment

A couple moment does work when it undergoes a rotational displacement. The rotational work can be expressed as

WM = T θ               (4)


WM = rotational work done (J, ft lb)

T = torque or moment (Nm, ft lb)

θ = displacement angle (radians)

Representations of Work 

Force can be exerted by weight or pressure:

W = ∫ F ds

    = ∫ m ag dh

    =∫ p A ds

    =∫ p dV                                             (5)


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

Related Documents

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