Work

Work is a general term that we use in our daily life to assess execution or completion of a task. The basic idea is to define a quantity that can be used to determine both "effort" and "result". In physics also, the concept of work follows the same basic idea. But, it is completely "physical" in the sense that it recognizes only force as the "effort" and only displacement as the "result". There is no recognition of mental or any other effort that does not involve physical movement of a body.

For a constant force (F) applied on a particle, work is defined as the product of "component of force along the direction of displacement" and "the magnitude of displacement of the particle". Mathematically,

$$\begin{array}{l}W=\left(F\cos \theta \right)r=Fr\cos \theta \end{array}$$

where "θ" is angle between force and displacement vectors and "r" is magnitude of displacement. The scalar component of force is also known as the projection of force. SI unit of work is Newton - meter (N-m). This is equivalent to the unit of energy $\mathrm{kg}-m^2{s}^{-2}$ i.e. Joule (J).

In order to appreciate the working of the formula to compute work, let us consider an example. A block is being pulled by an external force "F" on a smooth horizontal plane as shown in the figure. The work (W) by force (F) is :

$$\begin{array}{l}W=Fr\cos \theta \end{array}$$

The perpendicular forces i.e. normal force "N" and weight of block "mg" (not shown in the figure) do "no work" on the block as cosθ = cos90° = 0. An external force does the maximum work when it is applied in the direction of displacement. In that case, θ = 0°, cos0° = 1 (maximum) and Work $W_G=Fr$ .

Work as vector dot product

Work involves two vector quantities force and displacement, but work itself is a signed scalar quantity. Vector algebra provides framework for such multiplication of vectors, yielding scalar result via multiplication known as dot product. The work done by the constant force as dot product is :

$$\begin{array}{l}W=\mathbf{F}\mathbf{.}\mathbf{r}=Fr\cos \theta \end{array}$$

where "θ" is the angle between force and displacement vectors.

Computation of work

Sign of work

Work is a signed scalar quantity. It means that it can be positive or negative depending on the value of angle between force and displacement. We shall discuss the significance of the sign of work in a separate module. We should, however, be aware that the sign of work has specific meaning for the body on which force works. The sign determines the direction of energy exchange taking place between the body and its surrounding.

It is clear that the value of "cosθ" decides the sign of work. However, there is an easier method to determine sign of work. We determine the magnitude of work considering projection of force and displacement - without any consideration of the sign. Once magnitude is calculated, we simply check whether the component of force and displacement are in same direction or in opposite direction? If they are in opposite direction, then we put a negative sign before the magnitude of work.

Work by the named force

A body like the block on an incline is subjected to many forces viz weight, friction, normal force and other external forces. Which of the forces do work? Does the work is associated with net force or any of the forces mentioned? In physics, we can relate work with any force or the net force working on the body. The only requirement is that we should mention the force involved. For this reason, we may be required to calculate work by any of the named forces.