Question

What is zero work? When can work be done be zero?

Answer 1

Hint: Magnitude of work done by or on a body is the product of the force applied by or on the body and the displacement in that direction. This happens when the force applied on a body and its displacement are perpendicular to each other, or when the force applied on the body produces zero displacements of the body ( A person trying to lift a massively heavy rock, but it does not budge).
Formula used:
$W=\overrightarrow{F}.\text{ }\overrightarrow{s}$
$W=\left| \overrightarrow{F} \right|.\left| \overrightarrow{s} \right|\cos \theta $
where W is the work done,$\overrightarrow{F}$ and $\overrightarrow{s}$ are the force and displacement vectors respectively. $\theta $ is the angle between $\overrightarrow{F}$ and $\overrightarrow{s}$ vectors.

Complete step by step answer:
The magnitude of work done by or on a body is the product of the force applied by or on the body and the displacement in that direction. The mathematical expression of work is given by
$W=\overrightarrow{F}.\text{ }\overrightarrow{s}$
$W=\left| \overrightarrow{F} \right|.\left| \overrightarrow{s} \right|\cos \theta $
where W is the work done,$\overrightarrow{F}$ and $\overrightarrow{s}$ are the force and displacement vectors respectively. $\theta $ is the angle between $\overrightarrow{F}$ and $\overrightarrow{s}$ vectors.
When the work is done is zero in some situations, regardless of whether force is applied or not, is known as zero work is done.
Zero work is done when the displacement of a body is zero or perpendicular ($\theta ={{90}^{0}},\cos \theta =0$) to the direction of force applied, then work done is zero. For example, if a person tries to push a wall, he is applying force yet the wall does not move, so the displacement of the wall is zero, and hence work done is zero.
An example of the perpendicular force and displacement is the example of a porter in a railway station. He carries luggage on his head and moves forward, however, to hold the luggage up he applies a force in the vertical direction (against gravity). However, the displacement is in the horizontal plane. Thus, the force applied and the displacement are in perpendicular directions. So, the work done is zero.

Note: The zero work concept can be quite confusing. However, to determine zero work, one must always consider the force and displacement vectors mathematically. If any of them are zero or perpendicular to each other.

If starting and ending points are the same, then also the displacement is zero. However, this is only true when the force in question is a conservative force (like gravitational force). However, for non-conservative forces like frictional force, even if starting and ending points are the same, work done is not zero. It becomes a path function.
If the work done is zero, then the power expended will also be zero, since power is nothing but the rate of doing work and if work is done is zero, then obviously the rate of work done will be zero.