When is the work done minimum?

Question

Work $W$ is defined as the dot product of the vectors force $F$ and displacement $s$.

The dot product can be written as the products of the magnitude of the force, magnitude of the displacement and the cosine of the angle between the force and displacement.

W=|F|.|s|cos(θ)

I want to know whether the work done by a force is minimum when θ=90° or when θ=180°. I've found only one other thread that discusses this (Why is the minimum work done not negative in this case?) but the answers were inadequate.

• The first answer used differentiated between work and absolute work.
• The second answer did not answer the question.

Here's what I already know.

• at θ=0° the work done is +Fs
• at θ=90° the work done is 0
• at θ=180° the work done is -Fs
• When 0° < θ < 90° the work done is positive
• When 90° < θ < 180° the work done is negative

If it was just an integer, then the least 'work' done would be when θ=180° since a negative number is lesser than zero. But because it is work I want to know whether the sign ('+' or '-') should be considered when considering at which angle the work done is minimum.

Here are a couple of reasonings I have (I'm not sure which of these are correct, if any):

Reasoning 1:

The work-energy theorem states

Work-energy theorem states that the work done by the sum of all forces acting on a particle equals the change in the kinetic energy of the particle (or) The net work done on a system is equal to the change in Kinetic energy.

The second last step in the proof below shows that

work = final kinetic energy - initial kinetic energy

$$W = KE_f - KE_i.$$

In my opinion this shows that the work done is an increase in energy and not just a change in energy. A change in energy could be either an increase or decrease in energy.

An increase is final - initial. A decrease is initial - final.

By convention an increase in energy is considered to be positive work and a decrease in energy is considered to be negative work.

The definition of the work-energy theorem on Wikipedia states that:

The work–energy principle states that an increase in the kinetic energy of a rigid body is caused by an equal amount of positive work done on the body by the resultant force acting on that body. Conversely, a decrease in kinetic energy is caused by an equal amount of negative work done by the resultant force.

This already takes into account this convention.

Based on the above I consider work to be an increase in energy. Using this definition, the work done by a force is minimum when θ=180°.

Reasoning 2

However, work is a scalar. The sign of work ('+' or '-') represents the flow of energy. If the sign solely represents the flow of energy then a work of -Fs would be more than a work of 0. This would imply that the least possible work can be when the work done is 0 which is when θ=90°.

This is what I've done so far. I would appreciate any further input or corrections to what I've stated above.

Answer 1

I've found only one other thread that discusses this (Why is the minimum work done not negative in this case?) but the answers were inadequate.

The first answer used differentiated between work and absolute work.

The first answer in the other thread does answer your question as best as it is possible to answer.

The issue is that when you are minimizing something there is a purpose. Sometimes that purpose will require you to minimize the absolute value of the work done. That will be minimized at 90 degrees. Sometimes that purpose will require you to minimize the value of the work itself. That will be minimized at 180 degrees.

Since neither this question nor the linked question gives any context to decide which quantity should be minimized, this is the best answer possible. If we were to just say one or the other then someone trying to apply this general question to a specific scenario might be misled.