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.
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):
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°.
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.