Question

A force acts on a body and displaces it by a distance S in a direction at an angle $\theta$ with the direction of force. What should be the value of $\theta$ to get the maximum positive work ?

Answer 1

In order to solve above problem first write the formula of work done in terms of $\theta$ i.e.,
$W = FS\cos \theta$
Now use the concept that for maximum work done i.e., for maximum LHS, RHS also should be maximum.Hence, we will get a desired solution.

Complete step by step answer:
We know that if a body displaces by distance S on acting the force F, then work done by the force is
$W = FS\cos \theta$ …..(1)
Where $\theta$ is the angle between force F and displacement S.
Here, we have to calculate $\theta$ for maximum work done W. So, for maximum work done RHS of equation 1 will also be maximum.
We know that the maximum value of $\cos \theta$ is 1.
$\Rightarrow \cos \theta = 1$
$\Rightarrow \cos \theta = \cos 0^\circ$
$\therefore \theta = 0^\circ$

Thus, displacement of the body must be in the same direction as that of force to get the maximum positive work.

Note: In many problems, student may get confused between maximum and minimum values of $\sin \theta$ and $\cos \theta$ which are given as,
${(\cos \theta )_{\max }} = 1$ ${(\sin \theta )_{\max }} = 1$
${(\cos \theta )_{\min }} = - 1$ ${(\sin \theta )_{\min }} = 0$
And when $\theta$ is very small then,
$\sin \theta \approx \theta ,\cos \theta \approx 1$