A particle of mass m rests on a horizontal floor with which it has a coefficient of static friction `mu`. It is desired to make the body move by applying the minimum possible force F. Find the magnitude of F and the direction in which it has to be applied.

Explanation

Let the force F be applied at an angle `theta` with the horizontal as shown in figure.
For vertical equilibrium,
`N+F sin theta=mgrArrN=mg-F sin theta`
for horizontal motion `F cos theta gef_(1)rArr F cos thetagemuN ["as" f_(L)=muN]`
substituting value of N from equation (i) in (ii)
`F cos thetage mu(mg-F sin theta)rArrFge(mu mg)/((cos theta_mu sin theta))`
For the force F to be minimum `(cos theta+mu sin theta)` must be maximum,
maximum value of `cos theta+mu sin theta` is `sqrt(1+mu^(2))` so that `F_("min")=(mumg)/sqrt(1+mu^(2))` with `theta=tan^(-1) (mu)`