A block of mass `m` lying on a horizontal surface (coefficient of static friction `= mu_(s))` is to be brought into motion by a pulling force `F` At what angle `theta ` with the horizontal should the force `F` be applied so that its magnitude is minimum ? Also find the minimum magnitude

Explanation

`uarr:Fsinalpha+N=mgimpliesN=mg-Fsinalpha`
`f=muN=mu(mg-Fsinalpha)`
Since the block moves slowly i.e., net horizontal force `=0`
`Fcosalpha=f=mu(mg-Fsinalpha)`
`F=(mumg)/(cosalpha+musinalpha)`
Work done by the applied force during displacement `d`
`W=Fcosalphad=(mumgcosalphad)/(cosalpha+musinalpha)=(mumgd)/(1+mutanalpha)`
If `F` is minimum `mu=tanalpha` (see the chapter: friction)
`W=(mumgd)/(1+mu^2)`