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A body of mass m rests on a horizontal f...

A body 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 a minimum possible force `vecF` as shown in the diagram. The values of `theta` and `F_("min")` shall be respectively equal to

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Let the force F be applied at an angle `theta` with the forizontal as shown in figure. For veritical equilibrium,
`R+Fsintheta=mgimplies" "N=mg-Fsintheta" "(i)`
for horizontal motion F `cosgef_(1)impliesFcosthetagemuN" "[asf_(1)=muN]" "(i)`
substituting value of R from equation (i) in (ii),
`Fcosthetagemu(mg-Fsintheta)impliesFge(mumg)/((costheta+musintheta))" "(iii)`
For the force F to be minimum `(costheta+musintheta)` must be maximum,
maximum value of `costheta+musinthetaissqrt(1+mu^(2))"so that" F_(min)=(mumg)/(sqrt(1+mu^(2)))withtheta=tan^(-1)(mu)`
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