A block of weight W rests on a horizontal floor with coefficient of static friction . It is desired to make the block move by applying minimum amount of force. The angle from the horizontal at which the force should be applied and magnitude of the force F are respectively.

Step-by-step solution

Let the force F is applied at an angle with the horizontal. For horizontal equilibrium, F = R For vertical equilibrium, R+F = mg or, R=m g-F Substituting this value of R in eq. (i), we get aligned & F = ( mg -F ) & = mg - F & or, F( + )= mg & or, F = mg + aligned For F to be minimum, the denominator ( + ) should be maximum. d d ( + )=0 or, - + =0 or, = or, = ⁻¹( ) Then, = 1+ ^2 and = 1 1+ ^2 Hence, F_ min = w 1 1+ ^2 + ^2 1+ ^2 = w 1+ ^2