Question

find minimum force required

Answer 1

I am taking the angle theta as x and friction coefficient as u.

Normal force acting on the block, N = Mg - F sin x

Maximum Friction, f = uN = u(Mg - F sin x)

Net horizontal force = F cos x - f = F cos x - u(Mg - F sin x)

For the block to start sliding, the net horizontal force should be slightly more than 0.

Thus F cos x - u(Mg - F sin x) = 0

=> F cos x - uMg + uF sin x = 0

=> F(cos x + u sinx) = uMg

=> F = uMg/(cos x + u sin x)


For F to be minimum, (cos x + u sin x) should be maximum.

Maximum value of (cos x + u sin x) = √(1+u^2).

Thus, minimum value of F = uMg/√(1+u^2)

Answer is 3.