An object with weight W is dragged along a horizontal plane by a force acting along a rope?

Question

attached to the object. If the rope makes an angle θ with a plane, then the magnitude of the force is given below, where μ is a constant called the coefficient of friction.

F = (μ * W) / (μ * sin( μ ) + cos( μ ))

1) For what value of θ is F the smallest?

2) What is the smallest value of F?

Answer 1

Fcosθ = μ(W - Fsinθ)
F = (μW) / (μsin(θ) + cos(θ)), 0 ≤ θ < 90°
1)
F' = - ((μW)/(μsin(θ) + cos(θ))^2)(μcos(θ) - sin(θ))
μcos(θ) - sin(θ) = 0 for minimum F
sin(θ) = μcos(θ)
tan(θ) = μ
2)
F = (μW) / (μ^2 / √(1 + μ^2) + 1 / √(1 + μ^2))
F = (μW) / √(1 + μ^2)