Q.45. Application of derivatives

Question 45.
An object with weight W is dragged along a horizontal plane by a force acting along a rope attached to the object. If the rope makes an angle θ with the plane, then the magnitude of the force is...
F = (uW) / (u sin θ + cos θ), where u is a constant called the 'coefficient of friction'.
a) Find the rate of change of F with respect to θ.
b) When is this rate of change equal to 0?

Answer 45.
a)
Given equation is
F (usinθ + cosθ) = uW
Differentiating w.r.t. θ,
dF/dθ * (usinθ + cosθ) + F (ucosθ - sinθ) = 0
=> dF/dθ
= - F (ucosθ - sinθ) / ( usinθ + cosθ)
= - uW (usinθ + cosθ) (ucosθ - sinθ) / (usinθ + cosθ)
= - uW [ucos2θ + (u^2 - 1)sinθcosθ] / (usinθ + cosθ)

b)
dF/dθ = 0
=> ucos2θ + (u^2 - 1)sinθcosθ = 0
=> 2ucos2θ + (u^2 - 1)sin2θ = 0
=> tan2θ = 2u/(1 - u^2)
=> 2tanθ/(1 - tan^2 θ) = 2u/(1 - u^2)
=> tanθ = u
=> θ = arctan u.

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