   Chapter 4.1, Problem 72E ### Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343

#### Solutions

Chapter
Section ### Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343
Textbook Problem

# 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 = μ W μ sin θ + cos θ where μ is a positive constant called the coefficient of friction and where 0 ≤ θ ≤ π/2. Show that F is minimized when tan θ = μ.

To determine

To show: The magnitude of force F is minimized if tanθ=μ .

Explanation

Given:

The magnitude of the force, F=μWμsinθ+cosθ,0θπ2 where μ is a positive constant called the coefficient of friction and W is the weight of the object.

proof:

Consider the given function, F=μWμsinθ+cosθ .

Differentiate F with respect to θ .

F(θ)=ddθ(μWμsinθ+cosθ)

Apply the Quotient Rule and simplify further.

F(θ)=(μsinθ+cosθ)(0)μW(μcosθsinθ)(μsinθ+cosθ)2=μW(μcosθsinθ)(μsinθ+cosθ)2

Set F(θ)=0 and obtain the critical numbers.

μW(μcosθsinθ)(μsinθ+cosθ)2=0μW(μcosθsinθ)=0μcosθsinθ=0μ=sinθcosθ  =tanθ

Substitute the obtained critical number μ=tanθ in F ,

F(tanθ)=(tanθ)W(tanθ)sinθ+cosθ=(tanθ)W(sinθcosθ)sinθ+cosθ=(tanθ)Wsin2θ+cos2θcosθ=W(sinθcosθ)sin2θ+cos2θ(cosθ)

Simplify further as follows.

F(tanθ)=W(sinθsin2θ+cos2θ)=W(sinθ1)[sin2θ+cos2θ=1]=Wsinθ

Express sinθ in terms of μ as follows.

If μ=tanθ , use the Pythagoras formula to find the value for sinθ as follows.

From this, it is observed that opposite side is μ and adjacent side is 1

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