   Chapter 4.7, Problem 44E ### 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 a plane, then the magnitude of the force is F = μ W μ sin θ + cos θ where μ is a constant called the coefficient of friction. For what value of θ is F smallest?

To determine

To find: The value of θ for which, F is smallest.

Explanation

Given:

The magnitude of the force is, F=μWμsinθ+cosθ.

Where, μ is a constant, called the coefficient of friction; weight of an object W, angle make with a plane is θ.

Calculation:

Consider F=μWμsinθ+cosθ.

Differentiate F with respect to θ,

dFdθ=(μsinθ+cosθ)(0)(μW)(μcosθsinθ)(μsinθ+cosθ)2=(μW)(sinθμcosθ)(μsinθ+cosθ)2

Since F is minimum, dFdθ=0.

(μW)(sinθμcosθ)(μsinθ+cosθ)2=0μcosθ=sinθμ=tanθθ=tan1(μ)

For θ=tan1(μ), F attains the minimum.

When F(θ) is increasing, F(θ)>0

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