Suppose that an object with weight w is dragged along a horizontal plane by a force acting along a rope attracted to the objet. If the rope makes an angle θ with the plane then the magnitude of the force;
F=μωμsinθ+cosθ
where μ 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?
c.) If w=50lb and μ=0.6, draw the graph of F as a function of θ and use it to locate the value of θ for which dFdθ=0. Is the value consistent with your answer to part(b)?
a.) dFdθ=[μsinθ+cosθ]ddθ(μω)−μωddθ[μsinθ+cosθ][μsinθ+cosθ]2dFdθ=0−μω[μcosθ+(−sinθ)][μsinθ+cosθ]2dFdθ=μω[sinθ−μcosθ][μsinθ+cosθ]2
b.) dFdθ=0=μω[sinθ−μcosθ][μsinθ+cosθ]2
μω[sinθ−μcosθ]=0sinθ−μcosθ=0sinθ=μcosθsinθcosθ=μtanθ=μθ=tan−1μ
The rate of change is equal to 0 when θ=tan−1μ
c.) F=μωμsinθ+cosθ;ω=50h,μ=0.6F=50(0.6)0.6sinθ+cosθF=300.6sinθ+cosθ
We know that dFdθ=0 when θ=tan−1μ so
θ=tan−1(0.6)=0.5404F=300.6sinθ+cosθ;where θ=0.5404F(θ)=300.6sin(0.5404)+cos(0.5404)=25.725
Based from the graph, our answer in part(b) matches our answer in part(c) since the value of tangent line of that point is zero.
Monday, December 16, 2013
Single Variable Calculus, Chapter 3, 3.4, Section 3.4, Problem 38
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