Monday, August 26, 2019

Single Variable Calculus, Chapter 4, 4.1, Section 4.1, Problem 64

F=μWμsinθ+cosθ;0θπ2 where μ is a positive constant called the coefficient of friction.
W represents the weight of an object.
θ= angle
F= force
Show that F is minmized when tanθ=μ.

Taking the derivative of the function,
dFdθ=μWddθ(1μsinθ+cosθ)

Using Quotient Rule,


dFdθ=μW[μsinθ+cosθddθ(1)1ddθ(μsinθ+cosθ)(μsinθ+cosθ)2]dFdθ=μW[0(μcosθ+(sinθ))(μsinθ+cosθ)2]dFdθ=μW(μcosθ+(sinθ))(μsinθ+cosθ)2


When dFdθ=0,


0=μW(μcosθsinθ)(μsinθ+cosθ)20=μcosθsinθsinθcosθ=μ\cancelcosθ\cancelcosθtanθ=μ


It shows that F is minimized when tanθ=μ

No comments:

Post a Comment