F=μWμsinθ+cosθ;0≤θ≤π2 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=μW⋅ddθ(1μsinθ+cosθ)
Using Quotient Rule,
dFdθ=μW[μsinθ+cosθ⋅ddθ(1)−1⋅ddθ(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(μcosθ−sinθ)(μsinθ+cosθ)20=μcosθ−sinθsinθcosθ=μ\cancelcosθ\cancelcosθtanθ=μ
It shows that F is minimized when tanθ=μ
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