Am 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 of θ with the plane, the thmagnitude of the force is
F= μW/(μsinθ+cosθ)
where μ is a positive constant called the coefficient of friction and where 0 is less than or equal to θ and θ is less than or equal to (pi/2). Show that F is minimized when tanθ=μ.
2006-10-25 17:58:21 · 2 answers · asked by Anonymous in Science & Mathematics ➔ Mathematics
Show that F is minimized when tanθ=μ.
2006-10-25 17:58:29 · update #1
an object**
2006-10-25 17:58:42 · update #2
dF/dθ = -μW/(μsinθ+cosθ)^2 * (μcosθ-sinθ).
This derivative is zero when μcosθ-sinθ = 0. Solving for μ yields μ = tanθ.
To confirm that this is a minimum, you can compute d^2F/dθ^2 and substitute tanθ=μ. You obtain Wsinθ, which is positive for the given range of values for θ, therefore F is minimized at this angle.
2006-10-25 18:04:26 · answer #1 · answered by James L 5 · 5⤊ 0⤋
you sure that isn't physics?
2006-10-25 18:09:17 · answer #2 · answered by fancy unicorn 4 · 0⤊ 2⤋