Question

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 = μW μ sin(θ) + cos(θ) where μ is a constant called the coefficient of friction. For what value of θ is F smallest?

Answer 1

Answer:$\theta =\arctan \mu$

Step-by-step explanation:

we know force sin component would oppose the weight  of object thus normal reaction will not be W rather it would be

$N=W-Fsin\theta$

therefore force cos component will balance the friction force

F$cos\theta$ =$\left ( \mu N\right )$

F$cos\theta$ =$\left ( \mu \left ( W-Fsin\theta \right )\right )$

F=$\frac{\mu W}{cos\theta +\mu sin\theta}$

F will be smallest when $cos\theta +\mu sin\theta$ will be maximum

and it will be maximum when we differentiate it to get

$\theta =\arctan \mu$

Answer 2

The magnitude of the force, F, is varies with the angle the rope makes

with the plane according to the given equations.

F will be smallest when $\underline{\theta \ is \ arctan (\mu)}$.

Reason:

The given parameters are;

Angle the rope makes with the plane = θ

The magnitude of the force is, $F = \dfrac{ \mu \cdot W}{\mu \cdot sin(\theta) +cos(\theta) }$

The value of θ for which the value of F is smallest.

Solution;

When, F is smallest, we have;

$\dfrac{dF}{d \theta} = \dfrac{d}{d\theta} \left(\dfrac{ \mu \cdot W}{\mu \cdot sin(\theta) +cos(\theta) } \right) = \dfrac{-\mu \cdot W \cdot (\mu \cdot cos(\theta) -sin(\theta))}{\left( \mu \cdot sin(\theta) +cos(\theta) \right)^2} = 0$

Therefore;

-μ·W·(μ·cos(θ) - sin(θ))

μ·cos(θ) = sin(θ)

By symmetric property, we have;

sin(θ) = μ·cos(θ)

$\mathbf{\dfrac{sin(\theta)}{cos(\theta)} = tan (\theta) = \mu}$

Which gives;

θ = arctan(μ)

Therefore;

F, will be smallest when $\underline{\theta = arctan (\mu)}$.

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Question; The given equation of the magnitude of the force in relation to the angle the rope makes with the plane, θ, is presented as follows;

$F = \dfrac{ \mu \cdot W}{\mu \cdot sin(\theta) +cos(\theta) }$