What is the Optimal Angle for Minimizing Work in Kinetic Friction?

[patrickmoloney]

Homework Statement

Using the relation $tan \theta = \mu_k$ what angle should the rope make with the horizontal
in order to minimize the work done per unit distance traveled along the ground.

Homework Equations

$W = Fdcos\phi$

The Attempt at a Solution

I found $tan\theta = \mu_k$ by finding the derivative of a with respect to $\theta$ and let it equal 0.

$a = \frac{F}{m}cos\theta - \mu_k (g - \frac{F}{m}sin\theta) = 0$

I think $F = \mu_k F_N$ is a minimum when $\frac{dF}{dx} = 0$ and $\frac{d^2F}{dx^2} > 0$ . Is this correct? If so how do I use this?

 

[BvU]

What rope ?

 

[patrickmoloney]

A case is being pulled along horizontal ground by means of a rope

 

[Rellek]

Homework Statement

Using the relation $tan \theta = \mu_k$ what angle should the rope make with the horizontal
in order to minimize the work done per unit distance traveled along the ground.

Homework Equations

$W = Fdcos\phi$

The Attempt at a Solution

I found $tan\theta = \mu_k$ by finding the derivative of a with respect to $\theta$ and let it equal 0.

$a = \frac{F}{m}cos\theta - \mu_k (g - \frac{F}{m}sin\theta) = 0$

I think $F = \mu_k F_N$ is a minimum when $\frac{dF}{dx} = 0$ and $\frac{d^2F}{dx^2} > 0$ . Is this correct? If so how do I use this?

Someone else just asked this exact question in the forum already, and we answered it there if you want to look at it.

 

[patrickmoloney]

proving $tan\theta = \mu_k$ is the first part of the question. That is for a maximum not a minimum. The third part is 'what angle should the rope make with the horizontal in order to minimize the work done per unit distance traveled along the ground'

 

[SammyS]

Homework Statement

Using the relation $tan \theta = \mu_k$ what angle should the rope make with the horizontal
in order to minimize the work done per unit distance traveled along the ground.

Homework Equations

$W = Fdcos\phi$

The Attempt at a Solution

I found $tan\theta = \mu_k$ by finding the derivative of a with respect to $\theta$ and let it equal 0.

$a = \frac{F}{m}cos\theta - \mu_k (g - \frac{F}{m}sin\theta) = 0$

I think $F = \mu_k F_N$ is a minimum when $\frac{dF}{dx} = 0$ and $\frac{d^2F}{dx^2} > 0$ . Is this correct? If so how do I use this?

The derivatives should be with respect to θ, not w.r.t. x .

You most likely want the acceleration to be zero.

 

[patrickmoloney]

I get to $F = \frac{\mu_k mg}{cos\theta + \mu_k sin\theta}$ and I don't think it's correct cause I'm not able to get an elegant derivative for it.

 

[patrickmoloney]

how would I go about differentiating the work $W$ with respect to $\theta$ ?

 

[Rellek]

how would I go about differentiating the work $W$ with respect to $\theta$ ?

You would want to differentiate the force equation after you have written in terms of $\theta$.

 

[SammyS]

how would I go about differentiating the work $W$ with respect to $\theta$ ?

Work per unit distance traveled is Fcosθ , isn't it ?

That's a bit nicer looking expression.

Use maybe the quotient rule ?