# Kinetic Frictional Force

1. Apr 29, 2014

### patrickmoloney

1. The problem statement, all variables and given/known data

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.

2. Relevant equations

$W = Fdcos\phi$

3. 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?

2. Apr 30, 2014

### BvU

What rope ?

3. Apr 30, 2014

### patrickmoloney

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

4. Apr 30, 2014

### Rellek

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

5. Apr 30, 2014

### 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'

6. Apr 30, 2014

### SammyS

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

You most likely want the acceleration to be zero.

7. Apr 30, 2014

### 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.

8. Apr 30, 2014

### patrickmoloney

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

9. Apr 30, 2014

### Rellek

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

10. Apr 30, 2014

### SammyS

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

That's a bit nicer looking expression.

Use maybe the quotient rule ?