Derive the expression for the work done by the friction force

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1. Dec 1, 2017

Alexanddros81

1. The problem statement, all variables and given/known data
14.6 The coefficient of kinetic friction between the slider and the rod is μ, and the
free length of the spring is $L_0 = b$. Derive the expression for the work done by
the friction force on the slider as it moves from A to B. Neglect the weight of the slider.

2. Relevant equations

3. The attempt at a solution

I
am going the right direction with my solution?
I don't see to be getting the $-0.1186μkb^2$ solution.

2. Dec 1, 2017

BvU

$\mu$ stands for the ratio of the friction force and some normal force. What's that normal force as a function of position ?

Pity you don't give any relevant equations. I don't think the exercise wants you to look at the velocities.

3. Dec 2, 2017

Alexanddros81

Am going the right direction with the above calculation?

4. Dec 2, 2017

haruspex

It is preferred that you type in your algebra. This makes it easier to comment on specific lines. Or you could number the equations.
Your FBD omits the force dragging the slider from A to B. You should take acceleration to be negligible. This invalidates your force balance equation for the x direction.
The NA=P cos(φ) equation is the useful one, but you need to find P as a function of φ and integrate.

5. Dec 4, 2017

Alexanddros81

For the x direction:
$ΣFx=0 => F-F_k=0 => F=F_k$ (1)

For the y direction:
$ΣFy=0 => N_A-Pcosφ=0 => N_A=Pcos(φ)$ (2)

How do I do that? I am a bit confused...

6. Dec 4, 2017

BvU

How would you calculate $P$ at point B ?

Last edited: Dec 4, 2017
7. Dec 4, 2017

haruspex

It's the tension in a spring. What determines the value of that?

8. Dec 7, 2017

Alexanddros81

P at point B be should be $-kx$ where x is the elongation of the spring.

$x = L_B-L_0 = \sqrt {b^2+b^2} - b = 0.414b$

So P = -0.414kb

Am I correct? (it is the second problem with springs ever i am trying to solve )

9. Dec 7, 2017

PeroK

I think the problem is a lot more complicated than that. First, you need an expression for the length of the spring as a function of the distance from A to B.

10. Dec 7, 2017

Alexanddros81

So to do that I split the distance between A and B to 3 sections. from 0 to 1/3 of b, from 1/3b to 2/3b, and from 2/3b to b.

At 1/3b $sinφ = \frac {1b/3} {l} => sinφ = \frac {1b/3} {1.054b} = 0.316$ (where $l = b + L_{1/3b} - L_0$)

At 2/3b $sinφ = 0.55$
At 3/3b $sinφ = 0.707$ that verifies that when collar is at position B the angle φ is 45 degrees.

So from above we can say that $l = L_{AB} = \sqrt {b^2+(xb)^2}$ where x is between 0 and 1

11. Dec 7, 2017

PeroK

Well, I'd say just use good old Pythagoras:

$L = \sqrt{x^2 + b^2}$, where $0 \le x \le b$

Now you need to analyse the forces at each point $x$ between $A$ and $B$.

12. Dec 7, 2017

BvU

Remember the last line in #4: You need $P$ as a function of $\phi$. I think you do fine for $\phi$ at specific values, but you need a function for all $\phi$ in $[0,\pi/4]$

13. Dec 9, 2017

Alexanddros81

Well using Inkscape, FBD is:

in y-axis $N_A = P_y => N_A = Pcos(φ)$

in x-axis $-F_k - P_x < F$ (Is this correct?)

$P=Pcos(φ)$ (Is this correct?)

14. Dec 9, 2017

BvU

$\ P_y = P \cos\phi$ is right. Both factors change during the trip. So fill in something for $P$: a function of $\phi$.

15. Dec 11, 2017

Alexanddros81

The only thing i understand is that $P = -kl$
I can see that as the angle increases so does P. So P∝φ

should it be P related to $1/cos(φ) ? If angle is 0deg the 1/cos(φ) is 1; Then as angle increases so does 1/cos(φ); 16. Dec 11, 2017 BvU Write it out in full so you have something that you can integrate from$\phi=0$to$\phi = \pi/2$.... To help you:$k$is given. Now you need$l(\phi)## -- you did it already, see above

17. Dec 18, 2017

Alexanddros81

Hi. When you say "see above" which post you are refering to?
thanks

18. Dec 18, 2017

haruspex

Post #10, but you would do better to use PeroK's simplification in post #11. It will be more convenient to have x as the distance along the bar from A, so x ranges from 0 to b.