Finding coefficient of friction by conservation of energy

[Phyba]

A 20 Kg. mass slides up on an infinitely long inclined plane (that forms an angle of 30º with the horizontal) with a velocity of 12 m/s. It's known that the mass returns to the starting point with a velocity of 6 m/s. Find,μ, the coefficient of friction between the plane and the body.

This is the problem I'm struggling with. By the mechanical energy conservation principle I know that the work done by the friction force is the increment in kinetic energy. But, how can I relate that to Newton's equation to find μ.

I thought, since the frinction force is μ*mgcos30, we can say

μ*s*mgcos30 = ΔKinetic energy

where s is the distance, but we don't even know the distance!

Thanks in advance, I'm new here!

 

[nasu]

Welcome to PF. :)

So you found that work-energy theorem does not help here.
Try to write the work-energy theorem for the up motion and for the down motion. Separately.
Then you will be able to eliminate the unknown distance.

 

[Phyba]

Thanks, Nasu. So I now write the work-energy theorem for the up and the down motion separately:

up motion:

down motion:

But I can't solve for any of the unknown variables s and μ with these two equations.. What am I doing wrong?

 

[ShayanJ]

I should point out that you're not using Conservation of energy.In fact energy is not conserved when you have dissipation.
But for solving the problem,when the mass is sliding upward,there are two forces,friction and gravitation.You can find where the speed becomes zero.you also can write the equations of motion for the part that the mass is sliding downward and because you have the final velocity,you can find where the velocity was zero.This place is the same as the place you found before so you have two equations with two unknowns which can be solved easily.

 

[Phyba]

Thanks Shyan, I have a doubt: what I'm trying to apply is that the work of non.conservative forces is the variation of kinetic energy. So, why should I add gravitation to my equation if gravitation is a conservative and is not responsible of the variation of KE?

 

[ShayanJ]

Thanks Shyan, I have a doubt: what I'm trying to apply is that the work of non.conservative forces is the variation of kinetic energy. So, why should I add gravitation to my equation if gravitation is a conservative and is not responsible of the variation of KE?

Every force acting on a particle,is able to do work on it,whether conservative or not!

 

[nasu]

Thanks Shyan, I have a doubt: what I'm trying to apply is that the work of non.conservative forces is the variation of kinetic energy. So, why should I add gravitation to my equation if gravitation is a conservative and is not responsible of the variation of KE?

This is not true.
The variation of kinetic energy is the work of all forces.

 

[Phyba]

Thank you very much, you two! I got confused because of how I tried to solve it at first. I did it and obtained μ=0.34, which makes a lot of sense. Your help was very useful!

 

[Phyba]

Solved

I'm uploading the solution (although it's a simple problem), in case somebody that got stuck with something similar finds it useful in the future.

:) This forum is awesome