Find work given two conservative forces and a nonconservative force

[bremenfallturm]

Homework Statement

A particle starts from rest in the point ##A## to the point ##B## along four different paths according to the figure (see my post):
1. A conservative force ##F_{K1}## does work ##10J## on the particle
2. A conservative force ##F_{K2}## does work ##5## on the particle
3. A nonconservative force ##F_{IK}## does work ##-5J## on the particle.
Now assume all forces act at the same time on the particle, when it is moved along the path 4. Find the work done by ##F_{IK}## given that the particle stops at point ##B##

Relevant Equations

##U=\int_{\mathcal C} \vec F \cdot \vec dr##

##U=V(A)-V(B)## if a force is conservative and ##V## is a potential function for it

##U=\oint_{\gamma} \vec F \cdot \vec dr## for a closed curve ##\gamma##

The figure provided with the question is:

I set up the following equation for path 4
##U_{path 4}=U_{Fk1, path 4}+U_{Fk2, path 4}+U_{FIK, path 4}## where ##U_{FIK,AB}## is the unknown.
I know that the work will be the same regardless the path for conservative forces, so I have:
##U_{path 4}=10+5+U_{FIK, path 4} (J)##
The answer key says ##U_{FIK, path 4}=-15J## (no further solution given), but I do not understand why ##U_{path 4}## is ##0##, if I set up my equations correctly and interpret the answer key. I know that mechanical energy is conserved with conservative forces, and that the work done over a closed curve with conservative forces is 0.
How can I come to a reasoning that ##-15J## is correct?
Thanks!

 

[Orodruin]

Since the particle starts from rest and stops at B, the work-energy theorem requires the total work done on the particle is zero.

 

[kuruman]

Homework Statement: A particle starts from rest in the point ##A## to the point ##B## along four different paths according to the figure (see my post):
1. A conservative force ##F_{K1}## does work ##10J## on the particle
2. A conservative force ##F_{K1}## does work ##10J## on the particle
3. A nonconservative force ##F_{IK}## does work ##-5J## on the particle.

The work done by ##F_{K1}## is mentioned twice and the work done by ##F_{K2}## is not mentioned at all. Please specify.

 

[bremenfallturm]

The work done by ##F_{K1}## is mentioned twice and the work done by ##F_{K2}## is not mentioned at all. Please specify.

Megasuperdupersorry, copy paste let me down. I've clarified, see #1.

 

[kuruman]

Thank you for the clarification. You should cast the problem not in terms of potential energy changes because non-conservative forces cannot be derived from a potential energy function. Use the work-energy energy theorem approach as @Orodruin suggested in post #2. Note that

1. The total work done by the conservative forces is independent of path.
2. The work done by the non-conservative force along path 3 (= -5 J) is totally irrelevant to the answer.

 

[bremenfallturm]

Thank you for the clarification. You should cast the problem not in terms of potential energy changes because non-conservative forces cannot be derived from a potential energy function. Use the work-energy energy theorem approach as @Orodruin suggested in post #2. Note that

1. The total work done by the conservative forces is independent of path.
2. The work done by the non-conservative force along path 3 (= -5 J) is totally irrelevant to the answer.

Alright! So for path (4), we simply know that
$$U_{1-2}=\underbrace{\frac{1}{2}mv_b^2}_{=0}-\underbrace{\frac{1}{2}mv_a^2}_{=0}$$
$$\implies 0=5+10+U_{FIk, path 4}\implies U_{FIk, path 4}=-15 J$$

I think I missed the fact that the work-energy theorem applies to all forces, I don't know why I've thought the other way around, it seems obvious now when I think about it. With the reservation of this being a dumb question, for the other three paths we also start and end at rest, but there is not zero work. That's, what I assume, from other unknown forces acting on the particle. We assume that they don't do any work on it for path (4), since the work-energy theorem is for the sum of all forces, or did I misunderstand the concept of it?

 

[Orodruin]

for the other three paths we also start and end at rest

No we don't. There is no such statement in the problem.

 

[bremenfallturm]

No we don't. There is no such statement in the problem.

Hm, okay. You're right, I re-read the statement. It says that the particle "stops" at B in situation 4 and "moves" to B in the other scenarios. It was even clearer in the original problem statement, which is not in English so I had to translate it. Thank you, I think I get it now!

 

[bremenfallturm]

(Is there any way I can mark a thread as solved? Still new to this forum )

 

[kuruman]

(Is there any way I can mark a thread as solved? Still new to this forum )

We are not formal in that regard. Once the OP (original poster) provides a solution as you have in post #6, that’s it. Thank you for your contribution and do come back. We are here to help.