# Finding path for which line integral of force is zero.

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A particle is constrained to move from initial point $O$ to final point $C$ along three different smooth horizontal tracks namely $OBC, OPC, OAC$. If the particle moves under the influence of an external force $F$ such that the initial and final speeds are same then:
1. There necessarily exists a path along which line integral of force $F$ is zero.
2. $F$ is conservative
3. $F$ cannot be conservative
4. There is no closed path along which line integral of force $F$ is zero.

Only correct answer is $(1)$.

What does this mean?

edited Oct 6
I do not understand it either. The context of this quotation might offer some clues. Where have you quoted from? Can you provide a link, or post an image of the quote and the surrounding text?
It was a problem with 4 options if you need I can write other three options.
Yes that might help. Is there only one correct answer or can more than one option be correct?

The main difficulty with such questions is applying correct logic. The options could be true for some cases, but we must decide if they are necessarily true for all cases.

The line integral $W=\int_O^C F.ds$ is the work done on the particle by the external force. By the Work-Energy Theorem this equals the change in kinetic energy of the particle. We are told that there is no change in the KE of the particle for any of the 3 paths OAC, OBC, OPC. So for each of these paths the work done is $W=0$.

Therefore option 1 is correct : at least 3 such paths exist along which the line integral of force is zero.

A conservative force is one for which the work done between two points such as O and C is independent of the path taken. This is true for the 3 paths identified, because the work done is zero for each. But we cannot assume that it is true for all other paths between O and C. Even if it is true for all paths between O and C we do not know if it is true for all paths between any other pairs of points. So option 2 is not necessarily true.

Conversely, it is possible that the work done could be zero for all paths between all pairs of point. We don't have enough evidence to decide this issue. So option 3 is not necessarily true either.

If we were told that C coincides with O then we would know that option 4 would be false, because we would know there are at least 3 closed paths along which the line integral is zero. But we don't know if C coincides with O, so we don't know whether the line integral is zero for any closed path. We cannot be sure whether option 4 is true or false. All we can conclude is that option 4 is not necessarily true.

answered Oct 6 by (26,096 points)
selected Oct 7 by n3