Work done by a conservative force in a closed loop is zero. We will show that mathematically here.
Read this related topic: Conservative force vs Non-conservative force – a comparative study
Work done by a Conservative force in moving a particle around a closed loop is zero – how to prove it
As per the definition, we know that a force is conservative if the work done by the force in displacing a particle from one point to another is independent of the path followed by the particle and depends only on the endpoints.
Suppose a particle moves from point A to point B along either path 1 or path 2, as shown in Figure 1(a). If a conservative force F acts on the particle, then the work done on the particle is the same along the two paths.
Mathematically, we can write
WAB (along path1) = WAB (along path 2) …..(i)
Now suppose the particle moves in a round trip, from point A to point B along path 1 and then back to point A along path 2, as shown in figure 1 (b).
For a conservative force,
Work done on the particle along path 2 from A to B = – Work done on the particle along path 2 from B to A
WAB(along path 2) = – WBA(along path2)…(ii)
From (i) and (ii), we have
WAB(along path 1) = -WBA (along path2)
WAB(along path 1) + WBA(along path2) = 0
or Wclosed path = 0
Hence a force is conservative if the work done by the force in moving a particle around any closed path is zero.