Assume a particle travelling along a curve, the work done by any Force field on the particle while moving along a curve is given by the line integral of F → ⋅ d r → , but shouldn't the path be a straight line regardless of the given path as the work done ( W ) = F ⋅ s (disp between A and B), displacement being the straight line path between the two points?

Question

Assume a particle travelling along a curve, the work done by any Force field on the particle while moving along a curve is given by the line integral of                     F            →        ⋅                    d        r            →      , but shouldn&#039;t the path be a straight line regardless of the given path as the work done   (  W  )  =  F  ⋅  s (disp between A and B), displacement being the straight line path between the two points?

Phoebe Medina · Accepted Answer

The formula  W  =      ∫          path                  F      →        ⋅                    d        r            →      is the formula for the general case, when the force             F      →       doesn’t have to be constant. If however             F      →       is constant, you can move             F      →       out of the integral, and the equation simplifies to  W  =      ∫          path                  F      →        ⋅                    d        r            →        =            F      →        ⋅                                          ∫                          path                                                                          d                r                            →                                      ⏟                    =                        s          →                      =            F      →        ⋅            s      →        .So the formula you propose to use instead of the integral is for the case when the force is constant.

Work done by a vector field (Force field) on a particle travelling along a curve.

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