Problem 1: A particle of mass m is displaced from a position P1 to P2 with position vectors $ \displaystyle \vec{r_1} = a\hat{i} + b\hat{j} + c\hat{k} $ and $ \displaystyle \vec{r_2} = c\hat{i} + a\hat{j} + b\hat{k} $ by a force $ \displaystyle \vec{F} = b\hat{i} + c\hat{j} + a\hat{k} $ . Find the work done by the force.
Solution: If the displacement of the particle is $ \displaystyle \vec{S} $ , the work done W by the given force is equal to $ \displaystyle W = \vec{F}.\vec{S} $ , when F→ is a constant force.
$ \displaystyle W = \vec{F}.\vec{S} $ …(1)
Where , $ \displaystyle \vec{F} = b\hat{i} + c\hat{j} + a\hat{k} $
and , $ \displaystyle \vec{S} = \vec{r_2} – \vec{r_1} $
$ \displaystyle \vec{S} = (c\hat{i} + a\hat{j} + b\hat{k}) – (a\hat{i} + b\hat{j} + c\hat{k}) $
$ \displaystyle \vec{S} = (c-a)\hat{i} + (a-b)\hat{j} + (b-c)\hat{k} $
$ \displaystyle W = (b\hat{i} + c\hat{j} + a\hat{k}).((c-a)\hat{i} + (a-b)\hat{j} + (b-c)\hat{k}) $
W = (c-a)b + (a-b)c + (b-c)a
W = bc – ab + ac – bc + ab – ac = 0
Net work done by the force for the given displacement is zero.