# A particle of mass m is displaced from a position P1 to P2 with position vectors ….

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.

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