Question

A particle moves from a point \[\overrightarrow{r}_1 = \left( 2 m \right) \overrightarrow{ i } + \left( 3 m \right) \overrightarrow{ j } \] to another point

\[\overrightarrow{r}_2 = \left( 3 m \right) \overrightarrow{ i } + \left( 2 m \right) \overrightarrow{ j } \]  acts on it. Find the work done by the force on the particle during the displacement.

Answer 1

Initial position vector, \[\vec{r_1} = 2 \vec{i} + 3 \vec{j}\] Final position vector, \[\vec{r}_2 = 3 \vec{i} + 2 \vec{j}\]

So, displacement vector,

\[\vec{r} = \vec{r}_2 - \vec{r}_1 \]

\[ = \left( 3 \vec{i} + 2 \vec{j} \right) - \left( 2 \vec{i} + \vec{j} \right)\]

\[ = \vec{i} - \vec{j} \]

\[\text{ Force acting on the particle } , \vec{F} = 5 \vec{i} + 5 \vec{j} \]

\[\text{ So, work done }  = \vec{F} \cdot \vec{S} = 5 \times 1 + 5 \left( - 1 \right) = 0\]