Question

An object has a displacement from position vector $\overrightarrow{r_1}=(2\hat{i}+3\hat{j})m$
to $\overrightarrow{r_2}=(4\hat{i}+6\hat{j})m$ under a force $\overrightarrow{F}=(3x^2\hat{i}+2y\hat{j})N,$ then work done by the force is:

• A. $24J$
• B. $33J$
• C. $83J$
• D. $45J$

Answer 1

The correct option is C $83J$

$\begin{array}{c}w={\int }_{\overrightarrow{r_1}}^{\overrightarrow{r_2}}\overrightarrow{F}\left(dx\hat{i}+dy\hat{j}\right)\\ ={\int }_{\overrightarrow{r_1}}^{\overrightarrow{r_2}}\left(3x^2\hat{i}+2y\hat{j}\right)\left(dx\hat{i}+dy\hat{j}\right)\\ ={\int }_2^{4}3x^{2}dx+{\int }_3^{6}3ydy\\ ={\left[x^3\right]}_2^4+{\left[y^2\right]}_3^6\\ =\left(64-8\right)+\left(36-9\right)\\ =56+27\\ =83J\end{array}$

$\therefore$ Option $C$ is correct.