Question

An object moves along a straight line path from (0,0) to the point (3,3) under the action of a force

Calculate the work done. (You must include all steps to get full credit for this problem.)

Answer 1

$\\slope\;of\;the\;straight\;line=\frac{y_2-y_1}{x_2-_1}=\frac{3-0}{3-0}=1\\\\\\equation\;of\;straight\;line\;y=mx\\ y=x\\\\ displacement\;vector\;\vec{r}=x\hat{i}+y\hat{j}\\\\ d\vec{r}=dx\hat{i}+dy\hat{j}\\\\ dy=dx\\ work\;done\;W=\;\int \vec{F}\cdot d\vec{r}\\\\W=\int (4N/m)y\hat{i}+(4N/m)x\hat{j}\cdot (dx\hat{i}+dy\hat{j})\\\\ W=\int \left ((4N/m)y\;dx+(4N/m)xdy \right )\\\\ W=\int \left ((4N/m)x\;dx+(4N/m)xdx \right )\\\\ W=\left |4N/m\frac{x^2}{2} \right |_{0}^{3m}=4(N/m)\left ( \frac{3^2m^2}{2} \right )\;=18\;Nm=18\;J$