Question

A force $\overrightarrow{F}=2x\hat{j}$ acts on a particle. The particle starts from point $(0,0)$ and moves to the point $(3,9)$ following the path $y=x^2$. The total work done by the force $\overrightarrow{F}$ on the particle is

• A. $20\text{J}$
• B. $36\text{J}$
• C. $-26\text{J}$
• D. $27\text{J}$

Answer 1

The correct option is B $36\text{J}$

Given variable force $\overrightarrow{F}=2x\hat{j}$
Displacement vector $\overrightarrow{ds}=dy\hat{j}$
workdone $\left(W\right)=\int \overrightarrow{F}.\overrightarrow{ds}={\int }_0^3\left(2x\hat{j}\right).\left(dy\hat{j}\right)$
$W={\int }_0^{3}2xdy$
As the path followed is $y=x^2\Rightarrow dy=2xdx$
$\Rightarrow W={\int }_0^3\left(2x\right)\left(2xdx\right)$$\Rightarrow W={\int }_0^{3}4x^{2}dx$
$\Rightarrow W=\frac{4}{3}{x}^3{|}_0^3=36\text{J}$