Question

A force $\overrightarrow{F}=-x\hat{j}$ acts on a particle. The particle starts from point $(0,2)$ and moves to the point $(1,4)$ following the path $y=2x+2$. The total work done by the force $\overrightarrow{F}$ on the particle is

• A. $-1\text{J}$
• B. $2\text{J}$
• C. $1\text{J}$
• D. $-2\text{J}$

Answer 1

The correct option is A $-1\text{J}$

Given variable force $\overrightarrow{F}=-x\hat{j}$
Displacement vector $\overrightarrow{ds}=dy\hat{j}$
workdone $\left(W\right)=\int \overrightarrow{F}.\overrightarrow{ds}={\int }_0^1(-x\hat{j}).\left(dy\hat{j}\right)$
$W={\int }_0^1-xdy$
As the path followed is $y=2x+2\Rightarrow dy=2dx$
$\Rightarrow W={\int }_0^1(-x)\left(2dx\right)$$\Rightarrow W={\int }_0^1-2xdx$
$\Rightarrow W=-x^2{|}_0^1=-1\text{J}$