Question

An object is displaced from point $A(2,0,0)m$ to a point $B(1,0,4)m$ under the influence of a force $\overrightarrow{F}=4x\hat{i}-2z\hat{k}.$ The workdone by this force is

• A. $18\text{J}$
• B. $-22\text{J}$
• C. $-20\text{J}$
• D. $-18\text{J}$

Answer 1

The correct option is B $-22\text{J}$

Given variable force $\overrightarrow{F}=4x\hat{i}-2z\hat{k}$
Displacement vector $\overrightarrow{ds}=dx\hat{i}+dy\hat{j}+dz\hat{k}$
workdone $\left(W\right)=\int \overrightarrow{F}.\overrightarrow{ds}={\int }_{(2,0,0)}^{(1,0,4)}(4x\hat{i}-2z\hat{k}).(dx\hat{i}+dy\hat{j}+dz\hat{k})$$W={\int }_{(2,0,0)}^{(1,0,4)}4xdx+0-2zdzW=2x^2{|}_{\left(2\right)}^{\left(1\right)}-z^2{|}_{\left(0\right)}^{\left(4\right)}$$=(2\times 1^2-2\times 2^2)-(4^2-0^2)=-6-16=-22\text{J}$