Question

An object is displaced from point $A(0,0,0)m$ to a point $B(2,4,6)m$ under the influence of a force $\overrightarrow{F}=2x\hat{i}+3y^2\hat{j}+4z^3\hat{k}.$ The workdone by this force is

• A. $1380\text{J}$
• B. $1364\text{J}$
• C. $1220\text{J}$
• D. $1180\text{J}$

Answer 1

The correct option is B $1364\text{J}$

Given variable force $\overrightarrow{F}=2x\hat{i}+3y^2\hat{j}+4z^3\hat{k}$
Displacement vector $\overrightarrow{ds}=dx\hat{i}+dy\hat{j}+dz\hat{k}$
workdone $\left(W\right)=\int \overrightarrow{F}.\overrightarrow{ds}={\int }_{(0,0,0)}^{(2,4,6)}(2x\hat{i}+3y^2\hat{j}+4z^3\hat{k}).(dx\hat{i}+dy\hat{j}+dz\hat{k})$$W={\int }_{(0,0,0)}^{(2,4,6)}2xdx+3y^{2}dy+4z^{3}dzW=x^2{|}_{\left(0\right)}^{\left(2\right)}+y^3{|}_{\left(0\right)}^{\left(4\right)}+z^4{|}_{\left(0\right)}^{\left(6\right)}$$=(2^2-0^2)+(4^3-0^3)+(6^4-0^4)=4+64+1296=1364\text{J}$