Question

An object is displaced from point $A(2,3,4)m$ to a point $B(1,2,3)m$ under the influence of a force $\overrightarrow{F}=2\hat{i}+3y^2\hat{j}+5\hat{k}.$ The workdone by this force is

• A. $26\text{J}$
• B. $-80\text{J}$
• C. $-26\text{J}$
• D. $34\text{J}$

Answer 1

The correct option is C $-26\text{J}$

Given variable force $\overrightarrow{F}=2\hat{i}+3y^2\hat{j}+5\hat{k}$
Displacement vector $\overrightarrow{ds}=dx\hat{i}+dy\hat{j}+dz\hat{k}$
workdone $\left(W\right)=\int \overrightarrow{F}.\overrightarrow{ds}={\int }_{(2,3,4)}^{(1,2,3)}(2\hat{i}+3y^2\hat{j}+5\hat{k}).(dx\hat{i}+dy\hat{j}+dz\hat{k})W={\int }_{(2,3,4)}^{(1,2,3)}2dx+3y^{2}dy+5dzW=2x{|}_{\left(2\right)}^{\left(1\right)}+\frac{3y^3}{3}{|}_{\left(3\right)}^{\left(2\right)}+5z{|}_{\left(4\right)}^{\left(3\right)}=2(1-2)+(2^3-3^3)+(3-4)=-2-19-5=-26\text{J}$