Question

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

• A. $10\text{J}$
• B. $6\text{J}$
• C. $4\text{J}$
• D. $-2\text{J}$

Answer 1

The correct option is C $4\text{J}$

Given, force $\overrightarrow{F}=4\hat{i}-2\hat{j}$
Displacement vector $\overrightarrow{ds}=dx\hat{i}+dy\hat{j}$
workdone $\left(W\right)=\int \overrightarrow{F}.\overrightarrow{ds}={\int }_{(0,0)}^{(3,4)}(4\hat{i}-2\hat{j}).(dx\hat{i}+dy\hat{j})$
$W={\int }_{(0,0)}^{(3,4)}4dx-2dyW=4x{|}_{\left(0\right)}^{\left(3\right)}-2y{|}_{\left(0\right)}^{\left(4\right)}$
$W=4(3-0)-2(4-0)=4\text{J}$

Alternate:
From the question we have displacement as
$\overrightarrow{S}=3\hat{i}+4\hat{j}$
Given, force $\overrightarrow{F}=4\hat{i}-2\hat{j}$
From the definition of workdone, we have
$W=\overrightarrow{F}\cdot \overrightarrow{S}$
$\Rightarrow W=(4\hat{i}-2\hat{j})\cdot (3\hat{i}+4\hat{j})=12-8=4\text{J}$