Question

A particle is moved from $(0,0)$ to $(a,a)$ under a force $F=3i+4j$ from two paths. Path $1$ is $OP$ and path $2$ is $QOP$. Let $W_1$ and $W_2$ be the work done by this force in two paths. Then

• A. $W_1=2W_2$
• B. $W_1=W_2$
• C. $W_2=2W_1$
• D. $W_1=4W_2$

Answer 1

The correct option is B

$W_1=W_2$

Step 1. Given Data,

$F=3i+4j$

Position Vector, $P=a\hat{i}+a\hat{j}=S$

Step 2. Formula used,

$W=F.S$, S is the displacement

W is the work done and F is the force.

Step 3. Calculating the Path $1OP$,

Work done on path $1OP$

$$\begin{gathered} W_1=\hat{F}.\hat{S} \\ {W}_1=(3\hat{i}+4\hat{j}).(a\hat{i}+a\hat{j}) \\ {W}_1=3a+4a \\ {W}_1=7a \end{gathered}$$

Step 4. Calculating the Path $2QOP$,

Now work is done on two different paths,

$OQ=a\hat{i}$

$QP=a\hat{j}$

Work done $W_2$ is,

$$\begin{gathered} W_2=\overrightarrow{F}.\overrightarrow{OQ}+\overrightarrow{F}.\overrightarrow{OP} \\ {W}_2=(3\hat{i}+4\widehat{j)}.(a\hat{i})+(3\hat{i}+4\widehat{j)}.\left(a\hat{j}\right) \\ {W}_2=3a+4a \\ {W}_2=7a \end{gathered}$$

So $W_1=W_2$

Hence the work done is equal for both path.

Hence, option B is the correct answer.