A particle is moved along a path AB-BC-CD-DE-EF-FA, as shown in figure, in presence of a force F→=αyi^+2αxj^N, where x and y are in meter and α=−1Nm−1. The work done on the particle by this force F→ will be n×0.25 Joule.Find the value of n

For AB:y= const F→=−yi^−2xj^⇒F→=−i^−2xj^dr→=dxi^ So, WAB=∫x=01 F→⋅dr→=∫01 −dx=−1J For BC:x=constF→=−yi^−2j^dr→=dyjWBC=∫y=10.5 −2dy⇒WBC=+1J For CD:y=const F→=−0.5i^−2xj and dr→=dxi^WCD=∫10.5 −0.5dx=0.25J For DE:x= const F→=−yi^−j^dr→=dyj^WDE=∫y=0.50 −dy=+0.5J For EF:y=0 F→=−2×j^dr→=dxi^⇒WEF=0 For FA:x=0 F→=−yi^dr→=dyj^⇒WFA=0WNet=−1+1+0.25+0.5=0.75J

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A particle is moved along a path AB-BC-CD-DE-EF-FA, as shown in figure, in presence of a force $\vec{F} = (α y \hat{i} + 2 α x \hat{j}) N,$ where x and y are in meter and $α = - 1 N m^{- 1}$ . The work done on the particle by this force $\vec{F}$ will be $n \times 0.25$ Joule.Find the value of n

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Detailed Solution

$\begin{array}{l} For A B : y = const \\ \vec{F} = - y \hat{i} - 2 x \hat{j} \\ \Rightarrow \vec{F} = - \hat{i} - 2 x \hat{j} \\ \vec{d r} = d x \hat{i} \\ So, W_{A B} = \int_{x = 0}^{1} \vec{F} \cdot \vec{d r} = \int_{0}^{1} - d x = - 1 J \end{array}$

$\begin{array}{l} For BC : x = const \\ \vec{F} = - y \hat{i} - 2 \hat{j} \\ \vec{d r} = d y j \\ W_{B C} = \int_{y = 1}^{0.5} - 2 d y \\ \Rightarrow W_{B C} = + 1 J \end{array}$

$\begin{array}{l} For CD:y=const \\ \vec{F} = - 0.5 \hat{i} - 2 x j and d \vec{r} = d x \hat{i} \\ W_{C D} = \int_{1}^{0.5} - 0.5 d x = 0.25 J \end{array}$

$\begin{array}{l} For DE : x = const \\ \vec{F} = - y \hat{i} - \hat{j} \\ \vec{d r} = d y \hat{j} \\ W_{D E} = \int_{y = 0.5}^{0} - d y \\ = + 0.5 J \end{array}$

$\begin{array}{l} For EF : y = 0 \\ \vec{F} = - 2 \times \hat{j} \\ d \vec{r} = d x \hat{i} \\ \Rightarrow W_{E F} = 0 \\ For F A : x = 0 \\ \vec{F} = - y \hat{i} \\ d \vec{r} = d y \hat{j} \\ \Rightarrow W_{F A} = 0 \\ W_{Net} = - 1 + 1 + 0.25 + 0.5 = 0.75 J \end{array}$