A particle is moved along a path AB - BC - CD - DE - EF - FA , as shown in figure, in presence of a force F = ( y i + 2 x j ) ?N , where x and y are in meter and = - 1 ?N m - 1 . The work done on the particle by this force F will be _____ Joule.

Question

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 _____ Joule. [2019]

Smart Achievers · Accepted Answer

(0.75)

Given : Force, $\vec{F} = (α y \hat{i} + 2 α x \hat{j})$

$and α = - 1 {N m}^{- 1}$

$We know that d W = \vec{F} \cdot d \vec{r} = (α y \hat{i} + 2 α x \hat{j}) \cdot (d x \hat{i} + d y \hat{j})$

$∴$ $d W = α y d x + 2 α x d y$

$Work done$

$From A \to B, d y = 0, as y = 1$

$∴$ $W_{1} = \int_{0}^{1} α y d x = α \int_{0}^{1} d x = α$

$From B \to C, d x = 0, as x = 1$

$∴$ $W_{2} = \int_{1}^{0.5} 2 α x d y = \int_{1}^{0.5} 2 α d y = 2 α (0.5 - 1) = - α$

$From C \to D, d y = 0, as y = 0.5$

$∴$ $W_{3} = \int_{1}^{0.5} α \times 0.5 d x = \frac{α}{2} (0.5 - 1) = - \frac{α}{4}$

$From D \to E, d x = 0, as x = 0.5$

$∴$ $W_{4} = \int_{0.5}^{0} 2 α \times 0.5 d y = α (0 - 0.5) = - \frac{α}{2}$

$From E \to F, d y = 0, as y = 0$

$∴$ $W_{5} = \int α \times 0 d x = 0$

$From F \to A, d x = 0, as x = 0$

$∴$ $W_{6} = \int 2 α \times 0 d y = 0$

$∴$ $Total work done W = W_{1} + W_{2} + W_{3} + W_{4} + W_{5} + W_{6}$

$W = α - α - \frac{α}{4} - \frac{α}{2} = - \frac{3 α}{4} = - \frac{3 (- 1)}{4} = 0.75 J$

Solution

Q.
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 _____ Joule. [2019]

Want Us to Email you About Special Offers & Updates?

Solution

Q. 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=-1 N m-1. The work done on the particle by this force F→ will be _____ Joule. [2019]

Want Us to Email you About Special Offers & Updates?

Q.
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 _____ Joule. [2019]