First slide

Work

Question

A particular of mass ‘m’ moves along the quarter section of the circular path whose centre is at the origin. The radius of the circular path is ‘a’. A force $\vec{F} = y \hat{i} - x \hat{j}$ newton acts on the particle, where x,y denote the coordinates of position of the particle. Calculate the work done by this force in taking the particle from Point A (a,0) to Point B(0,a) along the circular path.

Moderate

A

$- \sqrt{2 π a^{2} J}$
B

$- \frac{π a^{2}}{2} J$
C

$- π a^{2} J$
D

$- \frac{\sqrt{3} π a^{2}}{2} J$

Solution

Work done by force F is
$W = \int \vec{F} . d r = \int (y \hat{i} - x \hat{j}) . (d x \hat{i} + d y \hat{i}) = \int (y d x - x d y)$
Equation of circular path
$X^{2} + y^{2} = a^{2}$
differentiate
$∴ x d x + y d y = 0$ $\Rightarrow d x = \frac{- y d y}{x}$
$\Rightarrow W = \int (y (\frac{- y d y}{x}) - x d y) = - \int \frac{(x^{2} + y^{2})}{x} d y$
substitute, $X^{2} + y^{2} = a^{2}$ in above integral
$W = - \int_{0}^{a} \frac{a^{2}}{\sqrt{a^{2} - y^{2}}} d y = - \frac{π a^{2}}{2} j o u l e$

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