First slide

Work done by Diff type of forces

Question

A force $F = - k (y \hat{i} + x \hat{j})$ where k is a positive constant, acts on a particle moving in the XY-plane. Starting from the origin, the particle is taken along the positive X-axis to the point (a, 0) and then parallel to the Y-axis to the point (a, a). The total work done by the force on the particle is [AIIMS 2017]

Difficult

A

$- 2 k a^{2}$
B

$2 k a^{2}$
C

$- k a^{2}$
D

$k a^{2}$

Solution

Given,    $F = - k (y \hat{i} + x \hat{j})$
As, work done,    $W = \int F \cdot d r$
So,    $W^{'} = - k \int (y \hat{i} + x \hat{j}) \cdot (d x \hat{i} + d y \hat{j})$
$\Rightarrow W = - k \int (y d x + x d y)$
$\Rightarrow W = - k \int_{(0, 0)}^{(a, a)} d (x y) [∵ y d x + x d y = \int d (x y)]$
Hence, $W = - k (x y)_{(0, 0)}^{(a, a)} = - k a^{2}$

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