Which of the following force is conservative force?

\to F = - 3 y^i - 4 x^j

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\to F = 5 y^i - 5 x^j

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\to F = 3 y^i + 4 x^j

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\to F = - 5 y^i - 5 x^j

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Solution

The correct option is D $\to F = - 5 y^i - 5 x^j$
We know any force, $\to F = F_{1}^i + F_{2}^j$ is conservative force if $\frac{\partial F_{1}}{\partial y} = \frac{\partial F_{2}}{\partial x}$
For, $\to F = - 5 y^i - 5 x^j$
$F_{1} = - 5 y$ and $F_{2} = - 5 x$
$\frac{\partial F_{1}}{\partial y} = - 5$ and $\frac{\partial F_{2}}{\partial x} = - 5$
As, $\frac{\partial F_{1}}{\partial y} = \frac{\partial F_{2}}{\partial x}$
Therefore, $\to F = - 5 y^i - 5 x^j$ is conservative force.

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List - I	List - II
a) non-conservative force	e) depends on mass
b) conservative force	f) work done is independent of path
c) deceleration due to friction on rough horizontal surface	g) workdone depends on path
d) accelerating force on a smooth inclined surface	h) independent of mass

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