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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