Q: The potential energy of a particle on the line y = 2 x are same at every point. The conservative force on the particle at (1 , 2) may be

(a) 4i + 3j

(b) 4i + 8j

(c) 8i + 4j

(d) – 8i + 4j

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Ans: (d)

Sol: As Potential Energy decreases in the direction of force and remains constant in the perpendicular direction of force .

$F = -\frac{dU}{dr}$

Slope in the direction of line = 2 and slope in perpendicular direction = -1/2 since m_{1} × m_{1} = -1

hence , $\vec{F} = – 8 \hat{i} + 4 \hat{j}$ ; because slope = 4/-8 = -1/2