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A car of mass $m$ is moving with speed $u$ on the concave circular path while another car of mass $= m / 2$ is moving on the convex path of radius $= 2 r$ , where $r$ is the radius of the concave path, with same speed. Ratio of their normal reaction at the lowest and highest points of their paths respectively is

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$\frac{1}{2}$

$\frac{(g + \frac{u^{2}}{r})}{2 (g - \frac{u^{2}}{r})}$

$\frac{2 (g + \frac{u^{2}}{r})}{(g - \frac{u^{2}}{2 r})}$

answer is D.

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Detailed Solution

In first case the centripetal force is in upward direction,

$F_{1} = m g + \frac{m u^{2}}{r}$

In second case the centripetal force is in the downward direction

$F_{2} = \frac{m}{2} g - \frac{m}{2} \frac{u^{2}}{r}$

Ratio of normal reactions= $\frac{2 (g + \frac{u^{2}}{r})}{(g - \frac{u^{2}}{2 r})}$

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