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Q.

A car moves uniformly along a horizontal sine curve $y = a \sin (\frac{x}{α})$ , where a and $α$ are certain constants. The coefficient of friction between the wheels and the road is equal to k. The maximum constant speed at which the car can travel without sliding is

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a

$α \sqrt{\frac{k g}{a}}$

b

$- α \sqrt{\frac{k g}{a}}$

c

$k g \sqrt{\frac{α}{a}}$

d

$- k g \sqrt{\frac{α}{a}}$

answer is A.

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Detailed Solution

The radius of curvature is

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$\begin{matrix} f \leq f_{2} or f \leq k m g \\ Or \frac{m v^{2}}{R} \leq k m g \\ or \frac{v^{2}}{R} \leq k g \\ ∴ \frac{v_{max}^{2}}{R_{min}} = k g \\ ∴ v_{max} = \sqrt{kg R_{min}} \end{matrix}$

$\begin{aligned} \frac{x}{a} = \frac{π}{2} \\ ∴ R_{min} = \frac{α^{2}}{a} (from equ (i)) \\ v_{max} = \sqrt{\frac{k g α^{2}}{a}} = α \sqrt{\frac{k g}{a}} \end{aligned}$

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