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A vehicle of mass $m$ starts moving along a horizontal circle without slipping, so that centripetal acceleration is $a_{r} = b t^{2}$ ,where $b$ is constant. At $t = t_{0}$ , the vehicle skids. The friction on the car as function of time $t (< t_{0})$ is $n m \sqrt{b^{2} t^{4} + b R}$ . The value of $n$ is

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Detailed Solution

$a_{r} = b t^{2}$ or $\frac{v^{2}}{R} = b t^{2}$ or $v = \sqrt{b R t^{2}}$

or $v = \sqrt{b R} t$ or $a_{t} = \frac{d v}{d t} = \sqrt{b R}$

The required acceleration for which vehicle is provided by friction

$∴ f = m a = m \sqrt{a_{r}^{2} + a_{t}^{2}} = m \sqrt{b^{2} t^{4} + b R} ∴ n = 1$

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