Question 167 of 210

An inclined plane is bent in such a way that the vertical cross-section is given by $y=\dfrac{{x}^{2}}{4}$ where $y$ is in vertical and $x$ in horizontal direction. If the upper surface of this curved plane is rough with coefficient of friction $\mu =0.5,$ the maximum height in $cm$ at which a stationary block will not slip downward is________$cm.$

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Answer: 25

At maximum ht. block will experience maximum friction force. Therefore if at this height slope of the tangent is $\tan \theta ,$ then $\theta =$Angle of repose
$∴\tan \theta =\dfrac{dy}{dx}=\dfrac{2x}{4}=\dfrac{x}{2}=0.5$
$\Rightarrow x=1$ and therefore $y=\dfrac{{x}^{2}}{4}=0.25m$
$=25cm$
$∴$ Answer is $25cm$
(Assuming that $x$ & $y$ in the equation are given in meter)