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A football of radius $R$ is kept on a hole of radius $r (r < R)$ made on a plank kept horizontally. One end of the plank is now lifted so that it gets tilted making an angle $\theta$ from the horizontal as shown in the figure below. The maximum value of $\theta$ so that the football does not start rolling down the plank satisfies (figure is schematic and not drawn to scale) -
$\sin \theta=\frac{r}{R}$
$\tan \theta=\frac{r}{R}$
$\sin \theta=\frac{r}{2 R}$
$\cos \theta=\frac{r}{2 R}$
Solution
For $\theta_{\max }$, the football is about to roll, then $N _2=0$ and all the forces $\left( Mg\right.$ and $\left.N _1\right)$ must pass through contact point
$\therefore \cos \left(90^{\circ}-\theta_{\max }\right)=\frac{ r }{ R } \Rightarrow \sin \theta_{\max }=\frac{ r }{ R }$
