Question

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)-

• A. $\sin \theta =\frac{r}{R}$
• B. $\tan \theta =\frac{r}{R}$
• C. $\sin \theta =\frac{r}{2R}$
• D. $\cos \theta =\frac{r}{2R}$

Answer 1

The correct option is A $\sin \theta =\frac{r}{R}$


For ${\theta }_{max}$ the football is about to roll , then $N_2$= 0 and all the other forces ( Mg and $N_1$ ) must pass through the
contact point .
$\therefore \cos ({90}^{\circ }-{\theta }_{max})=\frac{r}{R}$$\Rightarrow \sin {\theta }_{max}=\frac{r}{R}$