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. $\sin \theta=\frac{ r }{2 R }$
• D. $\cos \theta=\frac{ r }{2 R }$

Answer 1

Answer: (A)

At the verge of toppling, $N _{1}=0$
$mg \sin \theta h = mg \cos \theta r$ and $\cos \theta=\frac{ h }{ R }$
So, $m g \sin \theta h=m g \frac{h}{R} r$
$\Rightarrow \sin \theta= r / R$