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 (the 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

$\sin \theta =\frac{r}{R}$

Draw a free body diagram, where the normal reaction will act on point of contact. So,

In the limiting case, $mg$ will act along with a normal reaction. For ${\theta }_{max}$, the football is about to roll, then $N_2=0$ and all the forces $(mgand{N}_1)$must pass through the contact point. So,

Therefore,

$$\begin{gathered} Sin{\theta }_{max}=\frac{perpendicular}{Hypotenuse} \\ Sin{\theta }_{max}=\frac{r}{R} \end{gathered}$$

Hence, the correct option is (A.)