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)

Step 1: Identify the geometry of the contact points.

The football of radius $R$ rests on a hole of radius $r$ in the plank, touching the two edges of the hole.

Let $\alpha$ be the angle that the line connecting the center of the ball to either contact edge makes with the normal to the plank.

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

Step 2: Analyze the forces at the verge of rolling.

At the critical angle where the football is on the verge of rolling down, the normal force at the uphill edge becomes zero.

$$N_{1} = 0$$

Step 3: Apply the torque balance about the downhill edge.

For the ball to remain in equilibrium, the line connecting the center of the ball to the downhill contact edge must be vertical.

The normal to the plank makes an angle $\theta$ with the vertical, and the line from the center of the ball to the downhill edge makes an angle $\alpha$ with the normal to the plank.

$$\theta = \alpha$$

Step 4: Relate the critical angle to the radii.

Substitute the relation for $\alpha$ into the equilibrium condition.

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

Answer: The maximum value of $\theta$ satisfies $\sin\theta = \dfrac{r}{R}$.