Maximum Inclination Angle for Football on Plank Hole Without Rolling

To find the maximum angle of inclination $\theta$ such that the football does not start rolling down the plank, we analyze the condition for rotational equilibrium about the lowest point of contact.

1. Geometry of the Contact Points

Let:

• $O$ be the center of the circular hole of radius $r$ in the plane of the plank.
• $C$ be the center of mass of the sphere of radius $R$.
• $P$ be the lowest point of contact between the sphere and the edge of the hole.

Since the sphere is symmetric and rests uniformly in the circular hole, its center $C$ lies on the line passing through $O$ perpendicular to the plank's surface.

In the right-angled triangle $\triangle COP$:

• The angle at $O$ is a right angle ($\angle COP = 90^\circ$).
• The distance from the center of the hole to the contact point is $OP = r$.
• The radius of the sphere is $CP = R$.

The angle between the line $CO$ (which is normal to the plank) and the line $CP$ is $\angle OCP$. From the geometry of $\triangle COP$: $\sin(\angle OCP) = \frac{OP}{CP} = \frac{r}{R}$

2. Condition for Rolling Down

When the plank is inclined at an angle $\theta$ to the horizontal:

• The normal to the plank ($CO$) makes an angle $\theta$ with the vertical.
• The sphere will begin to roll down about the pivot point $P$ when the vertical line passing through its center of mass $C$ passes through or goes beyond $P$.

At the critical angle $\theta_{\text{max}}$, the vertical line from $C$ passes directly through the lower contact point $P$. This implies that the line segment $CP$ is perfectly vertical.

Since $CO$ is perpendicular to the inclined plank, the angle between the vertical line $CP$ and the normal $CO$ is equal to the angle of inclination of the plank, $\theta$: $\angle OCP = \theta$

3. Calculating $\theta$

Substituting $\angle OCP = \theta$ into the geometric relation derived above: $\sin\theta = \frac{r}{R}$

Thus, the maximum value of $\theta$ such that the football remains at rest without rolling down satisfies: $\sin\theta = \frac{r}{R}$

Correct Answer: A