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JEE Advanced 2020 · Paper 1, Physics question 1

A football of radius RR is kept on a hole of radius r (r<R)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 θ\theta from the horizontal as shown in the figure below. The maximum value of θ\theta so that the football does not start rolling down the plank satisfies (figure is schematic and not drawn to scale)

A plank tilted at angle θ to the horizontal, with a ball of radius R resting in a hole of diameter 2r in the plank.
Choose one option
Answer and worked solution

Official answer: A

Geometry. The ball touches the plank only along the rim of the hole. Its centre lies on the normal through the centre of the hole, at height R2−r2\sqrt{R^2-r^2} above the plank. The lowest rim point is a distance rr down the slope from the foot of that normal.

Condition for rolling. The ball tips over the lower rim point once its weight acts beyond that point, i.e. once the vertical through the centre passes the lower rim point:
tan⁡θ=rR2−r2  ⟺  sin⁡θ=rR.\tan\theta=\frac{r}{\sqrt{R^2-r^2}}\iff\sin\theta=\frac rR.
(A)

Topics:Equilibrium of rigid bodiesToppling
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