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 θ from the horizontal as shown in the figure below. The maximum value of θ so that the football does not start rolling down the plank satisfies (figure is schematic and not drawn to scale)-
(A)sinθ=Rr✓ Correct
(B)tanθ=Rr
(C)sinθ=2Rr
(D)cosθ=2Rr
Answer: (A)
Solution
At the maximum angle θ_max, the football is about to roll. At this critical point, the normal force N₂ from the rim becomes zero, and all forces (Mg and N₁) must pass through the contact point. From the geometry of the setup, when the plank is tilted at angle θ, the line of action of the normal force N₁ makes an angle (90° - θ) with the vertical through the center. For equilibrium at the critical angle: cos(90° - θ_max) = r/R, which gives sin θ_max = r/R.
JEE Advanced · 2020 · 27 Sep 2020 · Paper 1 · Single correct option
