A football of radius is kept on a hole of radius made on a plank kept horizontally. One end of the plank is now lifted so that it gets tilted making an angle from the horizontal as shown in the figure below. The maximum value of so that the football d

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 $θ$ 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) [2020]

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(1)

The maximum value of $r (r < R)$ i.e., $q_{\max}$ , the football is about to roll, then $N_{2} = 0$ and all the forces $(m g and N_{1})$ must pass through contact point 'P'.

$∴$ $\cos (90 ° - θ_{\max}) = \frac{O Q}{O R} = \frac{r}{R} or, \sin θ_{\max} = \frac{r}{R}$

Solution

1 $\sin θ = \frac{r}{R}$

2 $\tan θ = \frac{r}{R}$

3 $\sin θ = \frac{r}{2 R}$

4 $\cos θ = \frac{r}{2 R}$

Ans.
(1)

The maximum value of $q$ i.e., $q_{\max}$ , the football is about to roll, then $N_{2} = 0$ and all the forces $(m g and N_{1})$ must pass through contact point 'P'.

$∴$ $\cos (90 ° - θ_{\max}) = \frac{O Q}{O R} = \frac{r}{R} or, \sin θ_{\max} = \frac{r}{R}$

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Solution

1 sinθ=rR

2 tanθ=rR

3 sinθ=r2R

4 cosθ=r2R