A light rod of length L is hinged to a plank of mass m. The plank is lying on the edge of a horizontal table such that the rod can swing freely in the vertical plane without any hindrance from the table. A particle of mass m is attached to the end of the rod and system is released from $θ = 0^{0}$ position. Assume that friction between the plank and the table is large enough to prevent it from slipping. Then

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Friction on the plank by the table is zero when $θ = \cos^{- 1} (\frac{2}{3})$

Value of $θ$ at which restoring force in the rod changes from compression type to tensile type is $\cos^{- 1} (\frac{2}{3})$

Normal force applied by the plank on the table is maximum when $θ = \cos^{- 1} (\frac{1}{3})$

Minimum value of normal force between plank and table is $\frac{3 m g}{2}$

answer is A, C, D.

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Detailed Solution

When $θ = \cos^{- 1} (\frac{2}{3}) ω = 0$
$∴ f = 0$
$N = 2 m g - \frac{m v^{2}}{r} \cos θ$
$\frac{1}{2} m v^{2} = m g r (1 - \cos θ)$
$N = 2 m g - 2 m g (1 - \cos θ) \cos θ$
for minimum value $\cos θ = \frac{1}{2}$
$N = 2 m g - \frac{2 m g}{42} = \frac{3 m g}{2}$