A thin rod AB is hinged at end A and the block is moving with a constant velocity $V = 8 m / s$
on a horizontal surface as shown in the figure. Then choose the correct option(s). (Take h =1m)

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The angular velocity of the road at $θ = 30^{\circ}$ is $4 rad / s$

The angular retardation of the rod at $θ = 30^{\circ} i s 8 \sqrt{3} r a d / s^{2}$

The angular retardation of the rod at $θ = 30^{\circ} i s 4 \sqrt{3} r a d / s^{2}$

The angular velocity of the road at $θ = 30^{\circ}$ is $2 rad / s$

answer is A, D.

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Detailed Solution

$\frac{ω h}{\sin θ} = v \sin θ \Rightarrow ω = \frac{V \sin^{2} θ}{h} A t θ = 30^{0}, ω = \frac{8}{1} \times \frac{1}{4} = 2 r a d / s N o w, \frac{d ω}{d t} = \frac{v}{h} 2 \sin θ \cos θ \frac{d θ}{d t} α = \frac{v}{h} \sin 2 θ (- ω) α = - \frac{v ω \sin 2 θ}{h} A t θ = 30^{0}, α = \frac{- 8 \times 2}{1} \times \frac{\sqrt{3}}{2} = - 8 \sqrt{3} r a d / s^{2}$