A bead is connected with a fixed disc’s periphery of radius R by an inextensible massless string of length $l = \frac{π R}{2}$ in a smooth horizontal plane. If the bead is pushed with a velocity $v_{0}$ perpendicular to the string, the bead moves in a horizontal curve and consequently collapses on the disc after a time t. Then,

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Work done by the string on the bead is $m v_{0}^{2}$

average speed of the bead during its motion before is strikes the disc is $v_{0}$

kinetic energy of the bead increases gradually

Time taken by bead to strike the disc is $t = \frac{l^{2}}{2 R v_{0}}$

answer is B, C.

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Detailed Solution

Power delivered by tension to the bead , $P = T v \cos 90^{0} = 0$ .
So, work done by tension on bead is 0. bead will remain constant and so its average speed will be $v_{0}$ only.
Also, let the length of string that winds on the string at time t be x as shown in the figure.
Question Image
$θ = \frac{I - x}{R} \Rightarrow \frac{d θ}{d t} = - \frac{d x}{R d t}$
$\Rightarrow \frac{v_{0}}{x} = - \frac{d x}{R d t}$

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