Circular MotionHard

Question

A small both A of mass m is attached to a massless rigid rod of length 1 m pivoted at point P and kept at an angle of $60^{\circ}$ with vertical as shown in figure. At distance of 1 m below point P , an identical bob B is kept at rest on a smooth horizontal surface that extends to a circular track of radius R as shown in figure. If bob B just manages to complete the circular path of radius R upto a point Q after being hit elastically by bob A , then radius R is $\_\_\_\_$ m.

Options

A.$\frac{3}{5}$
B.$\frac{1}{5}$
C.$\frac{2 + \sqrt{3}}{5}$
D.$\frac{2 - \sqrt{3}}{5}$

Solution

$V_{A}$ at lowest point

$${V_{A} = \sqrt{2\text{ }g\mathcal{l}(1 - cos\theta)} }{V_{A} = \sqrt{2 \times 10 \times 1\left( 1 - \frac{1}{2} \right)} = \sqrt{10} }$$After collision velocity of B becomes,

$V_{A} = \sqrt{10} = V_{B}\ $ (Same mass)

Now to complete circular motion

$${V_{B} = \sqrt{5gR} }{R = \frac{1}{5}}$$

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