Two small masses m and M lie on a large horizontal frictionless...

Two small masses

m

and

M

lie on a large horizontal frictionless circular track of radius

R

. The two masses are free to slide on the track but constrained to move along a circle. Initially the two masses are tied by a thread with a compressed spring between them (spring of negligible length being attached with none of the two masses). The compressed spring stores a potential energy

\mathrm{U}_0

. At a certain time

\mathrm{t}=0

the thread is burnt and the two masses are released to run opposite to each other leaving the spring behind. The total mechanical energy remaining conserved. On the circular track the two masses make a head on perfectly elastic collision. Take

\mathrm{M}=2 \mathrm{~m}

for all calculations. Which of the following option(s) is / are correct?

Options

AThe angle turned by mass m before the collision is $\theta=4 \frac{\pi}{3}$
BThe velocity of mass m on the track is $u=\sqrt{\frac{4 U_0}{3 m}}$
CThe time taken to collide for the first time is $t_1=2 \pi R \sqrt{\frac{m}{3 U_0}}$
DThe time taken for the second collision is $t_2=2 \pi R \sqrt{\frac{2 m}{3 U_0}}$

Correct answer

C. The time taken to collide for the first time is $t_1=2 \pi R \sqrt{\frac{m}{3 U_0}}$

Step-by-step solution

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