A dumb-bell is made of two small spheres of equal mass connected by a light rigid rod of length 2 ℓ . Two such identical dumb-bells approach each other with zero angular velocity on a smooth horizonta

Question

A dumb-bell is made of two small spheres of equal mass connected by a light rigid rod of length $2 ℓ$ . Two such identical dumb-bells approach each other with zero angular velocity on a smooth horizontal table as shown and undergo an elastic collision of negligible duration. Which of the following statements is/are true?

TestHub · Accepted Answer

By symmetry, after the collision both dumb-bells have the same COM speed v' and angular speed $ω$

Energy: $\frac{1}{2} (4 m) v^{2} = \frac{1}{2} (4 m) v^{' 2} + \frac{1}{2} (2 m ℓ^{2} ω^{2})$ ; angular momentum: $4 m v ℓ = 4 m ℓ^{2} ω$

Solving: $v^{'} = 0, ω = \frac{v}{ℓ}$ , so $T = \frac{m v ^{2}}{ℓ}$ (A wrong, B correct).

Each COM stays at rest for $\frac{π}{ω} = \frac{π ℓ}{v}$ while the dumb-bells rotate, after which a second collision occurs (C, D correct).

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Options:(select one or more)

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