A uniform rod of mass 4m and length 2r can rotate in horizontal plane about a vertical axis passing through its centre O. Two small balls each of mass m are attached to its ends. A fixed gun fires identical balls with speed v in horizontal direction. The firing is being done at suitable intervals so that the fired balls either hit the ball at end A or B while moving in the direction of velocity of A or B. All collisions are elastic. (i) Initial angular velocity of the rod is zero and its angular velocity after `n^(th)` collision is `omega_(n)` Write `omega_(n + 1)` in terms of `omega_(n)` (ii) Solve the above equation to get `omega_(n) (iii) Find the limiting value of `omega`.

A uniform rod of mass 4m and length 2r can rotate in horizontal plane

1.	A uniform rod of mass 4m and length 2r can rotate in horizontal plane about a vertical axis passing through its centre O. Two small balls each of mass m are attached to its ends. A fixed gun fires identical balls with speed v in horizontal direction. The firing is being done at suitable intervals so that the fired balls either hit the ball at end A or B while moving in the direction of velocity of A or B. All collisions are elastic. (i) Initial angular velocity of the rod is zero and its angular velocity after `n^(th)` collision is `omega_(n)` Write `omega_(n + 1)` in terms of `omega_(n)` (ii) Solve the above equation to get `omega_(n) (iii) Find the limiting value of `omega`.
Answer» Correct Answer - (i) `omega_(n+1) = (6v)/(13r) + (7)/(13)omega_(n)` (ii) `omega_(n) = (v)/(r) [1-((7)/(13))^(n-1)]` (iii) `(v)/(r)`

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