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 ballat 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 ballat 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» ANSWER :(i) `omega_(n+1) = (6v)/(13r) + (7)/(13)omega_(n)` (II) `omega_(n) = (v)/(R) [1-((7)/(13))^(n-1)]` (iii) `(v)/(r)`