A uniform bar of length 12 cm and mass 20 m lies on a smooth horizontal table. Two point masses m and 2 m are moving in opposite directions with same speed of v and in the same plane as the bar, as shown in figure. These masses strike the bar simultaneously and get stuck to it. After collision the entire system is rotating with angular frequency . The ratio of v and is :

Step-by-step solution

Linear momentum conservation: mv - 2mv = 23m v_ cm , giving v_ cm = - v 23 . Angular momentum about center of mass from the collision geometry: Taking collision points 2 cm and 4 cm from the 12 cm bar's center at different positions, the angular momentum is L = m v r₁ + (-2m) v r₂ . Calculating moment of inertia: I = 1 12 (20m)(0.12)^2 + m(0.02)^2 + 2m(0.04)^2 = 0.0276m The angular frequency is obtained from = L I , giving ratio v = 33 .