M5 June 2014 (R) Q3
3. A uniform rectangular lamina \(ABCD\), where \(AB = a\) and \(BC = 2a\), has mass \(2m\). The lamina is free to rotate about its edge \(AB\), which is fixed and vertical. The lamina is at rest when it is struck at \(C\) by a particle \(P\) of mass \(m\). The particle \(P\) is moving horizontally with speed \(U\) in a direction which is perpendicular to the lamina. The coefficient of restitution between \(P\) and the lamina is 0.5
Find the angular speed of the lamina immediately after the impact. (8)
| Scheme | Marks |
|---|---|
| \(I_{AB} = \tfrac{4}{3}(2m)a^2 = \tfrac{8}{3}ma^2\) | B1 |
| CAM: \(\quad mU2a = \tfrac{8}{3}ma^2\omega + mv2a\) | M1 A2 |
| NIL: \(\quad \tfrac{1}{2}U = 2a\omega - v\) | M1 A1 |
| \(\omega = \dfrac{9U}{20a}\) | M1 A1 |
| (8 marks) |