M5 June 2007 Q3
3. A uniform rod \(AB\), of mass \(m\) and length \(2a\), is free to rotate about a fixed smooth axis which passes through \(A\) and is perpendicular to the rod. The rod has angular speed \(\omega\) when it strikes a particle \(P\) of mass \(m\) and adheres to it. Immediately before the rod strikes \(P\), \(P\) is at rest and at a distance \(x\) from \(A\). Immediately after the rod strikes \(P\), the angular speed of the rod is \(\tfrac{3}{4}\omega\).
Find \(x\) in terms of \(a\).
| Scheme | Marks |
|---|---|
| \(\tfrac{4}{3}ma^2\omega = \left(\tfrac{4}{3}ma^2 + mx^2\right)\tfrac{3}{4}\omega\) | M1 A1 A1 |
| \(\Rightarrow\ x = \tfrac{2}{3}a\) | DM1 A1 |
| (5) |