M1 January 2012 Q4
4.
A non-uniform rod \(AB\), of mass \(m\) and length \(5d\), rests horizontally in equilibrium on two supports at \(C\) and \(D\), where \(AC = DB = d\), as shown in Figure 1. The centre of mass of the rod is at the point \(G\). A particle of mass \(\dfrac{5}{2}m\) is placed on the rod at \(B\) and the rod is on the point of tipping about \(D\).
(a) Show that \(GD = \dfrac{5}{2}d\). (4)
The particle is moved from \(B\) to the mid-point of the rod and the rod remains in equilibrium.
(b) Find the magnitude of the normal reaction between the support at \(D\) and the rod. (5)
| Scheme | Marks |
|---|---|
| M\((D)\) \(mg \times GD = \dfrac{5}{2}mg \times d\) | M1 A1 |
| \(GD = \dfrac{5}{2}d\) * | DM1 A1 |
| (4) |
| Scheme | Marks |
|---|---|
| M\((C)\) \(mg \times \dfrac{d}{2} + \dfrac{5}{2}mg \times \dfrac{3}{2}d = Y \times 3d\) | M1 A2(1, 0) |
| Leading to \(Y = \dfrac{17}{12}mg\) | DM1 A1 |
| (5) | |
| (9 marks) |