M2 January 2013 Q1
1. Two uniform rods \(AB\) and \(BC\) are rigidly joined at \(B\) so that \(\angle ABC = 90^\circ\). Rod \(AB\) has length 0.5 m and mass 2 kg. Rod \(BC\) has length 2 m and mass 3 kg. The centre of mass of the framework of the two rods is at \(G\).
The distance of \(G\) from \(AB\) is 0.6 m.
The framework is suspended from \(A\) and hangs freely in equilibrium.
| Scheme | Marks |
|---|---|
 | |
| \(5\bar{y} = 2 \times 0.25\ (+0)\) | M1 |
| \(\bar{y} = \dfrac{2 \times 0.25}{5} = 0.1\) | A1 |
| (2) |
Notes
M1 Moments equation with lengths ¼, 1 and (ratio of) masses 2, 3. Allow moments about a parallel axis. Use of length for mass is M0.
A1 For distance from \(BC\)
(The equals sign in the first line is printed as a dash in the mark scheme.)
| Scheme | Marks |
|---|---|
 | |
| \(\tan\theta = \dfrac{0.6}{0.5 - 0.1}\) | M1 A1ft |
| \(\theta = \tan^{-1}\left(\dfrac{6}{4}\right) = 56.3^\circ = 56^\circ\) | A1 |
| (3) | |
| (5 marks) |
Notes
M1 Must suspend from \(A\). Use of tan with 0.6 and \(0.5 - \bar{y}\). Could be wrong way up. Must be using 0.6
A1ft Correct way up. ft their \(\bar{y}\).
A1 Accept awrt 56.3