A2 June 2025 Q1
1.
A uniform rod is cut into six pieces. The pieces are used to form the framework \(ABCDE\) shown in Figure 1.
- All the pieces of the framework lie in the same plane.
- \(AB = BC = CD = 2a\)
- \(AE = EC = ED = \sqrt{2}\,a\)
- Point \(E\) is the midpoint of \(AC\)
- Angle \(ABC\) = angle \(BCD = 90^\circ\)
The distance of the centre of mass of the framework from \(AB\) is \(d\)
The mass of the framework is \(M\). A particle of mass \(kM\) is attached to the framework at \(B\)
The distance of the centre of mass of the loaded framework from \(AB\) is \(a\)
| Scheme | Marks | AO |
|---|---|---|
| Correct method to find \(d\) | M1 | 3.1a |
| \((2a \times 0) + 2a \times a + 2a \times 2a + 2\sqrt{2}a \times a + \sqrt{2}a \times \dfrac{3a}{2} = d\left(6a + 3\sqrt{2}a\right)\) | A1 A1 | 1.1b 1.1b |
| \(\Rightarrow d = \dfrac{12 + 7\sqrt{2}}{12 + 6\sqrt{2}}\,a\) * | A1* | 2.2a |
| (4) |
Notes
M1: Take moments about \(AB\) or an axis parallel to \(AB\).
Need all terms and dimensionally correct (condone invisible brackets).
A1: Unsimplified equation with at most one error (repeated errors only count once; condone invisible brackets)
A1: Correct unsimplified equation (condone invisible brackets)
A1*: Obtain given answer including “\(d =\)” from correct exact working (A0 if there are missing brackets in the working)
| Scheme | Marks | AO |
|---|---|---|
| Correct method to form an equation in \(k\) | M1 | 3.1a |
| \(Mg \times \dfrac{12 + 7\sqrt{2}}{12 + 6\sqrt{2}}\,a = (kMg + Mg)a\) | A1 | 1.1b |
| \(k = \dfrac{\sqrt{2}}{12 + 6\sqrt{2}}\) or equivalent | A1 | 1.1b |
| (3) | ||
| (7 marks) |
Notes
M1: Take moments about \(AB\) or an axis parallel to \(AB\).
Accept \(M\) consistently replaced by, e.g. \(6 + 3\sqrt{2}\)
A1: Correct unsimplified equation. Allow without any of \(M\), \(g\) and \(a\).
A1: Accept 0.069 or better \(\left(\dfrac{-1 + \sqrt{2}}{6}\right)\)
Condone any exact form if given as a single fraction.