A2 June 2022 Q1
1. Three particles of masses \(2m\), \(3m\) and \(km\) are placed at the points with coordinates \((3a, 2a)\), \((a, -4a)\) and \((-3a, 4a)\) respectively.
The centre of mass of the three particles lies at the point with coordinates \((\bar{x}, \bar{y})\).
Given that the distance of the centre of mass of the three particles from the point \((0, 0)\) is \(\dfrac{1}{3}a\)
| Scheme | Marks | AO |
|---|---|---|
| Moments about \(y\)-axis | M1 | 3.4 |
| \(\big((5+k)m\bar{x} = -3kma + 6ma + 3ma\big) \qquad \bar{x} = \dfrac{(9-3k)a}{5+k}\) | A1 | 1.1b |
| Moments about \(x\)-axis | M1 | 3.4 |
| \(\big((5+k)m\bar{y} = 4kma + 4ma - 12ma\big) \qquad \bar{y} = \dfrac{(4k-8)a}{5+k}\) | A1 | 1.1b |
| (4) |
Notes
M1: Moments equation to find \(\bar{x}\) – need all terms and dimensionally correct
Allow with \(m\) cancelled throughout
Allow if they have a common factor of \(g\)
A1: Correct expression for \(\bar{x}\)
Any equivalent form. Allow recovery
M1: Moments equation to find \(\bar{y}\) – need all terms and dimensionally correct
Allow with \(m\) cancelled throughout
Allow if they have a common factor of \(g\)
A1: Correct expression for \(\bar{y}\)
Any equivalent form. Allow recovery
| Scheme | Marks | AO |
|---|---|---|
| \(\Rightarrow 9\left[(9-3k)^2 + (4k-8)^2\right] = (5+k)^2\) \(\big(224k^2 - 1072k + 1280 = 0\big)\) | M1 | 3.1a |
| \(\Rightarrow k = \dfrac{5}{2}\), or \(k = \dfrac{16}{7}\) | A1 | 2.2a |
| (2) | ||
| (6 marks) |
Notes
M1: Use their moments equations to form a quadratic equation in \(k\) only with no square root (need not simplify)
A1: Obtain both correct values.
Accept 2.5 and 2.3 or better (2.2857…)