A2 October 2020 Q1
1. Three particles of masses \(3m\), \(4m\) and \(2m\) are placed at the points \((-2, 2)\), \((3, 1)\) and \((p, p)\) respectively.
The value of \(p\) is such that the distance of the centre of mass of the three particles from the point \((0, 0)\) is as small as possible.
Find the value of \(p\). (7)
| Scheme | Marks | AO |
|---|---|---|
| Correct method to find an equation in \(\bar{x}\) | M1 | 1.1b |
| \(-3\times 2 + 4\times 3 + 2\times p = 9\bar{x} \qquad (6 + 2p = 9\bar{x})\) | A1 | 1.1b |
| Correct method to find an equation in \(\bar{y}\) | M1 | 1.1b |
| \(3\times 2 + 4\times 1 + 2\times p = 9\bar{y} \qquad 10 + 2p = 9\bar{y}\) | A1 | 1.1b |
| \((9\bar{x})^2 + (9\bar{y})^2 = (6 + 2p)^2 + (10 + 2p)^2\) \(\big(= 136 + 64p + 8p^2\big)\) | M1 | 1.1b |
| \(= 8\left[(p + 4)^2 + 17 - 16\right]\) | M1 | 3.1a |
| \(\Rightarrow p = -4\) | A1 | 2.2a |
| (7 marks) |
Notes
M1: Take moments about axis parallel to \(x = 0\). Need all terms and dimensionally correct.
A1: Correct unsimplified equation in \(\bar{x}\). Seen or implied
M1: Take moments about axis parallel to \(y = 0\). Need all terms and dimensionally correct.
A1: Correct unsimplified equation in \(\bar{y}\). Seen or implied
M1: Use of Pythagoras to find distance (or square of distance) from origin
M1: Correct strategy to find value of \(p\) to minimise the distance e.g. use of calculus or complete the square
A1: Correct answer only