A2 June 2023 Q1
1. Three particles of masses \(3m\), \(4m\) and \(km\) are positioned at the points with coordinates \((2a, 3a)\), \((a, 5a)\) and \((2\mu a, \mu a)\) respectively, where \(k\) and \(\mu\) are constants.
The centre of mass of the three particles is at the point with coordinates \((2a, 4a)\).
Find
(i) the value of \(k\)
(ii) the value of \(\mu\) (6)
| Scheme | Marks | AO |
|---|---|---|
| Moments about \(y\)-axis | M1 | 3.1a |
| \(2a(7 + k) = 3 \times 2a + 4 \times a + k \times 2\mu a\) \((14 + 2k = 10 + 2\mu k)\) | A1 | 1.1b |
| Moments about \(x\)-axis | M1 | 3.1a |
| \(4a(7 + k) = 3 \times 3a + 4 \times 5a + k \times \mu a\) \((28 + 4k = 29 + \mu k)\) | A1 | 1.1b |
| Obtain \(k = 1\) or \(\mu = 3\) | A1 | 2.2a |
| Obtain \(k = 1\) and \(\mu = 3\) | A1 | 2.2a |
| (6) | ||
| (6 marks) |
Notes
M1: Dimensionally correct equation with all terms. Allow with \(\bar{x}\) in place of \(2a\)
A1: Correct unsimplified equation.
M1: Dimensionally correct equation with all terms. Allow with \(\bar{y}\) in place of \(4a\)
A1: Correct unsimplified equation.
These two equations can be combined as a single vector equation.
A1: One correct value.
A1: Both correct values.