M2 June 2014 Q1
1. Three particles of mass \(3m\), \(2m\) and \(km\) are placed at the points whose coordinates are \((1, 5)\), \((6, 4)\) and \((a, 1)\) respectively. The centre of mass of the three particles is at the point with coordinates \((3, 3)\).
Find
| Scheme | Marks |
|---|---|
| Moments about \(x\) axis: \(3m \times 5 + 2m \times 4 + km \times 1 = (5 + k)m \times 3\) | M1 A1 |
| \(15 + 8 + k = 3k + 15,\ \ \ \ \ \ k = 4\) | A1 |
| (3) |
Notes
M1 Use moments to form an equation in k. All terms required. Condone sign errors on LHS. Condone \(6 + k\). \(m\) not required. Could be in fraction form.
A1 Correct unsimplified equation. Allow with a common factor of \(g\)
A1 cso
| Scheme | Marks |
|---|---|
| Moments about \(y\) axis: \(3m \times 1 + 2m \times 6 + km \times a = (5 + k)m \times 3\) | M1 A1 |
| \(3 + 12 + 4a = 27,\ \ \ \ \ \ a = 3\) | A1 |
| (3) | |
| (6 marks) |
Notes
M1 Use moments to form an equation in \(a\) and \(k\) (or their \(k\)) only. All terms required. \(m\) not required. Could be in fraction form. Condone \(6 + k\).
A1 Correct unsimplified equation (follow their \(k\) if \(k\) substituted) Accept with a common factor of \(g\).
A1 Allow after use of an incorrect value for \(k\).
Alternative moments equations using axes through \((3, 3)\):
(a) Parallel to \(y\) axis \(3m \times 2 + 2m \times 1 = km \times 2\)
(b) Parallel to \(x\) axis \(2m \times 3 = km(a - 3) + 3m \times 2\)
NB The vector equation on its own is not sufficient for M1 – they need to form separate equations. However, if they deduce the correct answer(s) from their vector equation full marks are available.