| Exam Board | Edexcel |
|---|---|
| Module | M5 (Mechanics 5) |
| Year | 2006 |
| Session | January |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Moments |
| Type | 3D force systems: reduction to single force |
| Difficulty | Challenging +1.3 This M5 question requires systematic vector manipulation (finding direction vectors, resolving forces, computing moments) across multiple steps, but follows a standard template for 3D force systems reducing to couples. The calculations are lengthy but methodical, with no novel geometric insight required—harder than typical C3/M1 questions due to 3D vectors and Further Maths content, but routine for M5 students who know the couple equilibrium conditions. |
| Spec | 6.03b Conservation of momentum: 1D two particles6.03f Impulse-momentum: relation6.06a Variable force: dv/dt or v*dv/dx methods |
| Answer | Marks | Guidance |
|---|---|---|
| (a) \(SQ = 9j - 8 = 4j - 3k \Rightarrow E_1 = 16j - 12k\) | M1, A1 | |
| \(SR = -3i - 2j \Rightarrow E_2 = -6i - 4j\) | M1, A1 | |
| Net couple alone \(\Rightarrow \sum E_i = 0 \Rightarrow F_3 = 6i - 10j + 12k\) | M1, A1 (6) | |
| (b) \(M(S), G = SF \times E_3\) | M1 | |
| \(= (-7i + 4j - 2k) \times (6i - 12j + 12k)\) | M1, A1 | |
| (or complete express'y along another pt) | ||
| \(G = 24i + 12j + 60k\) | M1, A1 | |
| \(= 12(2i + 6j + 5k)\) | ||
| \(\ | G\ | = 12\sqrt{65}\) |
(a) $SQ = 9j - 8 = 4j - 3k \Rightarrow E_1 = 16j - 12k$ | M1, A1 |
$SR = -3i - 2j \Rightarrow E_2 = -6i - 4j$ | M1, A1 |
Net couple alone $\Rightarrow \sum E_i = 0 \Rightarrow F_3 = 6i - 10j + 12k$ | M1, A1 (6) |
(b) $M(S), G = SF \times E_3$ | M1 |
$= (-7i + 4j - 2k) \times (6i - 12j + 12k)$ | M1, A1 |
(or complete express'y along another pt) | |
$G = 24i + 12j + 60k$ | M1, A1 |
$= 12(2i + 6j + 5k)$ | |
$\|G\| = 12\sqrt{65}$ | M1, A1 (6) |6. The vertices of a tetrahedron $P Q R S$ have position vectors $\mathbf { p } , \mathbf { q } , \mathbf { r }$ and $\mathbf { s }$ respectively, where
$$\mathbf { p } = - 3 \mathbf { i } + 4 \mathbf { j } - \mathbf { k } , \quad \mathbf { q } = 4 \mathbf { i } + 4 \mathbf { j } - 2 \mathbf { k } , \quad \mathbf { r } = \mathbf { i } - 2 \mathbf { j } + \mathbf { k } , \quad \mathbf { s } = 4 \mathbf { i } + \mathbf { k }$$
Forces of magnitude 20 N and $2 \sqrt { } 13 \mathrm {~N}$ act along $S Q$ and $S R$ respectively. A third force acts at $P$.\\
Given that the system of three forces reduces to a couple $\mathbf { G }$, find
\begin{enumerate}[label=(\alph*)]
\item the third force,
\item the magnitude of $\mathbf { G }$.\\
(6)\\
(Total 12 marks)
\end{enumerate}
\hfill \mbox{\textit{Edexcel M5 2006 Q6 [12]}}