| Exam Board | Edexcel |
|---|---|
| Module | M5 (Mechanics 5) |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Topic | Moments |
| Type | 3D force systems: reduction to single force |
| Difficulty | Standard +0.3 This is a standard M5 question on forces as vectors requiring routine techniques: finding unit vectors, expressing forces in vector form, vector addition, and using moments to find the line of action. While it involves multiple steps (10 marks total), each step follows textbook procedures with no novel insight required. The calculations are straightforward, making it slightly easier than average for A-level. |
| Spec | 1.10a Vectors in 2D: i,j notation and column vectors1.10b Vectors in 3D: i,j,k notation1.10c Magnitude and direction: of vectors1.10d Vector operations: addition and scalar multiplication3.04a Calculate moments: about a point |
The points $P$ and $Q$ have position vectors $4i - 6j - 12k$ and $2i + 4j + 4k$ respectively, relative to a fixed origin $O$.
Three forces, $\mathbf{F}_1$, $\mathbf{F}_2$ and $\mathbf{F}_3$, act along $\overrightarrow{OP}$, $\overrightarrow{OQ}$ and $\overrightarrow{QP}$ respectively, and have magnitudes $7$ N, $3$ N and $3\sqrt{10}$ N respectively.
\begin{enumerate}[label=(\alph*)]
\item Express $\mathbf{F}_1$, $\mathbf{F}_2$ and $\mathbf{F}_3$ in vector form.
[3]
\end{enumerate}
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{1}
\item Show that the resultant of $\mathbf{F}_1$, $\mathbf{F}_2$ and $\mathbf{F}_3$ is $(2i - 10j - 16k)$ N.
[2]
\end{enumerate}
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{2}
\item Find a vector equation of the line of action of this resultant, giving your answer in the form $\mathbf{r} = \mathbf{a} + \lambda\mathbf{b}$, where $\mathbf{a}$ and $\mathbf{b}$ are constant vectors and $\lambda$ is a parameter.
[5]
\end{enumerate}
\hfill \mbox{\textit{Edexcel M5 Q5 [10]}}