| Exam Board | Edexcel |
|---|---|
| Module | M1 (Mechanics 1) |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Forces, equilibrium and resultants |
| Type | Equilibrium of particle under coplanar forces |
| Difficulty | Moderate -0.3 This is a standard M1 equilibrium problem requiring resolution of forces in two perpendicular directions (North-South and East-West). While it involves multiple forces and some trigonometry (30° and 45° angles), the method is routine: resolve horizontally and vertically, then solve simultaneous equations. The 'show that' part in (a) guides students significantly, making it slightly easier than average for M1. |
| Spec | 1.05b Sine and cosine rules: including ambiguous case3.03n Equilibrium in 2D: particle under forces |
| Answer | Marks | Guidance |
|---|---|---|
| (a) resolve \(\rightarrow\): \(6 + X\cos45 - 18\sin30 = 0\) | M2 | |
| \(6 + X\frac{\sqrt{2}}{2} - 9 = 0\) so \(X = 3\sqrt{2}\) N | M1 A1 | |
| (b) resolve \(\uparrow\): \(Y + X\cos45 + 18\cos30 - 20 = 0\) | M2 | |
| \(Y + (3\sqrt{2})\frac{\sqrt{2}}{2} + 18\frac{\sqrt{3}}{2} - 20 = 0\) | M1 | |
| \(Y = 20 - 9\sqrt{3} - 3 = 17 - 9\sqrt{3}\) | A1 | (8 marks) |
**(a)** resolve $\rightarrow$: $6 + X\cos45 - 18\sin30 = 0$ | M2 |
$6 + X\frac{\sqrt{2}}{2} - 9 = 0$ so $X = 3\sqrt{2}$ N | M1 A1 |
**(b)** resolve $\uparrow$: $Y + X\cos45 + 18\cos30 - 20 = 0$ | M2 |
$Y + (3\sqrt{2})\frac{\sqrt{2}}{2} + 18\frac{\sqrt{3}}{2} - 20 = 0$ | M1 |
$Y = 20 - 9\sqrt{3} - 3 = 17 - 9\sqrt{3}$ | A1 | (8 marks)3.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{60b9db45-b48e-40a1-bd22-909e11877bc3-2_442_805_1023_719}
\captionsetup{labelformat=empty}
\caption{Fig. 1}
\end{center}
\end{figure}
Figure 1 shows the forces acting on a particle, $P$. These consist of a 20 N force to the South, a 6 N force to the East, an 18 N force $30 ^ { \circ }$ West of North and two unknown forces $X$ and $Y$ which act to the North-East and North respectively.
Given that $P$ is in equilibrium,
\begin{enumerate}[label=(\alph*)]
\item show that $X$ has magnitude $3 \sqrt { } 2 \mathrm {~N}$,
\item find the exact value of $Y$.
\end{enumerate}
\hfill \mbox{\textit{Edexcel M1 Q3 [8]}}