| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9794/3 (Pre-U Mathematics Paper 3) |
| Year | 2017 |
| Session | June |
| Marks | 6 |
| Topic | Forces, equilibrium and resultants |
| Type | Triangle of forces method |
| Difficulty | Standard +0.3 This is a standard triangle of forces equilibrium problem requiring application of the sine rule to find two configurations. While it involves some geometric reasoning about the two possible triangle configurations and careful angle work, it's a routine mechanics question testing a well-defined method with no novel insight required. Slightly easier than average due to being a textbook application of a standard technique. |
| Spec | 3.03n Equilibrium in 2D: particle under forces |
**Question 8(i)**
- A fully labelled triangle of forces, including angles ($\theta$ and $30°$) and arrows. **B1** (Triangle is ambiguous. Candidates not expected to consider/show this here.)
**Question 8(ii)**
- $\dfrac{10}{\sin 30} = \dfrac{16}{\sin \phi} \left(= \dfrac{P}{\sin \theta}\right)$ **M1** (Sine rule or Lami's Theorem used. $\phi$ is the third angle $(= 180 - 30 - \theta)$.)
- $\therefore \sin \phi = \dfrac{16 \sin 30}{10} = \dfrac{4}{5}$
- $\therefore \phi = 53.1°$ or $126.9°$ **A1** (Either value correct.)
- $\therefore \theta = 150 - \phi = 96.9$ or $23.1$ **A1** (Both correct values required.)
- $\therefore P = \dfrac{10 \sin\theta}{\sin 30} (= 20\sin\theta)$ **M1** (Sine rule or Lami's Theorem used or resolve horizontally.)
- $\therefore P = 19.856\ldots$ or $7.856\ldots$ **A1** (cao. Both values required.)
**Alternative methods** (involving resolving and/or the cosine rule):
- Correct elimination of either $P$ (or $\theta$). **M1**
- Either value of (e.g. $\theta + 30$ or $\theta - 60$ or $\cos\theta$) (or $P$) correct. **A1**
- Both correct values of $\theta$ (or $P$). **A1**
- Use of $P = 20\sin\theta$ as above. **M1**
- Both values of $P$ (or $\theta$). **A1** (cao. NB Beware of spurious values of $\theta$.)
**Total: 6 marks**8 An object of weight 16 N is supported in equilibrium by a force of $P \mathrm {~N}$ at $30 ^ { \circ }$ to the vertical and by another of 10 N at $\theta ^ { \circ }$ to the vertical as shown in the diagram.\\
\includegraphics[max width=\textwidth, alt={}, center]{85c5c346-8eb5-47ea-b94e-80b1a0038ce1-4_549_483_397_831}\\
(i) Draw a triangle to show that the forces acting on the object are in equilibrium.\\
(ii) Find the two possible values of $\theta$ and the corresponding values of $P$.
\hfill \mbox{\textit{Pre-U Pre-U 9794/3 2017 Q8 [6]}}