| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9794/3 (Pre-U Mathematics Paper 3) |
| Year | 2014 |
| Session | June |
| Marks | 6 |
| Topic | Forces, equilibrium and resultants |
| Type | Forces in vector form: equilibrium (find unknowns) |
| Difficulty | Moderate -0.8 This is a straightforward equilibrium problem requiring only vector addition and basic trigonometry. Part (i) involves summing three given forces and setting equal to -F₄ (2-3 steps), while part (ii) requires calculating magnitude using Pythagoras and direction using arctan—both standard A-level mechanics procedures with no problem-solving insight needed. |
| Spec | 1.10c Magnitude and direction: of vectors1.10d Vector operations: addition and scalar multiplication3.03b Newton's first law: equilibrium |
| Answer | Marks | Guidance |
|---|---|---|
| (i) \(F_1 + F_2 + F_3 + F_4 = 0\) | M1 | Sum of 4 forces set equal to 0 o.e. |
| \(\therefore (5\mathbf{i} - 8\mathbf{j}) + (-3\mathbf{i} - 4\mathbf{j}) + (6\mathbf{i} + 6\mathbf{j}) + F_4 = 0\) | ||
| \(\therefore F_4 = (-8\mathbf{i} + 6\mathbf{j})\) | A1 | c.a.o. [2] |
| (ii) \( | F_4 | = \sqrt{(-8)^2 + 6^2} = 10\) N |
| \(\theta = \text{inv tan}\left(\frac{6}{-8}\right) = 143.(13...)°\) | M1 A1 | Correct use of inverse tan (or cos or sin). ft (i), but not c's magnitude. Must have a clear reference direction. Allow sketch as evidence if convincing. [4] |
**(i)** $F_1 + F_2 + F_3 + F_4 = 0$ | M1 | Sum of 4 forces set equal to 0 o.e.
$\therefore (5\mathbf{i} - 8\mathbf{j}) + (-3\mathbf{i} - 4\mathbf{j}) + (6\mathbf{i} + 6\mathbf{j}) + F_4 = 0$ |
$\therefore F_4 = (-8\mathbf{i} + 6\mathbf{j})$ | A1 | c.a.o. [2]
**(ii)** $|F_4| = \sqrt{(-8)^2 + 6^2} = 10$ N | M1 A1 | Use of Pythagoras. ft (i).
$\theta = \text{inv tan}\left(\frac{6}{-8}\right) = 143.(13...)°$ | M1 A1 | Correct use of inverse tan (or cos or sin). ft (i), but not c's magnitude. Must have a clear reference direction. Allow sketch as evidence if convincing. [4]A particle is being held in equilibrium by the following set of forces (in newtons).
$$\mathbf{F}_1 = 5\mathbf{i} - 8\mathbf{j}, \quad \mathbf{F}_2 = -3\mathbf{i} - 4\mathbf{j}, \quad \mathbf{F}_3 = 6\mathbf{i} + 6\mathbf{j} \quad \text{and} \quad \mathbf{F}_4.$$
\begin{enumerate}[label=(\roman*)]
\item Find $\mathbf{F}_4$ in terms of $\mathbf{i}$ and $\mathbf{j}$. [2]
\item Hence find the magnitude and direction of $\mathbf{F}_4$. [4]
\end{enumerate}
\hfill \mbox{\textit{Pre-U Pre-U 9794/3 2014 Q8 [6]}}