| Exam Board | CAIE |
|---|---|
| Module | Further Paper 3 (Further Paper 3) |
| Year | 2022 |
| Session | November |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Moments |
| Type | Toppling and sliding of solids |
| Difficulty | Challenging +1.2 This is a Further Maths mechanics question on moments involving toppling/sliding conditions. Part (a) requires taking moments about a point to derive a given expression (standard 'show that' with algebraic manipulation), while part (b) likely involves applying limiting equilibrium conditions (μN = F). The multi-step nature and Further Maths context place it moderately above average, but the techniques are standard for this level. |
| Spec | 6.04b Find centre of mass: using symmetry6.04c Composite bodies: centre of mass |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Let \(F\) and \(R\) be friction and normal reaction at \(A\). Take moments about \(A\): \(N \times 3a = W \times 2a\cos\theta + kW \times 4a\cos\theta\) | M1 | Correct terms, allow sign errors and cos/sin mix |
| \(3N = (2+4k)W \times \frac{4}{5}\) | A1 | At least one intermediate line of working |
| \(N = \frac{8}{15}W(1+2k)\) | AG |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\uparrow N\cos\theta + R = W + kW\) | B1 | Resolve (to include \(R\)) for rod |
| \(\rightarrow F = N\sin\theta\) and \(F = \frac{6}{7}R\) | B1 | Both |
| \(R = \frac{28}{75}W(1+2k)\) or \(R = \frac{21}{45}W(1+k)\) | M1 | Find \(R\) or \(N\) |
| Eliminate to find \(k\) | M1 | Complete method |
| \(k = \frac{1}{3}\) | A1 |
## Question 3(a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Let $F$ and $R$ be friction and normal reaction at $A$. Take moments about $A$: $N \times 3a = W \times 2a\cos\theta + kW \times 4a\cos\theta$ | M1 | Correct terms, allow sign errors and cos/sin mix |
| $3N = (2+4k)W \times \frac{4}{5}$ | A1 | At least one intermediate line of working |
| $N = \frac{8}{15}W(1+2k)$ | | AG |
## Question 3(b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\uparrow N\cos\theta + R = W + kW$ | B1 | Resolve (to include $R$) for rod |
| $\rightarrow F = N\sin\theta$ and $F = \frac{6}{7}R$ | B1 | Both |
| $R = \frac{28}{75}W(1+2k)$ or $R = \frac{21}{45}W(1+k)$ | M1 | Find $R$ or $N$ |
| Eliminate to find $k$ | M1 | Complete method |
| $k = \frac{1}{3}$ | A1 | |\begin{enumerate}[label=(\alph*)]
\item Show that $\mathrm { N } = \frac { 8 } { 15 } \mathrm {~W} ( 1 + 2 \mathrm { k } )$.
\item Find the value of $k$.
\end{enumerate}
\hfill \mbox{\textit{CAIE Further Paper 3 2022 Q3 [7]}}