| Exam Board | CAIE |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2018 |
| Session | June |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Moments |
| Type | Rod against wall and ground |
| Difficulty | Standard +0.8 This is a multi-part statics problem requiring moments, equilibrium equations, and friction. Part (i) is standard moments about a point. Part (ii) requires setting up equations with the doubled reaction force and algebraic manipulation. Part (iii) involves applying the friction condition F = μR at limiting equilibrium. While systematic, it requires careful setup of multiple equations and algebraic manipulation across connected scenarios, making it moderately challenging but still within standard Further Maths mechanics scope. |
| Spec | 6.04e Rigid body equilibrium: coplanar forces |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(R_B \cdot 2a\sin\theta = Wa\cos\theta + 5Wxa\cos\theta\) | M1 A1 | Moments at \(A\) |
| \(R_B = W\dfrac{(5x+1)}{2\tan\theta}\) AG | A1 | Simplify to give \(R_B\) |
| 3 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(R_B' = W\dfrac{(5x+5+1)}{2\tan\theta}\) (AEF) | B1 | Find new \(R_B'\) by moments at \(A\) |
| \(5x + 6 = 2(5x+1)\), \(x = \frac{4}{5}\) AG | M1 A1 | Find/verify \(x\) from \(R_B' = 2R_B\) |
| 3 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(R_A = 6W\), \(F_A' = R_B'\) *or* \(2R_B \left[= \frac{10W}{2\tan\theta} = 4W\right]\) | B1, B1 | Find \(R_A\) and new \(F_A'\) by resolutions (may be earned earlier) |
| \(\dfrac{2}{3} = \dfrac{F_A'}{R_A}\) | M1 | Relate \(R_A\) and \(F_A'\) using \(\mu = \frac{F}{R}\) |
| \(= \dfrac{\left(\frac{5W}{\tan\theta}\right)}{6W}\) so \(\tan\theta = \frac{5}{4}\) *or* \(1.25\) | M1 A1 | Find \(\tan\theta\); SC Allow M1 M1 for ground smooth and wall rough (max \(\frac{2}{5}\)) |
| 5 |
## Question 4(i):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $R_B \cdot 2a\sin\theta = Wa\cos\theta + 5Wxa\cos\theta$ | M1 A1 | Moments at $A$ |
| $R_B = W\dfrac{(5x+1)}{2\tan\theta}$ **AG** | A1 | Simplify to give $R_B$ |
| | **3** | |
---
## Question 4(ii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $R_B' = W\dfrac{(5x+5+1)}{2\tan\theta}$ (AEF) | B1 | Find new $R_B'$ by moments at $A$ |
| $5x + 6 = 2(5x+1)$, $x = \frac{4}{5}$ **AG** | M1 A1 | Find/verify $x$ from $R_B' = 2R_B$ |
| | **3** | |
---
## Question 4(iii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $R_A = 6W$, $F_A' = R_B'$ *or* $2R_B \left[= \frac{10W}{2\tan\theta} = 4W\right]$ | B1, B1 | Find $R_A$ and new $F_A'$ by resolutions (may be earned earlier) |
| $\dfrac{2}{3} = \dfrac{F_A'}{R_A}$ | M1 | Relate $R_A$ and $F_A'$ using $\mu = \frac{F}{R}$ |
| $= \dfrac{\left(\frac{5W}{\tan\theta}\right)}{6W}$ so $\tan\theta = \frac{5}{4}$ *or* $1.25$ | M1 A1 | Find $\tan\theta$; **SC** Allow M1 M1 for ground smooth and wall rough (max $\frac{2}{5}$) |
| | **5** | |
---4 A uniform $\operatorname { rod } A B$ has length $2 a$ and weight $W$. The end $A$ rests on rough horizontal ground and the end $B$ rests against a smooth vertical wall. The rod is in a vertical plane that is perpendicular to the wall. The angle between the rod and the horizontal is $\theta$. A particle of weight $5 W$ hangs from the rod at the point $C$, with $A C = x a$, where $0 < x < 1$.\\
(i) By taking moments about $A$, show that the magnitude of the normal reaction at $B$ is $\frac { W ( 5 x + 1 ) } { 2 \tan \theta }$.\\[0pt]
[3]\\
The particle of weight $5 W$ is now moved a distance $a$ up the rod, so that $A C = ( x + 1 ) a$. This results in the magnitude of the normal reaction at $B$ being double its previous value. The system remains in equilibrium with the rod at angle $\theta$ with the horizontal.\\
(ii) Show that $x = \frac { 4 } { 5 }$.\\
The coefficient of friction between the rod and the ground is $\frac { 2 } { 3 }$.\\
(iii) Given that the rod is about to slip when the particle of weight $5 W$ is in its second position, find the value of $\tan \theta$.\\
\begin{figure}[h]
\begin{center}
\captionsetup{labelformat=empty}
\caption{Axis $l$}
\includegraphics[alt={},max width=\textwidth]{0eb3892f-628f-449a-b022-b38170754d89-08_462_693_301_731}
\end{center}
\end{figure}
Three thin uniform rings $A , B$ and $C$ are joined together, so that each ring is in contact with each of the other two rings. Ring $A$ has radius $2 a$ and mass $3 M$; rings $B$ and $C$ each have radius $3 a$ and mass $2 M$. The rings lie in the same plane and the centres of the rings are at the vertices of an isosceles triangle. The object consisting of the three rings is free to rotate about the horizontal axis $l$ which is tangential to ring $A$, in the plane of the object and perpendicular to the line of symmetry of the object (see diagram).\\
\hfill \mbox{\textit{CAIE FP2 2018 Q4 [11]}}