| Exam Board | CAIE |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2010 |
| Session | November |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Moments |
| Type | Rod on smooth peg or cylinder |
| Difficulty | Challenging +1.8 This is a challenging Further Maths mechanics problem requiring geometric insight to establish relationships between angles and distances, followed by systematic application of equilibrium conditions (resolving forces in two directions and taking moments). The geometry is non-trivial (relating θ and 2θ, finding perpendicular distances), and students must coordinate three interrelated results, but the techniques are standard for FM students who have practiced statics problems. |
| Spec | 6.04e Rigid body equilibrium: coplanar forces |
| Answer | Marks |
|---|---|
| 4 | Resolve in any two dirns. for rod, e.g. vertically: RA sin 2θ + RB cos θ = W |
| Answer | Marks |
|---|---|
| Substitute and simplify: 2r cos 2θ = a cos θ A.G. A1 | 2 |
| Answer | Marks |
|---|---|
| 3 | [9] |
Question 4:
4 | Resolve in any two dirns. for rod, e.g. vertically: RA sin 2θ + RB cos θ = W
or horizontally: RA cos 2θ – RB sin θ = 0
or parallel to rod: RA cos θ = W sin θ
or normal to rod: RA sin θ + RB = W cos θ B1 B1
(i) Solve to find R A, e.g.: RA = W sin θ /
(cos 2θ cos θ – sin 2θ sin θ)
= W tan θ A.G. M1 A1
(ii) Solve to find R B, e.g.: RB = W tan θ cos 2θ / sin θ
= W cos 2θ / cos θ A.G. M1 A1
(iii) Take moments for rod, e.g. about A: R B 2r cos θ = W a cos θ
or about B: RA 2r cos θ sin θ =
W (2r cos θ – a) cos θ M1 A1
Substitute and simplify: 2r cos 2θ = a cos θ A.G. A1 | 2
2
2
3 | [9]\includegraphics{figure_4}
A hemispherical bowl of radius $r$ is fixed with its rim horizontal. A thin uniform rod rests in equilibrium on the rim of the bowl with one end resting on the inner surface of the bowl at $A$, as shown in the diagram. The rod has length $2a$ and weight $W$. The point of contact between the rod and the rim is $B$, and the rim has centre $C$. The rod is in a vertical plane containing $C$. The rod is inclined at $\theta$ to the horizontal and the line $AC$ is inclined at $2\theta$ to the horizontal. The contacts at $A$ and $B$ are smooth.
In any order, show that
\begin{enumerate}[label=(\roman*)]
\item the contact force acting on the rod at $A$ has magnitude $W\tan\theta$,
\item the contact force acting on the rod at $B$ has magnitude $\frac{W\cos 2\theta}{\cos\theta}$,
\item $2r\cos 2\theta = a\cos\theta$.
\end{enumerate}
[9]
\hfill \mbox{\textit{CAIE FP2 2010 Q4 [9]}}