| Exam Board | CAIE |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2019 |
| Session | November |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Simple Harmonic Motion |
| Type | Small oscillations: rigid body compound pendulum |
| Difficulty | Challenging +1.8 This is a challenging Further Maths mechanics problem requiring parallel axis theorem applications to three different objects (hollow sphere, solid sphere, and rod), careful distance calculations from a non-standard axis, and then using SHM period formula to find an unknown parameter. The multi-component system and algebraic manipulation with λ elevate this above standard A-level, though the techniques themselves are systematic once identified. |
| Spec | 4.10f Simple harmonic motion: x'' = -omega^2 x6.04d Integration: for centre of mass of laminas/solids6.05f Vertical circle: motion including free fall |
| Answer | Marks | Guidance |
|---|---|---|
| 5(i) | I = 1 λMa2 + λM(a/2)2 [= (7/12) λMa2] | |
| rod 3 | B1 | Find or state MI of rod AB about axis L |
| Answer | Marks | Guidance |
|---|---|---|
| O 3 | M1 A1 | Find MI of hollow sphere centre O about axis L |
| Answer | Marks | Guidance |
|---|---|---|
| C | M1 A1 | Find MI of solid sphere centre C about axis L |
| Answer | Marks | Guidance |
|---|---|---|
| I = ((7λ + 408) / 12) Ma2 | A1 | Verify MI of object about axis L. AG |
| Answer | Marks | Guidance |
|---|---|---|
| 5(ii) | [–] I d2θ/dt2 = [– 3Mg × (5a/2) sin θ + 5Mg × (3a/2) sin θ] | |
| – λMg × (a/2) sin θ | M1 A1 | Use eqn of circular motion to find d2θ/dt2 where θ is angle |
| Answer | Marks | Guidance |
|---|---|---|
| d2θ/dt2 = – {6gλ / (7λ + 408)a} θ | M1* | Approximate sin θ by θ to give standard form of SHM eqn |
| T = 2π √{(7λ + 408)a / 6gλ} = 5π√(2a/g) | DM1A1 | Find possible values of λ by equating period T to 5π√(2a/g).AEF |
| λ = 6 | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Question | Answer | Marks |
Question 5:
--- 5(i) ---
5(i) | I = 1 λMa2 + λM(a/2)2 [= (7/12) λMa2]
rod 3 | B1 | Find or state MI of rod AB about axis L
I = 2 3M a2 + 3M (5a/2)2 [= (83/4) Ma2]
O 3 | M1 A1 | Find MI of hollow sphere centre O about axis L
I = (2/5) 5Ma2 + 5M (3a/2)2 [= (53/4) Ma2]
C | M1 A1 | Find MI of solid sphere centre C about axis L
I = (7λ/12 + 83/4 + 53/4) Ma2
I = ((7λ + 408) / 12) Ma2 | A1 | Verify MI of object about axis L. AG
6
--- 5(ii) ---
5(ii) | [–] I d2θ/dt2 = [– 3Mg × (5a/2) sin θ + 5Mg × (3a/2) sin θ]
– λMg × (a/2) sin θ | M1 A1 | Use eqn of circular motion to find d2θ/dt2 where θ is angle
of rod with vertical. AEF
d2θ/dt2 = – {6gλ / (7λ + 408)a} θ | M1* | Approximate sin θ by θ to give standard form of SHM eqn
T = 2π √{(7λ + 408)a / 6gλ} = 5π√(2a/g) | DM1A1 | Find possible values of λ by equating period T to 5π√(2a/g).AEF
λ = 6 | A1
6
Question | Answer | Marks | Guidance\includegraphics{figure_5}
A thin uniform rod $AB$ has mass $\lambda M$ and length $2a$. The end $A$ of the rod is rigidly attached to the surface of a uniform hollow sphere (spherical shell) with centre $O$, mass $3M$ and radius $a$. The end $B$ of the rod is rigidly attached to the surface of a uniform solid sphere with centre $C$, mass $5M$ and radius $a$. The rod lies along the line joining the centres of the spheres, so that $CBAO$ is a straight line. The horizontal axis $L$ is perpendicular to the rod and passes through the point of the rod that is a distance $\frac{1}{2}a$ from $B$ (see diagram). The object consisting of the rod and the two spheres can rotate freely about $L$.
\begin{enumerate}[label=(\roman*)]
\item Show that the moment of inertia of the object about $L$ is $\left(\frac{408 + 7\lambda}{12}\right)Ma^2$.
[6]
\end{enumerate}
The period of small oscillations of the object about $L$ is $5\pi\sqrt{\left(\frac{2a}{g}\right)}$.
\begin{enumerate}[label=(\roman*)]
\setcounter{enumi}{1}
\item Find the value of $\lambda$.
[6]
\end{enumerate}
\hfill \mbox{\textit{CAIE FP2 2019 Q5 [12]}}