| Exam Board | CAIE |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2019 |
| Session | June |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Moments of inertia |
| Type | Small oscillations period |
| Difficulty | Challenging +1.8 This compound pendulum problem requires calculating moment of inertia using parallel axis theorem for three connected bodies (non-trivial geometry), then applying SHM theory to find oscillation conditions and period. The multi-body setup, algebraic manipulation with parameter k, and need to determine stability conditions elevate this beyond standard SHM questions, though the techniques themselves are established Further Maths methods. |
| Spec | 4.10f Simple harmonic motion: x'' = -omega^2 x6.04e Rigid body equilibrium: coplanar forces |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(I_{rod} = \frac{1}{3}kMa^2\) | B1 | Find or state MI of rod \(AB\) about axis \(L\) |
| \(I_{sphere} = \frac{2}{3}kM(2a)^2 + kM(3a)^2\) \([= \frac{35k}{3}Ma^2]\) | M1 A1 | M1 for one term correct, A1 for both terms correct |
| \(I_{ring} = \frac{1}{2} \times Ma^2 + M(2a)^2\) \([= \frac{9}{2}Ma^2]\) | M1 A1 | M1 for one term correct, A1 for both terms correct |
| \(I = (\frac{k}{3} + \frac{35k}{3} + \frac{9}{2})Ma^2 = \frac{3}{2}(8k+3)Ma^2\) AG | A1 | MI of object about axis \(L\) |
| 6 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \([-]I\ddot{\theta} = -kMg \times 3a\sin\theta + Mg \times 2a\sin\theta\) \([= -(3k-2)Mga\sin\theta]\) | M1 A1 | Use equation of circular motion to find \(\ddot{\theta}\) where \(\theta\) is angle of rod with vertical |
| \(\ddot{\theta} = -\frac{2g(3k-2)}{3a(8k+3)}\theta\) | M1 | Approximate \(\sin\theta\) by \(\theta\) to give standard form of SHM equation (M0 if wrong sign or \(\cos\theta \approx \theta\) used) |
| SHM if \(3k - 2 > 0\), \(k > \frac{2}{3}\) | M1 A1 | Find possible values of \(k\) |
| \(T = 2\pi\sqrt{\frac{3a(8k+3)}{2g(3k-2)}}\) or \(\pi\sqrt{\frac{6a(8k+3)}{g(3k-2)}}\) | A1 | Find period \(T\) |
| 6 |
## Question 5(i):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $I_{rod} = \frac{1}{3}kMa^2$ | B1 | Find or state MI of rod $AB$ about axis $L$ |
| $I_{sphere} = \frac{2}{3}kM(2a)^2 + kM(3a)^2$ $[= \frac{35k}{3}Ma^2]$ | M1 A1 | M1 for one term correct, A1 for both terms correct |
| $I_{ring} = \frac{1}{2} \times Ma^2 + M(2a)^2$ $[= \frac{9}{2}Ma^2]$ | M1 A1 | M1 for one term correct, A1 for both terms correct |
| $I = (\frac{k}{3} + \frac{35k}{3} + \frac{9}{2})Ma^2 = \frac{3}{2}(8k+3)Ma^2$ AG | A1 | MI of object about axis $L$ |
| | **6** | |
## Question 5(ii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $[-]I\ddot{\theta} = -kMg \times 3a\sin\theta + Mg \times 2a\sin\theta$ $[= -(3k-2)Mga\sin\theta]$ | M1 A1 | Use equation of circular motion to find $\ddot{\theta}$ where $\theta$ is angle of rod with vertical |
| $\ddot{\theta} = -\frac{2g(3k-2)}{3a(8k+3)}\theta$ | M1 | Approximate $\sin\theta$ by $\theta$ to give standard form of SHM equation (M0 if wrong sign or $\cos\theta \approx \theta$ used) |
| SHM if $3k - 2 > 0$, $k > \frac{2}{3}$ | M1 A1 | Find possible values of $k$ |
| $T = 2\pi\sqrt{\frac{3a(8k+3)}{2g(3k-2)}}$ or $\pi\sqrt{\frac{6a(8k+3)}{g(3k-2)}}$ | A1 | Find period $T$ |
| | **6** | |
---5\\
\includegraphics[max width=\textwidth, alt={}, center]{2aaf3493-6509-4668-91a2-9f4708bbbb58-10_809_778_258_680}
A thin uniform $\operatorname { rod } A B$ has mass $k M$ and length $2 a$. The end $A$ of the rod is rigidly attached to the surface of a uniform hollow sphere with centre $O$, mass $k M$ and radius $2 a$. The end $B$ of the rod is rigidly attached to the circumference of a uniform ring with centre $C$, mass $M$ and radius $a$. The points $C , B , A , O$ lie in a straight line. The horizontal axis $L$ passes through the mid-point of the rod and is perpendicular to the rod and in the plane of the ring (see diagram). The object consisting of the rod, the ring and the hollow sphere can rotate freely about $L$.\\
(i) Show that the moment of inertia of the object about $L$ is $\frac { 3 } { 2 } ( 8 k + 3 ) M a ^ { 2 }$.\\
The object performs small oscillations about $L$, with the ring above the sphere as shown in the diagram.\\
(ii) Find the set of possible values of $k$ and the period of these oscillations in terms of $k$.\\
\hfill \mbox{\textit{CAIE FP2 2019 Q5 [12]}}