| Exam Board | CAIE |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2019 |
| Session | June |
| 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 calculation of moment of inertia for a composite system using parallel axis theorem, then applying SHM theory to find period and constraints. It demands careful geometric reasoning, multiple applications of standard formulas, and algebraic manipulation across 12 marks, but follows established techniques without requiring novel insights. |
| Spec | 4.10f Simple harmonic motion: x'' = -omega^2 x6.04d Integration: for centre of mass of laminas/solids |
| Answer | Marks | Guidance |
|---|---|---|
| 5(i) | I = ⅓ kMa2 | |
| rod | B1 | Find or state MI of rod AB about axis L |
| Answer | Marks | Guidance |
|---|---|---|
| sphere | M1 A1 | M1 for one term correct, A1 for both terms correct |
| Answer | Marks | Guidance |
|---|---|---|
| ring | M1 A1 | M1 for one term correct, A1 for both terms correct |
| I = (k/3 + 35k/3 + 9/2) Ma2 = (3/2) (8k + 3) Ma2 AG | A1 | MI of object about axis L |
| Answer | Marks | Guidance |
|---|---|---|
| 5(ii) | [–] I d2θ/dt2 = – kMg × 3a sin θ + Mg × 2a sin θ | |
| [ = – (3k – 2) Mga sin θ ] | M1 A1 | Use equation of circular motion to find d2θ/dt2 where θ is angle |
| Answer | Marks | Guidance |
|---|---|---|
| d2θ/dt2 = – {2g (3k – 2) / 3a(8k + 3)} θ (AEF) | M1 | Approximate sin θ by θ to give standard form of SHM equation |
| Answer | Marks | Guidance |
|---|---|---|
| SHM if 3k – 2 > 0, k > 2/3 | M1 A1 | Find possible values of k |
| Answer | Marks | Guidance |
|---|---|---|
| or π √{6a(8k + 3) / g (3k – 2)} | A1 | Find period T |
| Answer | Marks | Guidance |
|---|---|---|
| Question | Answer | Marks |
Question 5:
--- 5(i) ---
5(i) | I = ⅓ kMa2
rod | B1 | Find or state MI of rod AB about axis L
I = ⅔ kM (2a)2 + kM (3a)2 [= (35k/3) Ma2]
sphere | M1 A1 | M1 for one term correct, A1 for both terms correct
I = ½ × Ma2 + M (2a)2 [= (9/2) Ma2]
ring | M1 A1 | M1 for one term correct, A1 for both terms correct
I = (k/3 + 35k/3 + 9/2) Ma2 = (3/2) (8k + 3) Ma2 AG | A1 | MI of object about axis L
6
--- 5(ii) ---
5(ii) | [–] I d2θ/dt2 = – kMg × 3a sin θ + Mg × 2a sin θ
[ = – (3k – 2) Mga sin θ ] | M1 A1 | Use equation of circular motion to find d2θ/dt2 where θ is angle
of rod with vertical
d2θ/dt2 = – {2g (3k – 2) / 3a(8k + 3)} θ (AEF) | M1 | Approximate sin θ by θ to give standard form of SHM equation
(M0 if wrong sign or cos θ ≈ θ used)
SHM if 3k – 2 > 0, k > 2/3 | M1 A1 | Find possible values of k
T = 2π √{3a(8k + 3) / 2g (3k – 2)}
or π √{6a(8k + 3) / g (3k – 2)} | A1 | Find period T
6
Question | Answer | Marks | Guidance\includegraphics{figure_5}
A thin uniform rod $AB$ has mass $kM$ and length $2a$. The end $A$ of the rod is rigidly attached to the surface of a uniform hollow sphere with centre $O$, mass $kM$ and radius $2a$. 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 $\frac{5}{2}(8k + 3)Ma^2$. [6]
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 $k$. [6]
\hfill \mbox{\textit{CAIE FP2 2019 Q5 [12]}}