CAIE FP2 2016 June — Question 5 12 marks

Exam BoardCAIE
ModuleFP2 (Further Pure Mathematics 2)
Year2016
SessionJune
Marks12
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicMoments of inertia
TypeSmall oscillations period
DifficultyChallenging +1.8 This compound pendulum problem requires calculating moment of inertia using parallel axis theorem for three components (two discs and a rod), finding the center of mass, and applying small oscillations theory. While conceptually demanding for A-level, it follows a standard Further Maths template with clear steps: I = Σ(I_cm + md²) for each component, then T = 2π√(I/mgh). The calculation is lengthy but methodical, typical of FP2 mechanics questions.
Spec4.10f Simple harmonic motion: x'' = -omega^2 x4.10g Damped oscillations: model and interpret6.04a Centre of mass: gravitational effect6.04b Find centre of mass: using symmetry6.04c Composite bodies: centre of mass6.04d Integration: for centre of mass of laminas/solids
5 \includegraphics[max width=\textwidth, alt={}, center]{3e224c82-68df-427e-a59b-7dc2bfd716a2-3_727_517_258_813} A thin uniform \(\operatorname { rod } A B\) has mass \(\frac { 3 } { 4 } m\) and length \(3 a\). The end \(A\) of the rod is rigidly attached to a point on the circumference of a uniform disc with centre \(C\), mass \(m\) and radius \(a\). The end \(B\) of the rod is rigidly attached to a point on the circumference of a uniform disc with centre \(D\), mass \(4 m\) and radius \(2 a\). The discs and the rod are in the same plane and \(C A B D\) is a straight line. The mid-point of \(C D\) is \(O\). The object consisting of the two discs and the rod is free to rotate about a fixed smooth horizontal axis \(l\), through \(O\) in the plane of the object and perpendicular to the rod (see diagram). Show that the moment of inertia of the object about \(l\) is \(50 m a ^ { 2 }\). The object hangs in equilibrium with \(D\) vertically below \(C\). It is displaced through a small angle and released from rest, so that it makes small oscillations about the horizontal axis \(l\). Show that it will move in approximate simple harmonic motion and state the period of the motion.
Question 5:
Part 1 (finding \(I\)):
AnswerMarks Guidance
Answer/WorkingMarks Guidance
\(I_{\text{rod}} = \frac{1}{3} \cdot \frac{3}{4}m(3a/2)^2 + \frac{3}{4}m(\frac{1}{2}a)^2 [= \frac{3}{4}ma^2]\)B1 Find MI of rod about \(l\)
\(I_{\text{disc}\,C} = \frac{1}{2} \times \frac{1}{2}ma^2 + m(3a)^2 [= (37/4)\,ma^2]\)M1 A1 Find MI of disc \(C\) about \(l\) using both theorems
\(I_{\text{disc}\,D} = \frac{1}{2} \times \frac{1}{2}m(2a)^2 + 4m(3a)^2 [= 40\,ma^2]\)M1 A1 Find MI of disc \(D\) about \(l\) using both theorems
\(I = (\frac{3}{4} + 37/4 + 40)\,ma^2 = 50\,ma^2\)A1 Sum to find MI of system about \(l\): A.G.; Total: [6]
Part 2 (finding \(T\)):
AnswerMarks Guidance
Answer/WorkingMarks Guidance
\([-]\,I\,\mathrm{d}^2\theta/\mathrm{d}t^2 = 4mg \times 3a\sin\theta\)M1 A2 Use equation of circular motion; \(\theta\) is angle of \(CD\) with vertical; A1 only for two correct terms on RHS; A0 if \(\cos\theta\) used
\(-\frac{3}{4}mg \times \frac{1}{2}a\sin\theta - mg \times 3a\sin\theta\)
\([= (69/8)\,mga\sin\theta]\)
\(\mathrm{d}^2\theta/\mathrm{d}t^2 = -(69g/400a)\,\theta\)M1 A1 Approximate \(\sin\theta\) by \(\theta\) to show SHM; M0 if wrong sign or \(\cos\theta \approx \theta\) used
or \(-(0.1725\,g/a)\,\theta\)
\(T = 2\pi/\sqrt{(69g/400a)} = 40\pi\,\sqrt{(a/69g)}\)A1 Find period \(T\) (AEF); allow \(g = 9.8\) or \(9.81\); Total: [6]
\(= 15.1\sqrt{(a/g)}\) or \(4.78\sqrt{a}\) 
## Question 5:

### Part 1 (finding $I$):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $I_{\text{rod}} = \frac{1}{3} \cdot \frac{3}{4}m(3a/2)^2 + \frac{3}{4}m(\frac{1}{2}a)^2 [= \frac{3}{4}ma^2]$ | B1 | Find MI of rod about $l$ |
| $I_{\text{disc}\,C} = \frac{1}{2} \times \frac{1}{2}ma^2 + m(3a)^2 [= (37/4)\,ma^2]$ | M1 A1 | Find MI of disc $C$ about $l$ using both theorems |
| $I_{\text{disc}\,D} = \frac{1}{2} \times \frac{1}{2}m(2a)^2 + 4m(3a)^2 [= 40\,ma^2]$ | M1 A1 | Find MI of disc $D$ about $l$ using both theorems |
| $I = (\frac{3}{4} + 37/4 + 40)\,ma^2 = 50\,ma^2$ | A1 | Sum to find MI of system about $l$: **A.G.**; Total: [6] |

### Part 2 (finding $T$):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $[-]\,I\,\mathrm{d}^2\theta/\mathrm{d}t^2 = 4mg \times 3a\sin\theta$ | M1 A2 | Use equation of circular motion; $\theta$ is angle of $CD$ with vertical; A1 only for two correct terms on RHS; A0 if $\cos\theta$ used |
| $-\frac{3}{4}mg \times \frac{1}{2}a\sin\theta - mg \times 3a\sin\theta$ | | |
| $[= (69/8)\,mga\sin\theta]$ | | |
| $\mathrm{d}^2\theta/\mathrm{d}t^2 = -(69g/400a)\,\theta$ | M1 A1 | Approximate $\sin\theta$ by $\theta$ to show SHM; M0 if wrong sign or $\cos\theta \approx \theta$ used |
| or $-(0.1725\,g/a)\,\theta$ | | |
| $T = 2\pi/\sqrt{(69g/400a)} = 40\pi\,\sqrt{(a/69g)}$ | A1 | Find period $T$ (AEF); allow $g = 9.8$ or $9.81$; Total: [6] |
| $= 15.1\sqrt{(a/g)}$ or $4.78\sqrt{a}$ | | |

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\includegraphics[max width=\textwidth, alt={}, center]{3e224c82-68df-427e-a59b-7dc2bfd716a2-3_727_517_258_813}

A thin uniform $\operatorname { rod } A B$ has mass $\frac { 3 } { 4 } m$ and length $3 a$. The end $A$ of the rod is rigidly attached to a point on the circumference of a uniform disc with centre $C$, mass $m$ and radius $a$. The end $B$ of the rod is rigidly attached to a point on the circumference of a uniform disc with centre $D$, mass $4 m$ and radius $2 a$. The discs and the rod are in the same plane and $C A B D$ is a straight line. The mid-point of $C D$ is $O$. The object consisting of the two discs and the rod is free to rotate about a fixed smooth horizontal axis $l$, through $O$ in the plane of the object and perpendicular to the rod (see diagram). Show that the moment of inertia of the object about $l$ is $50 m a ^ { 2 }$.

The object hangs in equilibrium with $D$ vertically below $C$. It is displaced through a small angle and released from rest, so that it makes small oscillations about the horizontal axis $l$. Show that it will move in approximate simple harmonic motion and state the period of the motion.

\hfill \mbox{\textit{CAIE FP2 2016 Q5 [12]}}
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