CAIE FP2 2015 June — Question 5 11 marks

Exam BoardCAIE
ModuleFP2 (Further Pure Mathematics 2)
Year2015
SessionJune
Marks11
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicMoments of inertia
TypeFind unknown parameter from period
DifficultyChallenging +1.8 This compound pendulum problem requires calculating moment of inertia using parallel axis theorem for multiple bodies (rod, disc, particle), then applying SHM period formula and solving a quadratic. It demands careful bookkeeping of distances and masses, plus understanding of small oscillations theory. More challenging than standard SHM but follows established methods for Further Maths mechanics.
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]{833c338f-53c1-436e-a772-0cdaf17fa72d-3_316_949_1320_598} The end \(B\) of a uniform rod \(A B\), of mass \(3 M\) and length \(4 a\), is rigidly attached to a point on the circumference of a uniform disc. The disc has centre \(O\), mass \(2 M\) and radius \(a\), and \(A B O\) is a straight line. The disc and the rod are in the same vertical plane. A particle \(P\), of mass \(M\), is attached to the rod at a distance \(k a\) from \(A\), where \(k\) is a positive constant (see diagram). Show that the moment of inertia of this system, about a fixed horizontal axis \(l\) through \(A\) perpendicular to the plane of the disc, is \(\left( 67 + k ^ { 2 } \right) M a ^ { 2 }\). The system is free to rotate about \(l\) and performs small oscillations of period \(4 \pi \sqrt { } \left( \frac { a } { g } \right)\). Find the possible values of \(k\).
Question 5:
AnswerMarks Guidance
Answer/WorkingMarks Guidance
\(I_{\text{rod}} = (1 + \frac{1}{3})3M(2a)^2\) or \(\frac{1}{3}3M(4a)^2\) [= \(16Ma^2\)]B1 State or find MI of rod about \(A\)
\(I_{\text{disc}} = \frac{1}{2}2Ma^2 + 2M(5a)^2\) [= \(51\ Ma^2\)]M1 A1 Find MI of disc about \(A\)
\(I_{\text{particle}} = M(ka)^2\)B1 State MI of particle about \(A\)
\(I = (16 + 51 + k^2)\ Ma^2 = (67 + k^2)\ Ma^2\)A1 AG Sum to find MI of system
\([-]\ Id^2\theta/dt^2 = 3Mg \times 2a\sin\theta + 2Mg \times 5a\sin\theta + Mg \times ka\sin\theta\) Use equation of circular motion
\(= (16 + k)\ Mga\sin\theta\)M1 A1 (A0 if \(\cos\theta\) used)
\(\omega^2 = (16+k)g/(67+k^2)\ a\)M1 A1 Approximate \(\sin\theta\) by \(\theta\), find \(\omega^2\) in SHM eqn.
\(\omega^2 = \frac{1}{4}g/a\), \(16 + k = \frac{1}{4}(67 + k^2)\) Equate \(2\pi/\omega\) to \(4\pi\sqrt{a/g}\)
\(k^2 - 4k + 3 = 0\), \(k = 1, 3\)M1 A1
Part marks: 5, 6Total: 11 
## Question 5:

| Answer/Working | Marks | Guidance |
|---|---|---|
| $I_{\text{rod}} = (1 + \frac{1}{3})3M(2a)^2$ or $\frac{1}{3}3M(4a)^2$ [= $16Ma^2$] | B1 | State or find MI of rod about $A$ |
| $I_{\text{disc}} = \frac{1}{2}2Ma^2 + 2M(5a)^2$ [= $51\ Ma^2$] | M1 A1 | Find MI of disc about $A$ |
| $I_{\text{particle}} = M(ka)^2$ | B1 | State MI of particle about $A$ |
| $I = (16 + 51 + k^2)\ Ma^2 = (67 + k^2)\ Ma^2$ | A1 | **AG** Sum to find MI of system |
| $[-]\ Id^2\theta/dt^2 = 3Mg \times 2a\sin\theta + 2Mg \times 5a\sin\theta + Mg \times ka\sin\theta$ | | Use equation of circular motion |
| $= (16 + k)\ Mga\sin\theta$ | M1 A1 | (A0 if $\cos\theta$ used) |
| $\omega^2 = (16+k)g/(67+k^2)\ a$ | M1 A1 | Approximate $\sin\theta$ by $\theta$, find $\omega^2$ in SHM eqn. |
| $\omega^2 = \frac{1}{4}g/a$, $16 + k = \frac{1}{4}(67 + k^2)$ | | Equate $2\pi/\omega$ to $4\pi\sqrt{a/g}$ |
| $k^2 - 4k + 3 = 0$, $k = 1, 3$ | M1 A1 | |

**Part marks: 5, 6 | Total: 11**

---
5\\
\includegraphics[max width=\textwidth, alt={}, center]{833c338f-53c1-436e-a772-0cdaf17fa72d-3_316_949_1320_598}

The end $B$ of a uniform rod $A B$, of mass $3 M$ and length $4 a$, is rigidly attached to a point on the circumference of a uniform disc. The disc has centre $O$, mass $2 M$ and radius $a$, and $A B O$ is a straight line. The disc and the rod are in the same vertical plane. A particle $P$, of mass $M$, is attached to the rod at a distance $k a$ from $A$, where $k$ is a positive constant (see diagram). Show that the moment of inertia of this system, about a fixed horizontal axis $l$ through $A$ perpendicular to the plane of the disc, is $\left( 67 + k ^ { 2 } \right) M a ^ { 2 }$.

The system is free to rotate about $l$ and performs small oscillations of period $4 \pi \sqrt { } \left( \frac { a } { g } \right)$. Find the possible values of $k$.

\hfill \mbox{\textit{CAIE FP2 2015 Q5 [11]}}
This paper (11 questions)
View full paper
Q1 6 Q2 7 Q3 9 Q4 10 Q5 11 Q6 8 Q7 11 Q8 12 Q9 12 Q10 EITHER Q10 OR