Challenging +1.8 This is a challenging Further Maths mechanics problem requiring moment of inertia calculations using parallel axis theorem, followed by energy conservation for rotational motion. It involves multiple components (two discs plus particle), careful geometric setup, and multi-step reasoning, but follows standard techniques for this topic without requiring novel insight.
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\includegraphics[max width=\textwidth, alt={}, center]{3daca234-9b7f-41d4-bbaa-d35615a120fc-2_510_755_667_696}
A uniform circular disc with centre \(A\) has mass \(M\) and radius \(3 a\). A second uniform circular disc with centre \(B\) has mass \(\frac { 1 } { 9 } M\) and radius \(a\). The two discs are rigidly joined together so that they lie in the same plane with their circumferences touching. The line of centres meets the circumference of the larger disc at \(P\) and the circumference of the smaller disc at \(O\). A particle of mass \(\frac { 1 } { 3 } M\) is attached at \(P\) (see diagram). Show that the moment of inertia of the system about an axis through \(O\), perpendicular to the plane of the discs, is \(51 M a ^ { 2 }\).
The system is free to rotate about a fixed horizontal axis through \(O\), perpendicular to the plane of the discs. The system is held with \(O P\) horizontal and is then released from rest. Given that \(a = 0.5 \mathrm {~m}\), find the greatest speed of \(P\) in the subsequent motion, giving your answer correct to 2 significant figures. [0pt]
[5]
Find MI of large disc about \(O\): \(\frac{1}{2}M(3a)^2 + M(5a)^2\ \left[= \frac{59Ma^2}{2}\right]\)
M1 A1
Find MI of small disc about \(O\): \(\frac{1}{2}(M/9)a^2 + (M/9)a^2\ \left[= \frac{Ma^2}{6}\right]\)
M1 A1
Find MI of particle about \(O\): \((M/3)(8a)^2\ \left[= \frac{64Ma^2}{3}\right]\)
B1
Sum to find MI of system about \(O\): \(I = (177+1+128)\frac{Ma^2}{6} = 51Ma^2\)
A1
A.G.
State or imply that speed is max when \(OP\) vertical
M1
Use energy when \(OP\) vertical (or at general point): \(\frac{1}{2}I\omega^2 = \left(5+1+\frac{1}{9}+\frac{1}{3}8\right)Mga = \frac{70Mga}{9}\)
M1 A1
Substitute for \(a\), \(I\) and find max speed \(8a\omega\): \(\omega = \sqrt{6{\cdot}10} = 2{\cdot}47,\ 8a\omega = 9{\cdot}9\ [\text{ms}^{-1}]\)
M1 A1
[Subtotal: 5]
Total: 11
## Question 2:
| Working/Answer | Mark | Guidance |
|---|---|---|
| Find MI of large disc about $O$: $\frac{1}{2}M(3a)^2 + M(5a)^2\ \left[= \frac{59Ma^2}{2}\right]$ | M1 A1 | |
| Find MI of small disc about $O$: $\frac{1}{2}(M/9)a^2 + (M/9)a^2\ \left[= \frac{Ma^2}{6}\right]$ | M1 A1 | |
| Find MI of particle about $O$: $(M/3)(8a)^2\ \left[= \frac{64Ma^2}{3}\right]$ | B1 | |
| Sum to find MI of system about $O$: $I = (177+1+128)\frac{Ma^2}{6} = 51Ma^2$ | A1 | **A.G.** | [Subtotal: 6] |
| State or imply that speed is max when $OP$ vertical | M1 | |
| Use energy when $OP$ vertical (or at general point): $\frac{1}{2}I\omega^2 = \left(5+1+\frac{1}{9}+\frac{1}{3}8\right)Mga = \frac{70Mga}{9}$ | M1 A1 | |
| Substitute for $a$, $I$ and find max speed $8a\omega$: $\omega = \sqrt{6{\cdot}10} = 2{\cdot}47,\ 8a\omega = 9{\cdot}9\ [\text{ms}^{-1}]$ | M1 A1 | [Subtotal: 5] |
**Total: 11**
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\includegraphics[max width=\textwidth, alt={}, center]{3daca234-9b7f-41d4-bbaa-d35615a120fc-2_510_755_667_696}
A uniform circular disc with centre $A$ has mass $M$ and radius $3 a$. A second uniform circular disc with centre $B$ has mass $\frac { 1 } { 9 } M$ and radius $a$. The two discs are rigidly joined together so that they lie in the same plane with their circumferences touching. The line of centres meets the circumference of the larger disc at $P$ and the circumference of the smaller disc at $O$. A particle of mass $\frac { 1 } { 3 } M$ is attached at $P$ (see diagram). Show that the moment of inertia of the system about an axis through $O$, perpendicular to the plane of the discs, is $51 M a ^ { 2 }$.
The system is free to rotate about a fixed horizontal axis through $O$, perpendicular to the plane of the discs. The system is held with $O P$ horizontal and is then released from rest. Given that $a = 0.5 \mathrm {~m}$, find the greatest speed of $P$ in the subsequent motion, giving your answer correct to 2 significant figures.\\[0pt]
[5]
\hfill \mbox{\textit{CAIE FP2 2011 Q2 [11]}}