CAIE FP2 2014 June — Question 5

Exam BoardCAIE
ModuleFP2 (Further Pure Mathematics 2)
Year2014
SessionJune
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicMoments
TypeResultant force on lamina
DifficultyModerate -0.5 This is a standard moments/equilibrium problem involving a composite lamina. Students need to find centers of mass of rectangular and circular components, then apply the principle of moments about a pivot point. While it requires careful calculation and understanding of composite bodies, it follows a well-established procedure taught in Further Maths mechanics with no novel problem-solving required.
Spec3.03a Force: vector nature and diagrams3.04b Equilibrium: zero resultant moment and force
5 \includegraphics[max width=\textwidth, alt={}, center]{f8961f84-c080-4407-a178-45b76f200111-3_533_698_1343_721} A uniform rectangular lamina \(A B C D\), in which \(A B = 8 a\) and \(B C = 6 a\), has mass \(M\). A uniform circular lamina of radius \(\frac { 5 } { 2 } a\) has mass \(\frac { 1 } { 3 } M\). The two laminas are fixed together in the same plane with their centres coinciding at the point \(O\) (see diagram). A particle \(P\) of mass \(\frac { 1 } { 2 } M\) is attached at \(C\). The system is free to rotate about a fixed smooth horizontal axis through \(A\) and perpendicular to the plane \(A B C D\). Show that the moment of inertia of the system about this axis is \(\frac { 2225 } { 24 } M a ^ { 2 }\). The system is released from rest with \(A C\) horizontal and \(D\) below \(A C\). Find, in the form \(k \sqrt { } \left( \frac { g } { a } \right)\), the greatest angular speed in the subsequent motion, giving the value of \(k\) correct to 3 decimal places.
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Question 5:
AnswerMarks Guidance
Answer/WorkingMarks Guidance
\(I_{\square,O} = \frac{1}{3}M\{(4a)^2 + (3a)^2\} = \frac{25Ma^2}{3}\)B1 MI of rectangular lamina about \(O\)
\(I_{O,O} = \frac{1}{2} \cdot \frac{1}{3}M\left(\frac{5a}{2}\right)^2 = \frac{25Ma^2}{24}\)B1 MI of circular lamina about \(O\)
\(I_O = I_{\square,A} + I_{O,A} = \frac{225Ma^2}{24}\)M1 A1 Find MI of combined laminas about \(O\)
\(I_A = I_O + \frac{4M}{3} \cdot 25a^2 = \frac{1025Ma^2}{24}\)M1 A1 Find MI of combined laminas about \(A\)
OR: \(I_{\square,A} = I_{\square,O} + 25Ma^2 = \frac{100Ma^2}{3}\) and \(I_{O,A} = I_{O,O} + \frac{1}{3}M \cdot 25a^2 = \frac{75Ma^2}{8}\)M1 A1
\(I = I_A + 50Ma^2 = \frac{2225Ma^2}{24}\)M1 A1 Find MI of system about \(A\), A.G.
Speed is max when \(AC\) verticalM1 State or imply
\(\frac{1}{2}I\omega^2 = \frac{4Mg}{3} \times 5a + \frac{1}{2}Mg \times 10a\) or \(\frac{11Mg}{6} \times \frac{70a}{11} = \frac{35Mga}{3}\)M1 A1 Use energy
\(\omega^2 = \frac{112}{445}\frac{g}{a}\), \(k = 0.502\)A1 Substitute for \(I\) and find \(k\) 
## Question 5:

| Answer/Working | Marks | Guidance |
|---|---|---|
| $I_{\square,O} = \frac{1}{3}M\{(4a)^2 + (3a)^2\} = \frac{25Ma^2}{3}$ | B1 | MI of rectangular lamina about $O$ |
| $I_{O,O} = \frac{1}{2} \cdot \frac{1}{3}M\left(\frac{5a}{2}\right)^2 = \frac{25Ma^2}{24}$ | B1 | MI of circular lamina about $O$ |
| $I_O = I_{\square,A} + I_{O,A} = \frac{225Ma^2}{24}$ | M1 A1 | Find MI of combined laminas about $O$ |
| $I_A = I_O + \frac{4M}{3} \cdot 25a^2 = \frac{1025Ma^2}{24}$ | M1 A1 | Find MI of combined laminas about $A$ |
| **OR:** $I_{\square,A} = I_{\square,O} + 25Ma^2 = \frac{100Ma^2}{3}$ and $I_{O,A} = I_{O,O} + \frac{1}{3}M \cdot 25a^2 = \frac{75Ma^2}{8}$ | M1 A1 | |
| $I = I_A + 50Ma^2 = \frac{2225Ma^2}{24}$ | M1 A1 | Find MI of system about $A$, A.G. |
| Speed is max when $AC$ vertical | M1 | State or imply |
| $\frac{1}{2}I\omega^2 = \frac{4Mg}{3} \times 5a + \frac{1}{2}Mg \times 10a$ or $\frac{11Mg}{6} \times \frac{70a}{11} = \frac{35Mga}{3}$ | M1 A1 | Use energy |
| $\omega^2 = \frac{112}{445}\frac{g}{a}$, $k = 0.502$ | A1 | Substitute for $I$ and find $k$ |

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\includegraphics[max width=\textwidth, alt={}, center]{f8961f84-c080-4407-a178-45b76f200111-3_533_698_1343_721}

A uniform rectangular lamina $A B C D$, in which $A B = 8 a$ and $B C = 6 a$, has mass $M$. A uniform circular lamina of radius $\frac { 5 } { 2 } a$ has mass $\frac { 1 } { 3 } M$. The two laminas are fixed together in the same plane with their centres coinciding at the point $O$ (see diagram). A particle $P$ of mass $\frac { 1 } { 2 } M$ is attached at $C$. The system is free to rotate about a fixed smooth horizontal axis through $A$ and perpendicular to the plane $A B C D$. Show that the moment of inertia of the system about this axis is $\frac { 2225 } { 24 } M a ^ { 2 }$.

The system is released from rest with $A C$ horizontal and $D$ below $A C$. Find, in the form $k \sqrt { } \left( \frac { g } { a } \right)$, the greatest angular speed in the subsequent motion, giving the value of $k$ correct to 3 decimal places.\\[0pt]
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\hfill \mbox{\textit{CAIE FP2 2014 Q5}}
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