CAIE Further Paper 3 2022 June — Question 5 7 marks

Exam BoardCAIE
ModuleFurther Paper 3 (Further Paper 3)
Year2022
SessionJune
Marks7
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TopicCircular Motion 1
DifficultyChallenging +1.8 This is a challenging circular motion problem requiring simultaneous analysis of two connected particles with geometric constraints. Students must set up tension equations for both particles, apply circular motion principles (T = mrω²), use the geometric constraint that the string is inextensible (relating the radii via the angles), and solve a system involving trigonometric relationships. The 7-mark allocation and the need to coordinate multiple concepts (forces, circular motion, geometry, and algebra) places this significantly above average difficulty, though it follows a systematic approach once the setup is understood.
Spec6.05b Circular motion: v=r*omega and a=v^2/r6.05c Horizontal circles: conical pendulum, banked tracks
\includegraphics{figure_5} A light inextensible string \(AB\) passes through two small holes \(C\) and \(D\) in a smooth horizontal table where \(AC = 3a\) and \(DB = a\). A particle of mass \(m\) is attached at the end \(A\) and moves in a horizontal circle with angular velocity \(\omega\). A particle of mass \(\frac{3}{4}m\) is attached to the end \(B\) and moves in a horizontal circle with angular velocity \(k\omega\). \(AC\) makes an angle \(\theta\) with the downward vertical and \(DB\) makes an angle \(\theta\) with the horizontal (see diagram). Find the value of \(k\). [7]
Question 5:
AnswerMarks
5Where a candidate has misread a number in the question and used that value consistently throughout, provided that number does not alter the difficulty or
the method required, award all marks earned and deduct just 1 mark for the misread.
AnswerMarks Guidance
5For A: Tsinmr2 M1
r3asinB1 Correct expression for radius.
T m3a2A1
3
Similarly, for B: Tcos mrk22
AnswerMarks Guidance
4M1 N2L horizontal
3
T  mak22
AnswerMarks
4A1
3
m3a2  mak22
AnswerMarks Guidance
4M1 Equate expressions for T.
k2 4, k 2A1
QuestionAnswer Marks
5Alternative method for question 5
For A: Tcosmg, Tsinmr2M1 N2L horizontal and vertical.
r3asinB1 Correct expression for radius.
5 5 g
T m3a2  mg,  2  
AnswerMarks Guidance
4 12 aA1 Combine to obtain expression for 2.
3
Similarly, for B: Tcos mrk22
AnswerMarks Guidance
4M1 N2L horizontal.
3
T  mak22
AnswerMarks
4A1
5 3 5g
mg  mak2
AnswerMarks Guidance
4 4 12aM1 Substitute for T and .
k2 4, k 2A1
7
AnswerMarks Guidance
QuestionAnswer Marks 
Question 5:
5 | Where a candidate has misread a number in the question and used that value consistently throughout, provided that number does not alter the difficulty or
the method required, award all marks earned and deduct just 1 mark for the misread.
5 | For A: Tsinmr2 | M1 | N2L horizontal.
r3asin | B1 | Correct expression for radius.
T m3a2 | A1
3
Similarly, for B: Tcos mrk22
4 | M1 | N2L horizontal
3
T  mak22
4 | A1
3
m3a2  mak22
4 | M1 | Equate expressions for T.
k2 4, k 2 | A1
Question | Answer | Marks | Guidance
5 | Alternative method for question 5
For A: Tcosmg, Tsinmr2 | M1 | N2L horizontal and vertical.
r3asin | B1 | Correct expression for radius.
5 5 g
T m3a2  mg,  2  
4 12 a | A1 | Combine to obtain expression for 2.
3
Similarly, for B: Tcos mrk22
4 | M1 | N2L horizontal.
3
T  mak22
4 | A1
5 3 5g
mg  mak2
4 4 12a | M1 | Substitute for T and .
k2 4, k 2 | A1
7
Question | Answer | Marks | Guidance
\includegraphics{figure_5}

A light inextensible string $AB$ passes through two small holes $C$ and $D$ in a smooth horizontal table where $AC = 3a$ and $DB = a$. A particle of mass $m$ is attached at the end $A$ and moves in a horizontal circle with angular velocity $\omega$. A particle of mass $\frac{3}{4}m$ is attached to the end $B$ and moves in a horizontal circle with angular velocity $k\omega$. $AC$ makes an angle $\theta$ with the downward vertical and $DB$ makes an angle $\theta$ with the horizontal (see diagram).

Find the value of $k$. [7]

\hfill \mbox{\textit{CAIE Further Paper 3 2022 Q5 [7]}}
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Q1 5 Q2 5 Q3 5 Q4 8 Q5 7 Q6 9 Q7 11