CAIE Further Paper 3 2023 June — Question 5 7 marks

Exam BoardCAIE
ModuleFurther Paper 3 (Further Paper 3)
Year2023
SessionJune
Marks7
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicCircular Motion 1
TypeElastic string – horizontal circle on surface
DifficultyStandard +0.8 This is a two-scenario circular motion problem requiring students to set up tension equations using Hooke's law and centripetal force for both cases, then solve simultaneous equations to find x and λ. While the mechanics concepts are standard Further Maths content, the algebraic manipulation of two coupled equations with multiple parameters (m, v, x, a, λ) and the need to carefully track how quantities change between scenarios elevates this above routine exercises. The 7 total marks and multi-step nature confirm moderate-to-high difficulty within Further Maths.
Spec6.02h Elastic PE: 1/2 k x^26.05b Circular motion: v=r*omega and a=v^2/r
A light elastic string of natural length \(a\) and modulus of elasticity \(\lambda mg\) has one end attached to a fixed point \(O\) on a smooth horizontal surface. When a particle of mass \(m\) is attached to the free end of the string, it moves with speed \(v\) in a horizontal circle with centre \(O\) and radius \(x\). When, instead, a particle of mass \(2m\) is attached to the free end of the string, this particle moves with speed \(\frac{1}{2}v\) in a horizontal circle with centre \(O\) and radius \(\frac{4}{3}x\).
  1. Find \(x\) in terms of \(a\). [5]
  2. Given that \(v = \sqrt{12ag}\), find the value of \(\lambda\). [2]
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
5(a)mg mg3 
Hooke’s law, T 1  a xa or T 2  a  4 xa B1
mv2 mv2 mg
Also, T = and equate  xa
AnswerMarks Guidance
1 x x aM1 gxxa
v2 
a
Dimensionally correct terms.
2
3x  1 
mg a 2m v
 4  2 
Similarly: 
a 3
x
AnswerMarks Guidance
4M1 3gx3 
v2   xa
2a 4 
1 3x
Must have v and on the RHS.
2 4
Their dimensionally correct T .
2
AnswerMarks Guidance
Equate expressions for v2 and solve for x in terms of a.M1
x4aA1 WWW
5SC B3 for answer of 4a using  instead of mg .
AnswerMarks
5(b)a 2a
 v2 or  v2 and substitute x4a, v 12ag
xgxa 3 
3xg xa
AnswerMarks Guidance
4 M1 FT their expression for x.
1A1 CAO
2
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(a) ---
5(a) | mg mg3 
Hooke’s law, T 1  a xa or T 2  a  4 xa  | B1
mv2 mv2 mg
Also, T = and equate  xa
1 x x a | M1 | gxxa
v2 
a
Dimensionally correct terms.
2
3x  1 
mg a 2m v
 4  2 
Similarly: 
a 3
x
4 | M1 | 3gx3 
v2   xa
2a 4 
1 3x
Must have v and on the RHS.
2 4
Their dimensionally correct T .
2
Equate expressions for v2 and solve for x in terms of a. | M1
x4a | A1 | WWW
5 | SC B3 for answer of 4a using  instead of mg .
--- 5(b) ---
5(b) | a 2a
 v2 or  v2 and substitute x4a, v 12ag
xgxa 3 
3xg xa
4  | M1 | FT their expression for x.
1 | A1 | CAO
2
Question | Answer | Marks | Guidance
A light elastic string of natural length $a$ and modulus of elasticity $\lambda mg$ has one end attached to a fixed point $O$ on a smooth horizontal surface. When a particle of mass $m$ is attached to the free end of the string, it moves with speed $v$ in a horizontal circle with centre $O$ and radius $x$. When, instead, a particle of mass $2m$ is attached to the free end of the string, this particle moves with speed $\frac{1}{2}v$ in a horizontal circle with centre $O$ and radius $\frac{4}{3}x$.

\begin{enumerate}[label=(\alph*)]
\item Find $x$ in terms of $a$. [5]

\item Given that $v = \sqrt{12ag}$, find the value of $\lambda$. [2]
\end{enumerate}

\hfill \mbox{\textit{CAIE Further Paper 3 2023 Q5 [7]}}
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