Question 4 - A-Level Maths

CAIE Further Paper 3 2024 June — Question 4 7 marks

Exam Board	CAIE
Module	Further Paper 3 (Further Paper 3)
Year	2024
Session	June
Marks	7
Paper	Download PDF ↗
Mark scheme	Download PDF ↗
Topic	Moments
Type	Ring on wire with string
Difficulty	Challenging +1.8 This is a challenging Further Maths mechanics problem requiring resolution of forces in two directions, taking moments about a strategic point, and using limiting equilibrium conditions. The geometry with the ring, inclined plane, and horizontal string creates a non-trivial setup, but the solution follows systematic statics methods without requiring exceptional insight. The given tan α = 1/3 simplifies calculations considerably.
Spec	3.03v Motion on rough surface: including inclined planes 6.04e Rigid body equilibrium: coplanar forces

A ring of weight \(W\), with radius \(a\) and centre \(O\), is at rest on a rough surface that is inclined to the horizontal at an angle \(\alpha\) where \(\tan\alpha = \frac{1}{3}\). The plane of the ring is perpendicular to the inclined surface and parallel to a line of greatest slope of the surface. The point \(P\) on the circumference of the ring is such that \(OP\) is parallel to the surface. A light inextensible string is attached to \(P\) and to the point \(Q\), which is on the surface, such that \(PQ\) is horizontal (see diagram). The points \(O\), \(P\) and \(Q\) are in the same vertical plane. The system is in limiting equilibrium and the coefficient of friction between the ring and the surface is \(\mu\). \includegraphics{figure_4}

Find, in terms of \(W\), the tension in the string \(PQ\). [4]
Find the value of \(\mu\). [3]

Show mark scheme Show mark scheme source

Question 4:

Answer	Marks	Guidance
4(a)	Frictional force F and normal reaction R at point of contact of ring with plane.
Resolve parallel to plane: F TcosWsin	M1	Only allow cos/sin errors for T and W

components, sign errors.

Accept equations for vertical and horizontal (both

needed).

Answer	Marks	Guidance
Moments about O: FaTasin	B1
Combine and substitute for :	M1	Expression for T in terms of W .

T  1W

Answer	Marks	Guidance
3	A1	CAO

Alternative solution for question 4(a)

Answer	Marks	Guidance
Moments about point where ring touches plane: TasinTacosWasin	M1 A1	Only allow cos/sin errors, sign errors.

Must be dimensionally correct.

Answer	Marks	Guidance
Rearrange and substitute for :	M1	Expression for T in terms of W .

T  1W

Answer	Marks	Guidance
3	A1	CAO

4

Answer	Marks	Guidance
4(b)	Resolve perpendicular to plane: RTsinWcos	M1

components, sign errors.

Answer	Marks	Guidance
Use F R and combine to reach an equation in  only.	M1	From part (a), F TcosWsin or

F Tsin.

 1

Answer	Marks
7	A1

3

Answer	Marks	Guidance
Question	Answer	Marks

Question 4:
--- 4(a) ---
4(a) | Frictional force F and normal reaction R at point of contact of ring with plane.
Resolve parallel to plane: F TcosWsin | M1 | Only allow cos/sin errors for T and W
components, sign errors.
Accept equations for vertical and horizontal (both
needed).
Moments about O: FaTasin | B1
Combine and substitute for : | M1 | Expression for T in terms of W .
T  1W
3 | A1 | CAO
Alternative solution for question 4(a)
Moments about point where ring touches plane: TasinTacosWasin | M1 A1 | Only allow cos/sin errors, sign errors.
Must be dimensionally correct.
Rearrange and substitute for : | M1 | Expression for T in terms of W .
T  1W
3 | A1 | CAO
4
--- 4(b) ---
4(b) | Resolve perpendicular to plane: RTsinWcos | M1 | Only allow cos/sin errors for T and W
components, sign errors.
Use F R and combine to reach an equation in  only. | M1 | From part (a), F TcosWsin or
F Tsin.
 1
7 | A1
3
Question | Answer | Marks | Guidance

Show LaTeX source

A ring of weight $W$, with radius $a$ and centre $O$, is at rest on a rough surface that is inclined to the horizontal at an angle $\alpha$ where $\tan\alpha = \frac{1}{3}$. The plane of the ring is perpendicular to the inclined surface and parallel to a line of greatest slope of the surface. The point $P$ on the circumference of the ring is such that $OP$ is parallel to the surface.

A light inextensible string is attached to $P$ and to the point $Q$, which is on the surface, such that $PQ$ is horizontal (see diagram). The points $O$, $P$ and $Q$ are in the same vertical plane. The system is in limiting equilibrium and the coefficient of friction between the ring and the surface is $\mu$.

\includegraphics{figure_4}

\begin{enumerate}[label=(\alph*)]
\item Find, in terms of $W$, the tension in the string $PQ$. [4]
\item Find the value of $\mu$. [3]
\end{enumerate}

\hfill \mbox{\textit{CAIE Further Paper 3 2024 Q4 [7]}}

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