AQA Further Paper 3 Mechanics 2024 June — Question 8 10 marks

Exam BoardAQA
ModuleFurther Paper 3 Mechanics (Further Paper 3 Mechanics)
Year2024
SessionJune
Marks10
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicCentre of Mass 1
TypeCone stability and toppling conditions
DifficultyChallenging +1.2 This is a multi-part mechanics question requiring volume of revolution, centre of mass calculation via integration, and equilibrium analysis with friction. Part (a)(i) is a standard 'show that' proof using integration (4 marks), parts (b) and (c) are straightforward statements (1 mark each), and part (c) requires setting up inequalities for sliding vs toppling conditions. While it involves several techniques and careful reasoning about the toppling condition, these are well-practiced Further Maths mechanics topics without requiring particularly novel insight. The integration is routine, and the friction/toppling analysis follows standard methods taught in the syllabus.
Spec6.04d Integration: for centre of mass of laminas/solids6.04e Rigid body equilibrium: coplanar forces
The finite region enclosed by the line \(y = kx\), the \(x\)-axis and the line \(x = 5\) is rotated through 360° around the \(x\) axis to form a solid cone.
    1. Use integration to show that the position of the centre of mass of the cone is independent of \(k\) [4 marks]
    2. State the distance between the base of the cone and its centre of mass. [1 mark]
  2. State one assumption that you have made about the cone. [1 mark]
  3. The plane face of the cone is placed on a rough inclined plane. The coefficient of friction between the cone and the plane is 0.8 The angle between the plane and the horizontal is gradually increased from 0° Find the range of values of \(k\) for which the cone slides before it topples. [4 marks]
Question 8:
AnswerMarks
8(a)(i)Sets up integral(s) to find the
centre of mass.
AnswerMarks Guidance
Condone lack of ρ3.3 M1
π ρ(kx)2dx×x = π ρx(kx)2dx
0 0
125ρπk2 625ρπk2
×x =
3 4
625×3 15
x = =
4×125 4
which is independent of k
Forms two correct integrals.
AnswerMarks Guidance
Condone lack of ρ or missing dx1.1b A1
Evaluates at least one integral to
1 2 5 6 2 5
obtain or
AnswerMarks Guidance
3 41.1b A1
Completes a reasoned argument to
show that x is independent of k
AnswerMarks Guidance
Must include ρ2.1 R1
Subtotal4
QMarking Instructions AO
AnswerMarks
8(a)(ii)5
Obtains OE
4
AnswerMarks Guidance
Follow through their x3.4 B1F
Distance from base = 5 – =
4 4
AnswerMarks Guidance
Subtotal1
QMarking Instructions AO
AnswerMarks Guidance
8(b)States uniform 3.5b
Subtotal1
QMarking Instructions AO
AnswerMarks Guidance
8(c)Obtains tanα = 0.8 or AWRT 38.7
.3.3 B1
tanα = 0.8
On the point of toppling
5 k
t a n α = = 4 k
1 . 2 5
If the cone slides before it topples
0.8 < 4k
1
k >
5
Finds an expression for tanα on the
point of toppling using 5k and their
5
AnswerMarks Guidance
43.3 M1
Obtains a correct expression for
AnswerMarks Guidance
tanα on the point of toppling.1.1b A1
1
Deduces that k >
AnswerMarks Guidance
52.2a R1
Subtotal4
Question total10
QMarking Instructions AO 
Question 8:
--- 8(a)(i) ---
8(a)(i) | Sets up integral(s) to find the
centre of mass.
Condone lack of ρ | 3.3 | M1 | 5 5
π ρ(kx)2dx×x = π ρx(kx)2dx
0 0
125ρπk2 625ρπk2
×x =
3 4
625×3 15
x = =
4×125 4
which is independent of k
Forms two correct integrals.
Condone lack of ρ or missing dx | 1.1b | A1
Evaluates at least one integral to
1 2 5 6 2 5
obtain or
3 4 | 1.1b | A1
Completes a reasoned argument to
show that x is independent of k
Must include ρ | 2.1 | R1
Subtotal | 4
Q | Marking Instructions | AO | Marks | Typical Solution
--- 8(a)(ii) ---
8(a)(ii) | 5
Obtains OE
4
Follow through their x | 3.4 | B1F | 1 5 5
Distance from base = 5 – =
4 4
Subtotal | 1
Q | Marking Instructions | AO | Marks | Typical Solution
--- 8(b) ---
8(b) | States uniform | 3.5b | B1 | The cone is a uniform solid.
Subtotal | 1
Q | Marking Instructions | AO | Marks | Typical Solution
--- 8(c) ---
8(c) | Obtains tanα = 0.8 or AWRT 38.7
. | 3.3 | B1 | On the point of sliding
tanα = 0.8
On the point of toppling
5 k
t a n α = = 4 k
1 . 2 5
If the cone slides before it topples
0.8 < 4k
1
k >
5
Finds an expression for tanα on the
point of toppling using 5k and their
5
4 | 3.3 | M1
Obtains a correct expression for
tanα on the point of toppling. | 1.1b | A1
1
Deduces that k >
5 | 2.2a | R1
Subtotal | 4
Question total | 10
Q | Marking Instructions | AO | Marks | Typical Solution
The finite region enclosed by the line $y = kx$, the $x$-axis and the line $x = 5$ is rotated through 360° around the $x$ axis to form a solid cone.

\begin{enumerate}[label=(\alph*)]
\item \begin{enumerate}[label=(\roman*)]
\item Use integration to show that the position of the centre of mass of the cone is independent of $k$
[4 marks]

\item State the distance between the base of the cone and its centre of mass.
[1 mark]
\end{enumerate}

\item State one assumption that you have made about the cone.
[1 mark]

\item The plane face of the cone is placed on a rough inclined plane.

The coefficient of friction between the cone and the plane is 0.8

The angle between the plane and the horizontal is gradually increased from 0°

Find the range of values of $k$ for which the cone slides before it topples.
[4 marks]
\end{enumerate}

\hfill \mbox{\textit{AQA Further Paper 3 Mechanics 2024 Q8 [10]}}
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