CAIE Further Paper 3 2022 June — Question 4 8 marks

Exam BoardCAIE
ModuleFurther Paper 3 (Further Paper 3)
Year2022
SessionJune
Marks8
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicCentre of Mass 1
TypeConical or hemispherical shell composite
DifficultyChallenging +1.2 This is a standard centre of mass problem requiring knowledge of standard results (hemisphere COM at 3r/8 from base) and application of equilibrium conditions. Part (a) involves straightforward composite body COM calculation using moments. Part (b) requires toppling analysis with given angle, which is routine for Further Maths mechanics. The 8 marks and two-part structure indicate moderate difficulty, but the techniques are well-practiced in FM syllabi with no novel geometric insight required.
Spec6.04c Composite bodies: centre of mass6.04e Rigid body equilibrium: coplanar forces
\includegraphics{figure_4} An object is composed of a hemispherical shell of radius \(2a\) attached to a closed hollow circular cylinder of height \(h\) and base radius \(a\). The hemispherical shell and the hollow cylinder are made of the same uniform material. The axes of symmetry of the shell and the cylinder coincide. \(AB\) is a diameter of the lower end of the cylinder (see diagram).
  1. Find, in terms of \(a\) and \(h\), an expression for the distance of the centre of mass of the object from \(AB\). [4]
The object is placed on a rough plane which is inclined to the horizontal at an angle \(\theta\), where \(\tan \theta = \frac{2}{5}\). The object is in equilibrium with \(AB\) in contact with the plane and lying along a line of greatest slope of the plane.
  1. Find the set of possible values of \(h\), in terms of \(a\). [4]
Question 4:
AnswerMarks
4Unless otherwise indicated, marks once gained cannot subsequently be lost, e.g. wrong working following a correct form of answer is ignored (isw).
AnswerMarks
4(a)Area Centre of mass from AB
1
Cylinder 2πah2πa2 h
2
AnswerMarks Guidance
Shell 2π2a2 haM1 Moments equation, condone missing ends of cylinder.
One expression on the RHS correct.
Moments about AB
x  2πah2πa2 2π2a2 2π2a2 ha  2πah2πa2  1 h  
AnswerMarks Guidance
2 A1 A1 One correct expression on RHS correct scores A1.
2h10ax h2 ah8ah8a2
h2 9ah8a2
x 
AnswerMarks
2h5aA1
4
AnswerMarks
4(b)a
tan
AnswerMarks
xB1
h2 9ah8a2 3
x   a
2h5a 2
AnswerMarks Guidance
h2 6ah7a20M1 Form inequality and rearrange to quadratic, condone
equation.
AnswerMarks Guidance
hah7a0M1 Attempt to solve, condone equation.
7a haA1
4
AnswerMarks Guidance
AreaCentre of mass from AB
Cylinder2πah2πa2 1
h
2
AnswerMarks Guidance
Shell2π2a2 ha
QuestionAnswer Marks 
Question 4:
4 | Unless otherwise indicated, marks once gained cannot subsequently be lost, e.g. wrong working following a correct form of answer is ignored (isw).
--- 4(a) ---
4(a) | Area Centre of mass from AB
1
Cylinder 2πah2πa2 h
2
Shell 2π2a2 ha | M1 | Moments equation, condone missing ends of cylinder.
One expression on the RHS correct.
Moments about AB
x  2πah2πa2 2π2a2 2π2a2 ha  2πah2πa2  1 h  
2  | A1 A1 | One correct expression on RHS correct scores A1.
2h10ax h2 ah8ah8a2
h2 9ah8a2
x 
2h5a | A1
4
--- 4(b) ---
4(b) | a
tan
x | B1
h2 9ah8a2 3
x   a
2h5a 2
h2 6ah7a20 | M1 | Form inequality and rearrange to quadratic, condone
equation.
hah7a0 | M1 | Attempt to solve, condone equation.
7a ha | A1
4
Area | Centre of mass from AB
Cylinder | 2πah2πa2 | 1
h
2
Shell | 2π2a2 | ha
Question | Answer | Marks | Guidance
\includegraphics{figure_4}

An object is composed of a hemispherical shell of radius $2a$ attached to a closed hollow circular cylinder of height $h$ and base radius $a$. The hemispherical shell and the hollow cylinder are made of the same uniform material. The axes of symmetry of the shell and the cylinder coincide. $AB$ is a diameter of the lower end of the cylinder (see diagram).

\begin{enumerate}[label=(\alph*)]
\item Find, in terms of $a$ and $h$, an expression for the distance of the centre of mass of the object from $AB$. [4]
\end{enumerate}

The object is placed on a rough plane which is inclined to the horizontal at an angle $\theta$, where $\tan \theta = \frac{2}{5}$. The object is in equilibrium with $AB$ in contact with the plane and lying along a line of greatest slope of the plane.

\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{1}
\item Find the set of possible values of $h$, in terms of $a$. [4]
\end{enumerate}

\hfill \mbox{\textit{CAIE Further Paper 3 2022 Q4 [8]}}
This paper (7 questions)
View full paper
Q1 5 Q2 5 Q3 5 Q4 8 Q5 7 Q6 9 Q7 11