Edexcel M3 2018 June — Question 6 13 marks

Exam BoardEdexcel
ModuleM3 (Mechanics 3)
Year2018
SessionJune
Marks13
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicCentre of Mass 2
TypeDeriving standard centre of mass formulae by integration
DifficultyChallenging +1.2 This is a standard M3/Further Mechanics centre of mass question with three parts: (a) deriving a standard formula (3h/4) which is bookwork, (b) combining two shapes using the standard composite body formula, and (c) applying equilibrium conditions with moments. While it requires multiple techniques and careful algebra, all steps follow standard procedures taught in M3 with no novel insight required. The multi-part structure and need for accuracy across parts elevates it slightly above average difficulty.
Spec6.04d Integration: for centre of mass of laminas/solids6.04e Rigid body equilibrium: coplanar forces
6. A uniform solid right circular cone has base radius \(r\) and height \(h\).
  1. Use algebraic integration to show that the distance of the centre of mass of the cone from its vertex is \(\frac { 3 } { 4 } h\).
    [0pt] [You may assume that the volume of a cone of base radius \(r\) and height \(h\) is \(\frac { 1 } { 3 } \pi r ^ { 2 } h\) ] \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{2273ca38-1e16-44ab-ae84-f4c576cbb8f9-20_394_716_632_621} \captionsetup{labelformat=empty} \caption{Figure 3}
    \end{figure} A solid \(S\) is formed by joining a uniform right circular solid cone of mass \(5 m\) to a uniform solid hemisphere, of radius \(r\) and mass \(k m\) where \(k < 20\). The cone has base radius \(r\) and height \(6 r\). The plane face of the cone coincides with the plane face of the hemisphere. The centre of the plane face of the cone is \(O\) and the point \(A\) is on the circular edge of this plane face, as shown in Figure 3.
  2. Find the distance from \(O\) to the centre of mass of \(S\). The solid is suspended from \(A\) and hangs freely in equilibrium. The angle between the axis of the cone and the horizontal is \(30 ^ { \circ }\).
  3. Find, to the nearest whole number, the value of \(k\).
6(a)
AnswerMarks
M1Correct distances shown explicitly or used. Negative sign may be missing here.
dM1Moments equation including a minus sign. Must use the given masses (or ratio of them), not volumes.
A1Fully correct equation. Follow through their distances
M1A1csoCorrect expression for the required distance. Must be positive but can include modulus sign. \(k \cdot 20\) in the numerator without modulus signs scores A0. Equivalents accepted but not fractions within fractions.
[5]
6(b)
AnswerMarks
B1For either of the angles, shown explicitly or used. May be seen on a diagram.
M1A1ft\(\tan 30°\) or \(\tan 60°\) or \(\frac{x}{r}\) or \(\frac{r}{x}\) (i.e. \(30°\) or \(60°\) and fraction either way up). Fully correct equation. Follow through their \(x\)
A1\(k \geq 5\)
[4]
6(c)
AnswerMarks
B1For either of the angles, shown explicitly or used. May be seen on a diagram.
M1A1ft\(\tan 30°\) or \(\tan 60°\) or \(\frac{x}{r}\) or \(\frac{r}{x}\) (i.e. \(30°\) or \(60°\) and fraction either way up). Fully correct equation. Follow through their \(x\)
A1csoComplete to \(\cos = \frac{5}{6}\). Accept \(0.83\) or better.
[4]
[13]
**6(a)**

M1 | Correct distances shown explicitly or used. Negative sign may be missing here.

dM1 | Moments equation including a minus sign. Must use the given masses (or ratio of them), not volumes.

A1 | Fully correct equation. Follow through their distances

M1A1cso | Correct expression for the required distance. Must be positive but can include modulus sign. $k \cdot 20$ in the numerator without modulus signs scores A0. Equivalents accepted but not fractions within fractions.

[5]

**6(b)**

B1 | For either of the angles, shown explicitly or used. May be seen on a diagram.

M1A1ft | $\tan 30°$ or $\tan 60°$ or $\frac{x}{r}$ or $\frac{r}{x}$ (i.e. $30°$ or $60°$ and fraction either way up). Fully correct equation. Follow through their $x$

A1 | $k \geq 5$

[4]

**6(c)**

B1 | For either of the angles, shown explicitly or used. May be seen on a diagram.

M1A1ft | $\tan 30°$ or $\tan 60°$ or $\frac{x}{r}$ or $\frac{r}{x}$ (i.e. $30°$ or $60°$ and fraction either way up). Fully correct equation. Follow through their $x$

A1cso | Complete to $\cos = \frac{5}{6}$. Accept $0.83$ or better.

[4]

[13]

---
6. A uniform solid right circular cone has base radius $r$ and height $h$.
\begin{enumerate}[label=(\alph*)]
\item Use algebraic integration to show that the distance of the centre of mass of the cone from its vertex is $\frac { 3 } { 4 } h$.\\[0pt]
[You may assume that the volume of a cone of base radius $r$ and height $h$ is $\frac { 1 } { 3 } \pi r ^ { 2 } h$ ]

\begin{figure}[h]
\begin{center}
  \includegraphics[alt={},max width=\textwidth]{2273ca38-1e16-44ab-ae84-f4c576cbb8f9-20_394_716_632_621}
\captionsetup{labelformat=empty}
\caption{Figure 3}
\end{center}
\end{figure}

A solid $S$ is formed by joining a uniform right circular solid cone of mass $5 m$ to a uniform solid hemisphere, of radius $r$ and mass $k m$ where $k < 20$. The cone has base radius $r$ and height $6 r$. The plane face of the cone coincides with the plane face of the hemisphere. The centre of the plane face of the cone is $O$ and the point $A$ is on the circular edge of this plane face, as shown in Figure 3.
\item Find the distance from $O$ to the centre of mass of $S$.

The solid is suspended from $A$ and hangs freely in equilibrium. The angle between the axis of the cone and the horizontal is $30 ^ { \circ }$.
\item Find, to the nearest whole number, the value of $k$.

\begin{center}

\end{center}
\end{enumerate}

\hfill \mbox{\textit{Edexcel M3 2018 Q6 [13]}}
This paper (7 questions)
View full paper
Q1 4 Q2 11 Q3 12 Q4 8 Q5 13 Q6 13 Q7 14