| Exam Board | OCR MEI |
|---|---|
| Module | Further Mechanics B AS (Further Mechanics B AS) |
| Year | 2021 |
| Session | November |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Centre of Mass 1 |
| Type | Composite solid with hemisphere and cylinder/cone |
| Difficulty | Standard +0.8 This is a multi-part centre of mass problem requiring (a) proving the standard cone COM result using integration or formula, and (b) solving for a density ratio using the condition that the combined COM lies on the boundary. Part (a) is routine for Further Maths students; part (b) requires setting up and solving an equation involving volumes, densities, and COM positions, which is moderately challenging but follows standard techniques for composite bodies. |
| Spec | 6.04a Centre of mass: gravitational effect6.04d Integration: for centre of mass of laminas/solids6.04e Rigid body equilibrium: coplanar forces |
| Answer | Marks | Guidance |
|---|---|---|
| 3 | (a) | Vol of C is 𝜋𝑎3 |
| Answer | Marks |
|---|---|
| 0 | M1 |
| A1 | 2.3 |
| 1.1 | Allow missing π or limits or bracket |
| Answer | Marks | Guidance |
|---|---|---|
| 0 | A1FT | 1.1 |
| Answer | Marks | Guidance |
|---|---|---|
| 4 | A1 | 1.1 |
| Answer | Marks |
|---|---|
| (b) | 3 |
| Answer | Marks |
|---|---|
| 𝐶 4 𝐵 | M1 |
| A1FT | 3.1b |
| 1.1 | Allow ft from (a) for M1A1; allow |
| errors in cube for M1 only | Density of cone is 𝐷 and of |
| Answer | Marks | Guidance |
|---|---|---|
| 3𝜋 | A1FT | 1.1 |
Question 3:
3 | (a) | Vol of C is 𝜋𝑎3 | B1 | 2.1
3𝑎 3𝑎−𝑦 2
(𝜋𝑎3×𝑦̅) = 𝜋∫ ( ) 𝑦d𝑦
3
0 | M1
A1 | 2.3
1.1 | Allow missing π or limits or bracket
wrong, for M1 but y essential. Left
side can be implied, or wrong.
𝜋 9 6 1 3𝑎
(𝜋𝑎3×𝑦̅ =) [ 𝑎2𝑦2− 𝑎𝑦3+ 𝑦4]
9 2 3 4
0 | A1FT | 1.1 | Or equivalent
Must be a follow through from their
attempt at rearrangement for x2
3
𝑦̅ = 𝑎
4 | A1 | 1.1 | AG
[5]
(b) | 3
𝜋𝑎3𝐷 × 𝑎 = 8𝑎3𝐷 ×𝑎
𝐶 4 𝐵 | M1
A1FT | 3.1b
1.1 | Allow ft from (a) for M1A1; allow
errors in cube for M1 only | Density of cone is 𝐷 and of
𝐶
cube 𝐷
𝐵
32
Fraction is
3𝜋 | A1FT | 1.1
[3]3 In this question you must show detailed reasoning.
[In this question you may use the formula: Volume of cone $= \frac { 1 } { 3 } \times$ base area × height.]\\
The region between the line $\mathrm { y } = - 3 \mathrm { x } + 3 \mathrm { a }$, where $a > 0$, the $x$-axis and the $y$-axis is rotated about the $y$-axis to form a uniform right circular cone C with base radius $a$.
\begin{enumerate}[label=(\alph*)]
\item Show that the centre of mass of C is $\frac { 3 } { 4 } a$ from its base.
The cone C is fixed on top of a uniform cube, B , of edge length $2 a$, so that there is no overlap. Fig. 3.1 shows a side view of C and B fixed together; Fig. 3.2 shows a view of C and B from above.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{37798594-8cb0-48aa-8401-090f09e25dff-3_570_323_785_246}
\captionsetup{labelformat=empty}
\caption{Fig. 3.1}
\end{center}
\end{figure}
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{37798594-8cb0-48aa-8401-090f09e25dff-3_309_319_982_753}
\captionsetup{labelformat=empty}
\caption{Fig. 3.2}
\end{center}
\end{figure}
The centre of mass of the combined shape lies on the boundary of C and B .\\
The density of $B$ is not equal to the density of $C$.
\item Determine the exact value of $\frac { \text { density of } \mathrm { C } } { \text { density of } \mathrm { B } }$.\\[0pt]
[3]
\end{enumerate}
\hfill \mbox{\textit{OCR MEI Further Mechanics B AS 2021 Q3 [8]}}