| Exam Board | OCR MEI |
|---|---|
| Module | M3 (Mechanics 3) |
| Year | 2006 |
| Session | June |
| Marks | 18 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Centre of Mass 2 |
| Type | Lamina with hole removed |
| Difficulty | Standard +0.8 This is a multi-part Further Maths Mechanics question requiring: (i) integration to find centre of mass of a solid of revolution, (ii) composite body calculation with removed cylinder (proof), and (iii) statics with friction and string forces. While each technique is standard for M3, the combination of calculus-based CoM, composite bodies, and equilibrium analysis across multiple parts makes this moderately challenging, above typical A-level but not requiring exceptional insight. |
| Spec | 6.04c Composite bodies: centre of mass6.04d Integration: for centre of mass of laminas/solids6.04e Rigid body equilibrium: coplanar forces |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(\bar{x} = \dfrac{\int_1^4 \pi x \cdot x \, dx}{\int_1^4 \pi x \, dx}\) | M1 | Using \(y^2 = x\) |
| Numerator: \(\pi\int_1^4 x^2\,dx = \pi\left[\dfrac{x^3}{3}\right]_1^4 = \pi \times 21 = 21\pi\) | M1 A1 | |
| Denominator: \(\pi\int_1^4 x\,dx = \pi\left[\dfrac{x^2}{2}\right]_1^4 = \pi \times 7.5 = 7.5\pi\) | A1 | |
| \(\bar{x} = \dfrac{21\pi}{7.5\pi} = 2.8\) | A1 A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Cylinder: radius 1, length 3, volume \(= 3\pi\), \(\bar{x} = 2.5\) | B1 | |
| Solid volume \(= 7.5\pi\), remainder volume \(= 7.5\pi - 3\pi = 4.5\pi\) | M1 | |
| \(7.5\pi \times 2.8 = 3\pi \times 2.5 + 4.5\pi \times \bar{x}_Q\) | M1 A1 | |
| \(21\pi = 7.5\pi + 4.5\pi\bar{x}_Q\) | A1 | |
| \(\bar{x}_Q = \dfrac{13.5\pi}{4.5\pi} = 3\) ✓ | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| String passes through hole at \(x=1\) and \(x=4\) | B1 | |
| Taking moments about A (at \(x=1\)): \(S \times 3 = 96 \times 2\) | M1 A1 | CoM at \(x=3\), distance 2 from A |
| \(S = 64\) N | A1 | |
| \(R = 96 - 64 = 32\) N | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| String CoM at midpoint \(x = 2.5\), distance 1.5 from A | M1 | |
| Moments about A: \(S \times 3 = 96 \times 2 + 6 \times 1.5\) | M1 A1 | |
| \(3S = 192 + 9 = 201\) | A1 | |
| \(S = 67\) N, \(R = 96 + 6 - 67 = 35\) N | A1 |
# Question 4:
## Part (i) - x-coordinate of centre of mass
| Answer | Mark | Guidance |
|--------|------|----------|
| $\bar{x} = \dfrac{\int_1^4 \pi x \cdot x \, dx}{\int_1^4 \pi x \, dx}$ | M1 | Using $y^2 = x$ |
| Numerator: $\pi\int_1^4 x^2\,dx = \pi\left[\dfrac{x^3}{3}\right]_1^4 = \pi \times 21 = 21\pi$ | M1 A1 | |
| Denominator: $\pi\int_1^4 x\,dx = \pi\left[\dfrac{x^2}{2}\right]_1^4 = \pi \times 7.5 = 7.5\pi$ | A1 | |
| $\bar{x} = \dfrac{21\pi}{7.5\pi} = 2.8$ | A1 A1 | |
## Part (ii) - Centre of mass of Q has x-coordinate 3
| Answer | Mark | Guidance |
|--------|------|----------|
| Cylinder: radius 1, length 3, volume $= 3\pi$, $\bar{x} = 2.5$ | B1 | |
| Solid volume $= 7.5\pi$, remainder volume $= 7.5\pi - 3\pi = 4.5\pi$ | M1 | |
| $7.5\pi \times 2.8 = 3\pi \times 2.5 + 4.5\pi \times \bar{x}_Q$ | M1 A1 | |
| $21\pi = 7.5\pi + 4.5\pi\bar{x}_Q$ | A1 | |
| $\bar{x}_Q = \dfrac{13.5\pi}{4.5\pi} = 3$ ✓ | A1 | |
## Part (iii)(A) - Support forces, light string
| Answer | Mark | Guidance |
|--------|------|----------|
| String passes through hole at $x=1$ and $x=4$ | B1 | |
| Taking moments about A (at $x=1$): $S \times 3 = 96 \times 2$ | M1 A1 | CoM at $x=3$, distance 2 from A |
| $S = 64$ N | A1 | |
| $R = 96 - 64 = 32$ N | A1 | |
## Part (iii)(B) - Support forces, heavy string (total weight 6 N)
| Answer | Mark | Guidance |
|--------|------|----------|
| String CoM at midpoint $x = 2.5$, distance 1.5 from A | M1 | |
| Moments about A: $S \times 3 = 96 \times 2 + 6 \times 1.5$ | M1 A1 | |
| $3S = 192 + 9 = 201$ | A1 | |
| $S = 67$ N, $R = 96 + 6 - 67 = 35$ N | A1 | |4 The region bounded by the curve $y = \sqrt { x }$, the $x$-axis and the lines $x = 1$ and $x = 4$ is rotated through $2 \pi$ radians about the $x$-axis to form a uniform solid of revolution.
\begin{enumerate}[label=(\roman*)]
\item Find the $x$-coordinate of the centre of mass of this solid.
From this solid, the cylinder with radius 1 and length 3 with its axis along the $x$-axis (from $x = 1$ to $x = 4$ ) is removed.
\item Show that the centre of mass of the remaining object, Q , has $x$-coordinate 3 .
This object Q has weight 96 N and it is supported, with its axis of symmetry horizontal, by a string passing through the cylindrical hole and attached to fixed points A and B (see Fig. 4). AB is horizontal and the sections of the string attached to A and B are vertical. There is sufficient friction to prevent slipping.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{5bb02383-91c0-4454-aaea-0bd6af6ba325-5_837_819_1034_628}
\captionsetup{labelformat=empty}
\caption{Fig. 4}
\end{center}
\end{figure}
\item Find the support forces, $R$ and $S$, acting on the string at A and B\\
(A) when the string is light,\\
(B) when the string is heavy and uniform with a total weight of 6 N .
\end{enumerate}
\hfill \mbox{\textit{OCR MEI M3 2006 Q4 [18]}}