| Exam Board | OCR |
|---|---|
| Module | M2 (Mechanics 2) |
| Year | 2013 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Centre of Mass 1 |
| Type | Conical or hemispherical shell composite |
| Difficulty | Standard +0.3 This is a straightforward centre of mass problem requiring standard formulas for cone/shell centres of mass (given in formula booklets), taking moments about a point, then a simple equilibrium problem with two vertical forces. The calculations are routine with no conceptual challenges beyond applying memorized techniques. |
| Spec | 6.04c Composite bodies: centre of mass |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(4.4x_G = 4 \times \frac{1}{4} \times 8\) | M1, A1 | Table of moments idea; moments about other axes acceptable |
| \(- 0.4 \times \frac{1}{3} \times 10\) | A1 | |
| \(x_G = 1.52\) cm | A1 | Allow \(\frac{50}{33}\) |
| [4] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(T_\text{shell} \times 18 = 4.4g \times (8 - 1.52)\) or \(T_\text{cone} \times 18 = 4.4g \times (10 + 1.52)\) | M1, A1ft | Or any other correct moment equation; ft on \(x_G\) from (i) |
| \(T_\text{shell} + T_\text{cone} = 4.4g\) | M1 | May use a second moments equation |
| \(T_\text{shell} = 15.5\) and \(T_\text{cone} = 27.6\) | A1 | For both |
| [4] |
## Question 4:
### Part (i)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $4.4x_G = 4 \times \frac{1}{4} \times 8$ | M1, A1 | Table of moments idea; moments about other axes acceptable |
| $- 0.4 \times \frac{1}{3} \times 10$ | A1 | |
| $x_G = 1.52$ cm | A1 | Allow $\frac{50}{33}$ |
| **[4]** | | |
### Part (ii)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $T_\text{shell} \times 18 = 4.4g \times (8 - 1.52)$ or $T_\text{cone} \times 18 = 4.4g \times (10 + 1.52)$ | M1, A1ft | Or any other correct moment equation; ft on $x_G$ from (i) |
| $T_\text{shell} + T_\text{cone} = 4.4g$ | M1 | May use a second moments equation |
| $T_\text{shell} = 15.5$ and $T_\text{cone} = 27.6$ | A1 | For both |
| **[4]** | | |
---4 A solid uniform cone has height 8 cm , base radius 5 cm and mass 4 kg . A uniform conical shell has height 10 cm , base radius 5 cm and mass 0.4 kg . The two shapes are joined together so that the circumferences of their circular bases coincide.\\
(i) Find the distance of the centre of mass of the shape from the common circular base.\\
\includegraphics[max width=\textwidth, alt={}, center]{74eaa61a-1507-4cef-8f97-5c1860bdc36a-3_974_1141_484_463}
The object is suspended with a string attached to the vertex of the cone and another string attached to the vertex of the conical shell. The object is in equilibrium with the strings vertical and the axis of symmetry of the object horizontal (see diagram).\\
(ii) Find the tension in each string.
\hfill \mbox{\textit{OCR M2 2013 Q4 [8]}}