| Exam Board | CAIE |
|---|---|
| Module | M2 (Mechanics 2) |
| Year | 2018 |
| Session | November |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Centre of Mass 1 |
| Type | Composite solid with hemisphere and cylinder/cone |
| Difficulty | Standard +0.3 This is a standard two-part centre of mass question requiring (i) calculation using the formula with volumes and individual centres of mass for cone and cylinder, then (ii) a toppling condition using moments about the edge. Both parts follow routine procedures taught in M2 with no novel insight required, making it slightly easier than average. |
| 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 | Marks | Guidance |
| M1 | Attempt to take moments about the vertex of the cone | |
| \((\pi \times 0.3^2 \times 0.4/3) \times (3 \times 0.4/4) + (\pi \times 0.3^2 \times 0.4 \times (0.4 + 0.2))\) | A1 | |
| \(= (\pi \times 0.3^2 \times 0.4/3 + \pi \times 0.3^2 \times 0.4)\,\bar{x}\) | A1 | |
| \(\bar{x} = 0.525 \text{ m}\) | A1 | |
| Total: 4 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| M1 | Attempt to take moments about a point on the circumference of the base of the cone | |
| \(kW\cos30 \times 0.3 + kW\sin30 \times 0.8 = 0.3W\) | A1 | |
| \(k = 0.455\) | A1 | |
| Total: 3 |
**Question 2(i):**
| Answer | Marks | Guidance |
|--------|-------|----------|
| | M1 | Attempt to take moments about the vertex of the cone |
| $(\pi \times 0.3^2 \times 0.4/3) \times (3 \times 0.4/4) + (\pi \times 0.3^2 \times 0.4 \times (0.4 + 0.2))$ | A1 | |
| $= (\pi \times 0.3^2 \times 0.4/3 + \pi \times 0.3^2 \times 0.4)\,\bar{x}$ | A1 | |
| $\bar{x} = 0.525 \text{ m}$ | A1 | |
| **Total: 4** | | |
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**Question 2(ii):**
| Answer | Marks | Guidance |
|--------|-------|----------|
| | M1 | Attempt to take moments about a point on the circumference of the base of the cone |
| $kW\cos30 \times 0.3 + kW\sin30 \times 0.8 = 0.3W$ | A1 | |
| $k = 0.455$ | A1 | |
| **Total: 3** | | |2 A uniform solid object is made by attaching a cone to a cylinder so that the circumferences of the base of the cone and a plane face of the cylinder coincide. The cone and the cylinder each have radius 0.3 m and height 0.4 m .\\
(i) Calculate the distance of the centre of mass of the object from the vertex of the cone.\\[0pt]
[The volume of a cone is $\frac { 1 } { 3 } \pi r ^ { 2 } h$.]\\
The object has weight $W \mathrm {~N}$ and is placed with its plane circular face on a rough horizontal surface. A force of magnitude $k W \mathrm {~N}$ acting at $30 ^ { \circ }$ to the upward vertical is applied to the vertex of the cone. The object does not slip.\\
(ii) Find the greatest possible value of $k$ for which the object does not topple.\\
\hfill \mbox{\textit{CAIE M2 2018 Q2 [7]}}