| Exam Board | CAIE |
|---|---|
| Module | M2 (Mechanics 2) |
| Year | 2006 |
| Session | June |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Moments |
| Type | Toppling and sliding of solids |
| Difficulty | Standard +0.3 This is a standard mechanics problem involving center of mass of a cone (standard result: h/4 from base), toppling condition (vertical through COM passes through edge of base), and friction inequality. All steps are routine applications of well-known principles with straightforward trigonometry and algebra. Slightly easier than average due to the structured parts guiding the solution. |
| Spec | 3.03r Friction: concept and vector form3.03t Coefficient of friction: F <= mu*R model6.04b Find centre of mass: using symmetry6.04e Rigid body equilibrium: coplanar forces |
| Answer | Marks | Guidance |
|---|---|---|
| \(\tan 31° = m/9.5\) | B1, M1 | For using the idea that the centre of mass is vertically above the lowest point of the base |
| Answer | Marks | Guidance |
|---|---|---|
| \(\mu \geq 0.601\) | A1, B1 | |
| B1 | 2 | From \(F \leq \mu R\) or \(\mu \geq \tan \alpha\) |
**(i)** Distance is 9.5 cm
$\tan 31° = m/9.5$ | B1, M1 | For using the idea that the centre of mass is vertically above the lowest point of the base
**(iii)** Radius is 5.71 cm
$F = mg \sin 31°$ and $R = mg \cos 31°$
or $\mu = \tan \theta$ when on the point of slipping (may be implied)
$\mu \geq 0.601$ | A1, B1
B1 | 2 | From $F \leq \mu R$ or $\mu \geq \tan \alpha$
---2 A uniform solid cone has height 38 cm .\\
(i) Write down the distance of the centre of mass of the cone from its base.\\
\includegraphics[max width=\textwidth, alt={}, center]{ece63d46-5e56-4668-939a-9dbbcfc1a77a-2_497_547_1224_840}
The cone is placed with its axis vertical on a rough horizontal plane. The plane is slowly tilted, and the cone remains in equilibrium until the angle of inclination of the plane reaches $31 ^ { \circ }$ (see diagram), when the cone topples.\\
(ii) Find the radius of the cone.\\
(iii) Show that $\mu \geqslant 0.601$, correct to 3 significant figures, where $\mu$ is the coefficient of friction between the cone and the plane.
\hfill \mbox{\textit{CAIE M2 2006 Q2 [5]}}