| Exam Board | CAIE |
|---|---|
| Module | M2 (Mechanics 2) |
| Year | 2008 |
| Session | November |
| Marks | 4 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Moments |
| Type | Toppling and sliding of solids |
| Difficulty | Standard +0.8 This question requires understanding of toppling conditions for different solids (cylinder and cone), finding their centers of mass, applying the toppling criterion (vertical through COM passes through edge of base), and solving simultaneous geometric conditions. It combines mechanics with 3D geometry and requires insight that both solids topple at the same angle, leading to a relationship between their dimensions. More challenging than standard single-body toppling problems. |
| Spec | 6.04e Rigid body equilibrium: coplanar forces |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(r/(h/4) = r/(24/2)\) | M1 | For using \(r/\bar{y}_{cone} = r/\bar{y}_{cylinder}\) |
| \(h = 48\) | A1 | [2] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\tan\alpha = 10/12\) | M1 | For using \(\tan\alpha^\circ = r/\bar{y}\) |
| \(\alpha = 39.8\) | A1 | [2] |
## Question 2:
### Part (i):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $r/(h/4) = r/(24/2)$ | M1 | For using $r/\bar{y}_{cone} = r/\bar{y}_{cylinder}$ |
| $h = 48$ | A1 | **[2]** |
### Part (ii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\tan\alpha = 10/12$ | M1 | For using $\tan\alpha^\circ = r/\bar{y}$ |
| $\alpha = 39.8$ | A1 | **[2]** |
---2\\
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A uniform solid cylinder has height 24 cm and radius $r \mathrm {~cm}$. A uniform solid cone has base radius $r \mathrm {~cm}$ and height $h \mathrm {~cm}$. The cylinder and the cone are both placed with their axes vertical on a rough horizontal plane (see diagram, which shows cross-sections of the solids). The plane is slowly tilted and both solids remain in equilibrium until the angle of inclination of the plane reaches $\alpha ^ { \circ }$, when both solids topple simultaneously.\\
(i) Find the value of $h$.\\
(ii) Given that $r = 10$, find the value of $\alpha$.
\hfill \mbox{\textit{CAIE M2 2008 Q2 [4]}}