| Exam Board | CAIE |
|---|---|
| Module | M2 (Mechanics 2) |
| Year | 2010 |
| 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 toppling problem requiring knowledge that the centre of mass of a cone is at h/4 from the base, then applying the toppling condition (vertical through CM passes through edge of base). Part (i) uses basic trigonometry, part (ii) applies the friction inequality F ≤ μR with resolved forces. Straightforward application of standard mechanics principles with no novel insight required, making it slightly easier than average. |
| Spec | 3.03r Friction: concept and vector form6.04e Rigid body equilibrium: coplanar forces |
| Answer | Marks | Guidance |
|---|---|---|
| \(\tan 35° = r/7.5\) \(r = 5.25\) | M1 A1ft A1 [3] | For using the idea that the c.m. is vertically above the lowest point of contact. ft using their c of m from the base |
| Answer | Marks | Guidance |
|---|---|---|
| \([\mu \cos 35° > mg\sin 35°]\) \(\mu > \tan 35°\) → Coefficient is greater than 0.7 | M1 A1 [2] | For using 'no sliding \(\rightarrow \mu R >\) weight component'. Do not allow \(\mu ⩾ 0.7\) AG |
## (i)
$\tan 35° = r/7.5$ $r = 5.25$ | M1 A1ft A1 [3] | For using the idea that the c.m. is vertically above the lowest point of contact. ft using their c of m from the base
## (ii)
$[\mu \cos 35° > mg\sin 35°]$ $\mu > \tan 35°$ → Coefficient is greater than 0.7 | M1 A1 [2] | For using 'no sliding $\rightarrow \mu R >$ weight component'. Do not allow $\mu ⩾ 0.7$ AG
---\includegraphics{figure_2}
A uniform solid cone has height 30 cm and base radius $r$ cm. 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 $35°$, when the cone topples. The diagram shows a cross-section of the cone.
\begin{enumerate}[label=(\roman*)]
\item Find the value of $r$. [3]
\item Show that the coefficient of friction between the cone and the plane is greater than 0.7. [2]
\end{enumerate}
\hfill \mbox{\textit{CAIE M2 2010 Q2 [5]}}