| Exam Board | CAIE |
|---|---|
| Module | M2 (Mechanics 2) |
| Year | 2010 |
| Session | June |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Centre of Mass 1 |
| Type | Toppling on inclined plane |
| Difficulty | Standard +0.8 This question requires understanding of toppling conditions (centre of mass position relative to pivot), knowledge that a uniform cone's COM is at h/4 from base, geometric reasoning to find the critical angle, and friction analysis. It combines multiple mechanics concepts with non-trivial geometry, making it moderately challenging but still within standard M2 scope. |
| Spec | 3.03u Static equilibrium: on rough surfaces6.04e Rigid body equilibrium: coplanar forces |
| Answer | Marks | Guidance |
|---|---|---|
| \(r = 5.25\) | \(M1, A1 \text{ft}, A1\) [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 |
|---|---|---|
| \(\mu > \tan 35° \to\) Coefficient is greater than 0.7 | \(M1, A1\) [2] | For using 'no sliding \(\to \mu R >\) weight component'. Do not allow \(\mu \geq 0.7\). AG |
**(i)**
$\tan 35° = r/7.5$
$r = 5.25$ | $M1, A1 \text{ft}, A1$ [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)**
$[\mu g\cos 35° > mg\sin 35°]$
$\mu > \tan 35° \to$ Coefficient is greater than 0.7 | $M1, A1$ [2] | For using 'no sliding $\to \mu R >$ weight component'. Do not allow $\mu \geq 0.7$. AG
---2\\
\begin{tikzpicture}[scale=1.2, >=latex]
% Define the angle of inclination
\def\angle{35}
% Draw the horizontal ground reference line
\draw[dashed] (0,0) -- (5,0);
% Draw the inclined plane (ramp)
\draw[very thick] (0,0) -- (\angle:7.5);
% Draw the 35 degree angle arc and label
\draw (0.8,0) arc (0:\angle:0.8);
\node at (17.5:1.2) {$35^\circ$};
% Use a rotated scope to easily draw the triangle relative to the ramp
\begin{scope}[rotate=\angle]
% Coordinates on the ramp
\coordinate (F) at (4.5,0); % Foot of the altitude
\coordinate (P) at (4.5,5.5); % Top vertex (representing 30 cm height)
\coordinate (V1) at (3.2,0); % Left base vertex
\coordinate (V2) at (6.3,0); % Right base vertex
% Draw the main triangle sides
\draw[thick] (V1) -- (P) -- (V2);
% Draw the dashed altitude line
\draw[dashed] (P) -- (F);
% Draw the right-angle symbol at the foot of the altitude
\draw (F) ++(-0.25,0) -- ++(0,0.25) -- ++(0.25,0);
% Label: 30 cm (parallel to the altitude, on its right side)
% Drawing from P to F ensures 'above' places the label to the right
\path (P) -- (F) node[pos=.7, rotate=-55, above, font=\small] {30 cm};
% Label: r cm (parallel to the base, between foot F and vertex V2)
\path (F) -- (V2) node[midway, rotate=35, above, font=\small, yshift=2pt] {$r$ cm};
\end{scope}
\end{tikzpicture}
A uniform solid cone has height 30 cm and base radius $r \mathrm {~cm}$. 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 $35 ^ { \circ }$, when the cone topples. The diagram shows a cross-section of the cone.\\
(i) Find the value of $r$.\\
(ii) Show that the coefficient of friction between the cone and the plane is greater than 0.7 .
\hfill \mbox{\textit{CAIE M2 2010 Q2 [5]}}