| Exam Board | CAIE |
|---|---|
| Module | M2 (Mechanics 2) |
| Year | 2011 |
| Session | November |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Moments |
| Type | Toppling and sliding of solids |
| Difficulty | Challenging +1.2 This is a multi-part moments problem requiring understanding of toppling conditions (vertical line through center of mass must pass through base of support) and center of mass calculations for composite bodies. Part (i) is straightforward geometry with the cone's COM at h/4 from base. Part (ii) requires similar analysis for the cylinder. Part (iii) involves finding the combined COM of a composite body and applying toppling conditions, requiring more steps but still standard M2 techniques. The problem is moderately challenging due to multiple parts and composite body analysis, but uses well-established methods without requiring novel insight. |
| Spec | 3.04b Equilibrium: zero resultant moment and force6.04e Rigid body equilibrium: coplanar forces |
| Answer | Marks | Guidance |
|---|---|---|
| (i) \(\tan\theta = 0.7/(2.4/4)\) | M1 | |
| \(\theta = 49.4°\) | A1 | [2] |
| (ii) \(h/2 = 2.4/4\) | M1 | |
| \(h = 1.2\) | A1 | [2] |
| (iii) \(\text{Table of values idea, accept } w = 1\) | M1 | |
| \(4wVG = w \times 2.4 \times 3/4 + 3w(2.4 + h/2)\) | A1 | |
| M1 | Centre of mass above common circumference | |
| \(VG = [\sqrt{(0.7^2 + 2.4^2)}]/\cos\alpha\) | A1 | \(\cos\alpha = 2.4/2.5 = 0.96\) |
| \(h = 0.944\) | A1 | [5] |
**(i)** $\tan\theta = 0.7/(2.4/4)$ | M1
$\theta = 49.4°$ | A1 | [2]
**(ii)** $h/2 = 2.4/4$ | M1
$h = 1.2$ | A1 | [2]
**(iii)** $\text{Table of values idea, accept } w = 1$ | M1
$4wVG = w \times 2.4 \times 3/4 + 3w(2.4 + h/2)$ | A1
| M1 | Centre of mass above common circumference
$VG = [\sqrt{(0.7^2 + 2.4^2)}]/\cos\alpha$ | A1 | $\cos\alpha = 2.4/2.5 = 0.96$
$h = 0.944$ | A1 | [5]
---4 A uniform solid cylinder has radius 0.7 m and height $h \mathrm {~m}$. A uniform solid cone has base radius 0.7 m and height 2.4 m . The cylinder and the cone both rest in equilibrium each with a circular face in contact with a horizontal plane. The plane is now tilted so that its inclination to the horizontal, $\theta ^ { \circ }$, is increased gradually until the cone is about to topple.\\
(i) Find the value of $\theta$ at which the cone is about to topple.\\
(ii) Given that the cylinder does not topple, find the greatest possible value of $h$.
The plane is returned to a horizontal position, and the cone is fixed to one end of the cylinder so that the plane faces coincide. It is given that the weight of the cylinder is three times the weight of the cone. The curved surface of the cone is placed on the horizontal plane (see diagram).\\
\includegraphics[max width=\textwidth, alt={}, center]{d1f1f036-1676-443e-b733-ca1fe79972d4-3_476_1211_836_466}\\
(iii) Given that the solid immediately topples, find the least possible value of $h$.
\hfill \mbox{\textit{CAIE M2 2011 Q4 [9]}}