| Exam Board | CAIE |
|---|---|
| Module | M2 (Mechanics 2) |
| Year | 2012 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Moments |
| Type | Hemisphere or sphere resting on plane or wall |
| Difficulty | Standard +0.8 This question requires knowledge of standard center of mass formulas for hemispheres (shell at r/2, solid at 3r/8 from base), applying the principle of moments to find the combined center of mass, then solving a non-trivial equilibrium problem with the sphere resting on a curved surface. The geometry of finding moment arms when the axis is horizontal adds complexity beyond routine mechanics problems. |
| Spec | 6.04c Composite bodies: centre of mass6.04e Rigid body equilibrium: coplanar forces |
| Answer | Marks | Guidance |
|---|---|---|
| \(12 \times 3 \times 0.2/8 - 8 \times 0.2/2 = (8 + 12)d\) | M1 | Table of values or moment equation |
| \(0.9 - 0.8 = 20d\) | A1 | \(0.9 - 0.8 = 20d\) |
| \(d = (0.1/20) = 0.005\,\mathrm{m}\) | A1 | Accept \(d = -0.005\) |
| AG | [3] |
| Answer | Marks | Guidance |
|---|---|---|
| \(F \times (2 \times 0.2) = (12 + 8) \times 0.005\) | M1 | Moments about point of contact |
| \(F = 0.25\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| \(F \times (2 \times 0.2) + 8 \times 0.1 = 12 \times 0.075\) | M1 | Moments about point of contact |
| \(F = 0.25\) | A1 | [3] |
**(i)**
$12 \times 3 \times 0.2/8 - 8 \times 0.2/2 = (8 + 12)d$ | M1 | Table of values or moment equation
$0.9 - 0.8 = 20d$ | A1 | $0.9 - 0.8 = 20d$
$d = (0.1/20) = 0.005\,\mathrm{m}$ | A1 | Accept $d = -0.005$
AG | [3]
**(ii)**
$F \times (2 \times 0.2) = (12 + 8) \times 0.005$ | M1 | Moments about point of contact
$F = 0.25$ | A1 |
OR
$F \times (2 \times 0.2) + 8 \times 0.1 = 12 \times 0.075$ | M1 | Moments about point of contact
$F = 0.25$ | A1 | [3] | [6]2 A uniform hemispherical shell of weight 8 N and a uniform solid hemisphere of weight 12 N are joined along their circumferences to form a non-uniform sphere of radius 0.2 m .\\
(i) Show that the distance between the centre of mass of the sphere and the centre of the sphere is 0.005 m .
This sphere is placed on a horizontal surface with its axis of symmetry horizontal. The equilibrium of the sphere is maintained by a force of magnitude $F \mathrm {~N}$ acting parallel to the axis of symmetry applied to the highest point of the sphere.\\
(ii) Calculate $F$.
\hfill \mbox{\textit{CAIE M2 2012 Q2 [6]}}