| Exam Board | CAIE |
|---|---|
| Module | M2 (Mechanics 2) |
| Year | 2017 |
| Session | November |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Centre of Mass 1 |
| Type | Composite solid with hemisphere and cylinder/cone |
| Difficulty | Standard +0.8 This is a multi-part centre of mass problem requiring: (i) using the condition that COM is at O to find cylinder height (involves setting up moment equation with standard COM formulas for hemisphere and cylinder), (ii) recalculating COM after removing half the cylinder, and (iii) applying equilibrium conditions with forces at an angle. While it uses standard COM formulas and equilibrium principles, the multi-step nature, need to recall specific COM positions for standard shapes, and the geometric setup with inclined forces makes it moderately challenging—above average but not requiring exceptional insight. |
| Spec | 6.04c Composite bodies: centre of mass6.04e Rigid body equilibrium: coplanar forces |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(20 \times 3 \times 0.4/8 = 20 \times h/2\) | M1 | Takes moments about the common surface |
| \(h = 0.3 \text{ m}\) | A1 | AG |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Cylinder moment \(= 10 \times 0.15/2\) | B1 | |
| \(20 \times 3 \times 0.4/8 - 10 \times 0.15/2 = 30x\) | M1A1 | Takes moments about the base of the cylinder |
| \(x = 0.075 \text{ m}\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(30 \times 0.075\sin60 = P \times 0.4\sin60\) | M1A1 | Takes moments about point of contact of the cylinder with the surface |
| \(P = 5.625\) | A1 |
## Question 6(i):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $20 \times 3 \times 0.4/8 = 20 \times h/2$ | M1 | Takes moments about the common surface |
| $h = 0.3 \text{ m}$ | A1 | AG |
## Question 6(ii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Cylinder moment $= 10 \times 0.15/2$ | B1 | |
| $20 \times 3 \times 0.4/8 - 10 \times 0.15/2 = 30x$ | M1A1 | Takes moments about the base of the cylinder |
| $x = 0.075 \text{ m}$ | A1 | |
## Question 6(iii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $30 \times 0.075\sin60 = P \times 0.4\sin60$ | M1A1 | Takes moments about point of contact of the cylinder with the surface |
| $P = 5.625$ | A1 | |
---6 A solid object consists of a uniform hemisphere of radius 0.4 m attached to a uniform cylinder of radius 0.4 m so that the circumferences of their circular faces coincide. The hemisphere and cylinder each have weight 20 N . The centre of mass of the object lies at the centre $O$ of their common circular face.\\
(i) Show that the height of the cylinder is 0.3 m .\\
A new object is made by cutting the cylinder in half and removing the half not attached to the hemisphere. The cut is perpendicular to the axis of symmetry, so the new object consists of a hemisphere and a cylinder half the height of the original cylinder.\\
(ii) Find the distance of the centre of mass of the new object from $O$.\\
The new object is placed with its hemispherical part on a rough horizontal surface. The new object is held in equilibrium by a force of magnitude $P \mathrm {~N}$ acting along its axis of symmetry, which is inclined at $30 ^ { \circ }$ to the horizontal.\\
(iii) Find $P$.\\
\hfill \mbox{\textit{CAIE M2 2017 Q6 [9]}}