| Exam Board | CAIE |
|---|---|
| Module | M2 (Mechanics 2) |
| Year | 2016 |
| Session | November |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Centre of Mass 1 |
| Type | Frame with circular arc or semicircular arc components |
| Difficulty | Standard +0.3 This is a standard centre of mass problem for composite uniform wires requiring application of standard formulae (centre of mass of semicircular arc at 2r/π from diameter), setting up moments about a line, and a simple equilibrium calculation. The 'show that' part guides students to the answer, and the geometry is straightforward with no novel insight required. |
| Spec | 6.04c Composite bodies: centre of mass6.04e Rigid body equilibrium: coplanar forces |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\text{CoM(large)} = 0.6/(\pi/2)\) or \(\text{CoM(small)} = 0.3/(\pi/2)\) | B1 | |
| \((\pi \times 0.6 + \pi \times 0.3)D = \pi \times 0.6(1.2/\pi) - \pi \times 0.3(0.6/\pi)\) | M1 | OR \((2+1)D = 2(1.2/\pi) - 1(0.6/\pi)\), Moments about ACB |
| \(D = 0.191\) m | A1 AG | Total: 3 marks |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \((\pi \times 0.6 + \pi \times 0.3)H = \pi \times 0.6 \times 0.6 + \pi \times 0.3 \times 0.9\) | M1 | OR \(3H = 2 \times 0.6 + 1 \times 0.9\), Moments about A |
| \(H = 0.7\) | A1 | |
| \(\tan\theta = 0.191/0.7\) | M1 | |
| \(\theta = 15.3°\) | A1 | Total: 4 marks |
## Question 2:
### Part (i):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\text{CoM(large)} = 0.6/(\pi/2)$ or $\text{CoM(small)} = 0.3/(\pi/2)$ | B1 | |
| $(\pi \times 0.6 + \pi \times 0.3)D = \pi \times 0.6(1.2/\pi) - \pi \times 0.3(0.6/\pi)$ | M1 | OR $(2+1)D = 2(1.2/\pi) - 1(0.6/\pi)$, Moments about ACB |
| $D = 0.191$ m | A1 AG | Total: 3 marks |
### Part (ii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $(\pi \times 0.6 + \pi \times 0.3)H = \pi \times 0.6 \times 0.6 + \pi \times 0.3 \times 0.9$ | M1 | OR $3H = 2 \times 0.6 + 1 \times 0.9$, Moments about A |
| $H = 0.7$ | A1 | |
| $\tan\theta = 0.191/0.7$ | M1 | |
| $\theta = 15.3°$ | A1 | Total: 4 marks |
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\includegraphics[max width=\textwidth, alt={}, center]{0a80f46b-b37e-46ce-8907-9d10e4f62f6d-2_318_495_484_824}
A uniform wire is bent to form an object which has a semicircular arc with diameter $A B$ of length 1.2 m , with a smaller semicircular arc with diameter $B C$ of length 0.6 m . The end $C$ of the smaller arc is at the centre of the larger arc (see diagram). The two semicircular arcs of the wire are in the same plane.\\
(i) Show that the distance of the centre of mass of the object from the line $A C B$ is 0.191 m , correct to 3 significant figures.
The object is freely suspended at $A$ and hangs in equilibrium.\\
(ii) Find the angle between $A C B$ and the vertical.
\hfill \mbox{\textit{CAIE M2 2016 Q2 [7]}}