| Exam Board | CAIE |
|---|---|
| Module | M2 (Mechanics 2) |
| Year | 2016 |
| Session | November |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Circular Motion 1 |
| Type | Particle on cone surface – no string (normal reaction only) |
| Difficulty | Standard +0.8 This is a multi-part circular motion problem requiring resolution of forces in 3D (cone geometry with 60° angle), application of Newton's second law for circular motion, and then extension to include elastic string forces with Hooke's law. The geometric setup is non-trivial, requiring careful consideration of angles and force components, plus the second part adds complexity with the elastic string creating a coupled system. This goes beyond standard horizontal/vertical circle problems. |
| Spec | 6.02i Conservation of energy: mechanical energy principle6.05b Circular motion: v=r*omega and a=v^2/r6.05c Horizontal circles: conical pendulum, banked tracks |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Mark | Guidance |
| \(R\cos 30 = 0.5g\) | M1 | |
| \(R = 5.77(35\ldots)\) | A1 AG | |
| \(R\sin 30 = 0.5\omega^2 \times 0.4\) | M1 | |
| \(\omega = 3.8(0)\ \text{rad s}^{-1}\) | A1 | |
| [4] |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Mark | Guidance |
| \(T = 36(0.5 - 0.45)/0.45\) | M1 | 4 N |
| Vert cmpt \(= 4 \times 0.3/0.5 = 2.4\) | A1 | |
| Horiz cmpt \(= 4 \times 0.4/0.5 = 3.2\) | A1 | |
| \(R\cos 30 + 2.4 = 0.5g\) | M1 | |
| \(R = 3(.00\ldots)\) N | ||
| \(0.5v^2/0.4 = 3.2 + R\sin 30\) | M1 | |
| \(v = 1.94\ \text{ms}^{-1}\) | A1 | |
| [6] |
## Question 7:
### Part (i):
| Working/Answer | Mark | Guidance |
|---|---|---|
| $R\cos 30 = 0.5g$ | M1 | |
| $R = 5.77(35\ldots)$ | A1 AG | |
| $R\sin 30 = 0.5\omega^2 \times 0.4$ | M1 | |
| $\omega = 3.8(0)\ \text{rad s}^{-1}$ | A1 | |
| | [4] | |
### Part (ii):
| Working/Answer | Mark | Guidance |
|---|---|---|
| $T = 36(0.5 - 0.45)/0.45$ | M1 | 4 N |
| Vert cmpt $= 4 \times 0.3/0.5 = 2.4$ | A1 | |
| Horiz cmpt $= 4 \times 0.4/0.5 = 3.2$ | A1 | |
| $R\cos 30 + 2.4 = 0.5g$ | M1 | |
| $R = 3(.00\ldots)$ N | | |
| $0.5v^2/0.4 = 3.2 + R\sin 30$ | M1 | |
| $v = 1.94\ \text{ms}^{-1}$ | A1 | |
| | [6] | |7\\
\includegraphics[max width=\textwidth, alt={}, center]{d9970ad1-a7f4-429a-bad1-43e8d114b968-4_213_811_260_667}
A small ball $B$ of mass 0.5 kg moves in a horizontal circle with centre $O$ and radius 0.4 m on the smooth inner surface of a hollow cone fixed with its vertex down. The axis of the cone is vertical and the semi-vertical angle is $60 ^ { \circ }$ (see diagram).\\
(i) Show that the magnitude of the force exerted by the cone on $B$ is 5.77 N , correct to 3 significant figures, and calculate the angular speed of $B$.
One end of a light elastic string of natural length 0.45 m and modulus of elasticity 36 N is attached to $B$. The other end of the string is attached to the point on the axis 0.3 m above $O$. The ball $B$ again moves on the surface of the cone in the same horizontal circle as before.\\
(ii) Calculate the speed of $B$.
\hfill \mbox{\textit{CAIE M2 2016 Q7 [10]}}