| Exam Board | CAIE |
|---|---|
| Module | M2 (Mechanics 2) |
| Year | 2013 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Circular Motion 1 |
| Type | Particle on cone surface – with string attached to vertex or fixed point |
| Difficulty | Standard +0.3 This is a standard circular motion problem on a cone surface requiring resolution of forces, centripetal force equation, and basic geometry. Part (i) is straightforward center of mass calculation, part (ii) involves resolving forces in two directions (standard technique), and part (iii) is a simple application of v=rω. All steps are routine textbook exercises with no novel insight required, making it slightly easier than average. |
| Spec | 6.04c Composite bodies: centre of mass6.05b Circular motion: v=r*omega and a=v^2/r6.05c Horizontal circles: conical pendulum, banked tracks |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| M1 | Taking moments with 3 terms | |
| \(OG(0.5+0.2) = 0.5\times0.6/4 + 0.2\times(0.6 - 0.4\cos60)\) | A1 | Correct equation |
| \(OG = 0.221\) m | A1 [3] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(T\cos60 + R\sin60 = 0.2g\) | M1 | Either for resolving horizontally or vertically |
| \(T\sin60 - R\cos60 = 0.2\times4^2\times(0.4\sin60)\) | A1 | Both equations correct |
| Solves 2 simultaneous equations | M1 | 2 equations, 2 unknowns |
| \(T = 1.96\) N | A1 (AG) | \(g = 10\) only |
| \(R = 1.18\) N | A1 [5] | Allow values from \(g\) not 10 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(0.2\times4^2\times0.4\sin60\cos30 = T - 0.2g\cos60\) | M1 | Resolves \(\text{acc}^n\) and weight parallel to the slope |
| \(T = 1.96\) N | A1 (AG) | From \(g = 10\) only |
| \(0.2\times4^2\times0.4\sin60\cos60 = 0.2g\sin60 - R\) | M1 | Resolves \(\text{acc}^n\) and weight perpendicular to slope |
| A1 | Both equations correct | |
| \(R = 1.18\) N | A1 | Allow values from \(g\) not 10 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(v = 1.39\) ms\(^{-1}\) | B1 [1] [9] | \(r\omega = 1.3856...\) |
## Question 6:
### Part (i)
| Answer/Working | Mark | Guidance |
|---|---|---|
| | M1 | Taking moments with 3 terms |
| $OG(0.5+0.2) = 0.5\times0.6/4 + 0.2\times(0.6 - 0.4\cos60)$ | A1 | Correct equation |
| $OG = 0.221$ m | A1 [3] | |
### Part (ii)
| Answer/Working | Mark | Guidance |
|---|---|---|
| $T\cos60 + R\sin60 = 0.2g$ | M1 | Either for resolving horizontally or vertically |
| $T\sin60 - R\cos60 = 0.2\times4^2\times(0.4\sin60)$ | A1 | Both equations correct |
| Solves 2 simultaneous equations | M1 | 2 equations, 2 unknowns |
| $T = 1.96$ N | A1 (AG) | $g = 10$ only |
| $R = 1.18$ N | A1 [5] | Allow values from $g$ not 10 |
**OR:**
| Answer/Working | Mark | Guidance |
|---|---|---|
| $0.2\times4^2\times0.4\sin60\cos30 = T - 0.2g\cos60$ | M1 | Resolves $\text{acc}^n$ and weight parallel to the slope |
| $T = 1.96$ N | A1 (AG) | From $g = 10$ only |
| $0.2\times4^2\times0.4\sin60\cos60 = 0.2g\sin60 - R$ | M1 | Resolves $\text{acc}^n$ and weight perpendicular to slope |
| | A1 | Both equations correct |
| $R = 1.18$ N | A1 | Allow values from $g$ not 10 |
### Part (iii)
| Answer/Working | Mark | Guidance |
|---|---|---|
| $v = 1.39$ ms$^{-1}$ | B1 [1] [9] | $r\omega = 1.3856...$ |
---6\\
\includegraphics[max width=\textwidth, alt={}, center]{c85aa042-7b8c-44cc-b579-a5deef91e7e5-3_291_993_1238_575}
A uniform solid cone of height 0.6 m and mass 0.5 kg has its axis of symmetry vertical and its vertex $V$ uppermost. The semi-vertical angle of the cone is $60 ^ { \circ }$ and the surface is smooth. The cone is fixed to a horizontal surface. A particle $P$ of mass 0.2 kg is connected to $V$ by a light inextensible string of length 0.4 m (see diagram).\\
(i) Calculate the height, above the horizontal surface, of the centre of mass of the cone with the particle.\\
$P$ is set in motion, and moves with angular speed $4 \mathrm { rad } \mathrm { s } ^ { - 1 }$ in a circular path on the surface of the cone.\\
(ii) Show that the tension in the string is 1.96 N , and calculate the magnitude of the force exerted on $P$ by the cone.\\
(iii) Find the speed of $P$.
\hfill \mbox{\textit{CAIE M2 2013 Q6 [9]}}