| Exam Board | CAIE |
|---|---|
| Module | M2 (Mechanics 2) |
| Year | 2010 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Circular Motion 1 |
| Type | Conical pendulum – horizontal circle in free space (no surface) |
| Difficulty | Moderate -0.8 This is a straightforward conical pendulum problem requiring standard application of Newton's second law in circular motion. Students resolve forces vertically and horizontally, use the given acceleration to find tan θ (which is shown), then find tension and speed using basic trigonometry and circular motion formulas. All steps are routine with no problem-solving insight required. |
| Spec | 3.03d Newton's second law: 2D vectors3.03e Resolve forces: two dimensions6.05b Circular motion: v=r*omega and a=v^2/r6.05c Horizontal circles: conical pendulum, banked tracks |
| Answer | Marks | Guidance |
|---|---|---|
| \(T = 0.24 \times 10/\cos\theta = 3\) | \(B1, B1, B1, B1\) [4] | SR \(B1\) not \(B2\) for \(\tan\theta = \sqrt{3}/gr\) or \(a/g\) used. AG. For using \(T\cos\theta = mg\) to find \(T\) |
| Answer | Marks | Guidance |
|---|---|---|
| Speed is 3ms\(^{-1}\) | \(M1, A1\) [2] | For using \(v^2 = ar\) to find \(v\) |
**(i)**
$mg = T\cos\theta$
$ma = T\sin\theta$
$\tan\theta = a/g = 0.75$
$T = 0.24 \times 10/\cos\theta = 3$ | $B1, B1, B1, B1$ [4] | SR $B1$ not $B2$ for $\tan\theta = \sqrt{3}/gr$ or $a/g$ used. AG. For using $T\cos\theta = mg$ to find $T$
**(ii)**
$[v^2 = 7.5 \times 2\sin\theta]$
Speed is 3ms$^{-1}$ | $M1, A1$ [2] | For using $v^2 = ar$ to find $v$
---3\\
\includegraphics[max width=\textwidth, alt={}, center]{ae809dfc-c5af-4c0a-9c88-009949d3e9f9-3_456_511_260_817}
A particle of mass 0.24 kg is attached to one end of a light inextensible string of length 2 m . The other end of the string is attached to a fixed point. The particle moves with constant speed in a horizontal circle. The string makes an angle $\theta$ with the vertical (see diagram), and the tension in the string is $T \mathrm {~N}$. The acceleration of the particle has magnitude $7.5 \mathrm {~m} \mathrm {~s} ^ { - 2 }$.\\
(i) Show that $\tan \theta = 0.75$ and find the value of $T$.\\
(ii) Find the speed of the particle.
\hfill \mbox{\textit{CAIE M2 2010 Q3 [6]}}