| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9795/2 (Pre-U Further Mathematics Paper 2) |
| Year | 2012 |
| Session | June |
| Marks | 9 |
| Topic | Circular Motion 1 |
| Type | Conical pendulum – horizontal circle in free space (no surface) |
| Difficulty | Standard +0.3 This is a standard conical pendulum problem requiring resolution of forces and circular motion equations. Part (i) involves straightforward limiting arguments, part (ii) is direct force resolution, and part (iii) requires eliminating tension between equations. While it has multiple parts and requires careful algebra, it follows a well-established template with no novel insights needed, making it slightly easier than average. |
| Spec | 6.05a Angular velocity: definitions6.05b Circular motion: v=r*omega and a=v^2/r |
**Question 9(i)**
$T\cos\theta = mg$ (Must be seen to score in part **(i)**.) **B1**
$\theta = 90° \Rightarrow mg = 0 \quad \therefore AP$ can never be horizontal. **B1**
$\therefore 0 < \cos\theta < 1 \Rightarrow T\left(= \frac{mg}{\cos\theta}\right) > mg$ **B1** [3]
**Question 9(ii)**
$T\sin\theta = ml\sin\theta\,\omega^2 \Rightarrow T = ml\omega^2$ **M1A1** [2]
**Question 9(iii)**
$T\frac{h}{l} = mg \Rightarrow T = \frac{mgl}{h}$ **B1**
$\Rightarrow ml\omega^2 = \frac{mgl}{h}$ **B1**
$\Rightarrow \omega^2 h = g$ **B1**
$h = 0.5 \Rightarrow \omega = \sqrt{20} = 4.47$ Angular speed is 4.47 rad/s. **B1**
(Allow 4.43 from $g = 9.8$.) [4] **[9]**9\\
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A particle $P$ of mass $m$ is attached to one end of a light inextensible string of length $l$. The other end of the string is attached to a fixed point $A$. The particle moves with constant angular speed $\omega$ in a horizontal circle whose centre is at a distance $h$ vertically below $A$ (see diagram).\\
(i) Show that however fast the particle travels $A P$ will never become horizontal, and that the tension in the string is always greater than the weight of the particle.\\
(ii) Find the tension in the string in terms of $m , l$ and $\omega$.\\
(iii) Show that $\omega ^ { 2 } h = g$ and calculate $\omega$ when $h$ is 0.5 m .
\hfill \mbox{\textit{Pre-U Pre-U 9795/2 2012 Q9 [9]}}