3 A particle \(P\) of mass \(m\) is attached to one end of a light inextensible string of length \(a\). The other end of the string is attached to a fixed point \(O\). The particle is held with the string taut and horizontal and is then released. When the string is vertical, it comes into contact with a small smooth peg \(A\) which is vertically below \(O\) and at a distance \(x ( < a )\) from \(O\). In the subsequent motion, when \(A P\) makes an angle \(\theta\) with the downward vertical, the tension in the string is \(T\). Show that $$T = m g \left( 3 \cos \theta + \frac { 2 x } { a - x } \right)$$ Given that \(P\) completes a vertical circle about \(A\), find the least possible value of \(\frac { x } { a }\).

## Question 3:

| Answer/Working | Marks | Guidance |
|---|---|---|
| $\frac{1}{2}mv^2 = mga\ [v^2 = 2ga]$ | B1 | Energy to find speed $v$ when $AP$ vertical |
| $\frac{1}{2}mw^2 = \frac{1}{2}mv^2 - mg(a-x)(1-\cos\theta)$ | M1 A1 | Energy to find speed $w$ when $AP$ at angle $\theta$ |
| $[mw^2 = 2mg\{x + (a-x)\cos\theta\}]$ | | |
| $T - mg\cos\theta = \frac{mw^2}{(a-x)}$ | M1 A1 | $F = ma$ radially for tension $T$ |
| $T = mg\{3\cos\theta + \frac{2x}{(a-x)}\}$ | M1 A1 | **A.G.** Substitute for $w^2$ |
| $2x = 3(a-x),\quad \frac{x}{a} = \frac{3}{5}$ | M1 A1 | Find $x/a$ if $T=0$ when $\theta = \pi$ |

**Total: [9]** (Part marks: 7 + 2)

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3 A particle $P$ of mass $m$ is attached to one end of a light inextensible string of length $a$. The other end of the string is attached to a fixed point $O$. The particle is held with the string taut and horizontal and is then released. When the string is vertical, it comes into contact with a small smooth peg $A$ which is vertically below $O$ and at a distance $x ( < a )$ from $O$. In the subsequent motion, when $A P$ makes an angle $\theta$ with the downward vertical, the tension in the string is $T$. Show that

$$T = m g \left( 3 \cos \theta + \frac { 2 x } { a - x } \right)$$

Given that $P$ completes a vertical circle about $A$, find the least possible value of $\frac { x } { a }$.

\hfill \mbox{\textit{CAIE FP2 2012 Q3 [9]}}