A particle \(P\) of mass \(m\) kg moves in a horizontal circle at one end of a light inextensible string of length 40 cm, as shown. The other end of the string is attached to a fixed point \(O\). The angular velocity of \(P\) is \(\omega\) rad s\(^{-1}\). \includegraphics{figure_2} If the angle \(\theta\) which the string makes with the vertical must not exceed 60°, calculate the greatest possible value of \(\omega\). [7 marks]

$T\cos\theta = mg$, $T\sin\theta = m(0.4\sin\theta)\omega^2$ | M1 A1 M1 A1 | 
$g = 0.4\omega^2\cos\theta$ | | 
$\theta \leq 60°$, so $\cos\theta \geq 0.5$ | | 
$g \geq 0.2\omega^2$ | | 
$\omega^2 \leq 49$ | B1 M1 A1 | 
$\omega \leq 7$ | | 

**Total: 7 marks**

A particle $P$ of mass $m$ kg moves in a horizontal circle at one end of a light inextensible string of length 40 cm, as shown. The other end of the string is attached to a fixed point $O$. The angular velocity of $P$ is $\omega$ rad s$^{-1}$.

\includegraphics{figure_2}

If the angle $\theta$ which the string makes with the vertical must not exceed 60°, calculate the greatest possible value of $\omega$. [7 marks]

\hfill \mbox{\textit{Edexcel M3  Q2 [7]}}