7.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{1f39620e-c10f-4344-89f1-626fff36d187-24_639_593_246_737}
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\caption{Figure 5}
\end{figure}
A package \(P\) of mass \(m\) is attached to one end of a string of length \(\frac { 2 a } { 5 }\). The other end of the string is attached to a fixed point \(O\). The package hangs at rest vertically below \(O\) with the string taut and is then projected horizontally with speed \(u\), as shown in Figure 5.
When \(O P\) has turned through an angle \(\theta\) and the string is still taut, the tension in the string is \(T\)
The package is modelled as a particle and the string as being light and inextensible.
- Show that \(T = 3 m g \cos \theta - 2 m g + \frac { 5 m u ^ { 2 } } { 2 a }\)
Given that \(P\) moves in a complete vertical circle with centre \(O\)
- find, in terms of \(a\) and \(g\), the minimum possible value of \(u\)
Given that \(u = 2 \sqrt { a g }\)
- find, in terms of \(g\), the magnitude of the acceleration of \(P\) at the instant when \(O P\) is horizontal.
- Apart from including air resistance, suggest one way in which the model could be refined to make it more realistic.