5.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{ffe0bc72-3136-48d9-9d5b-4a364d134070-07_371_800_262_573}
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\caption{Figure 2}
\end{figure}
A particle \(P\) of mass \(m\) is attached to one end of a light elastic string, of natural length \(l\) and modulus of elasticity \(\lambda\). The other end of the string is attached to a fixed point \(A\) on a smooth plane inclined at angle \(\alpha\) to the horizontal, where \(\sin \alpha = \frac { 3 } { 5 }\). The particle rests in equilibrium on the plane at the point \(B\) with the string lying along a line of greatest slope of the plane, as shown in Figure 2.
Given that \(A B = \frac { 6 } { 5 } l\)
- show that \(\lambda = 3 \mathrm { mg }\)
The particle is pulled down the line of greatest slope to the point \(C\), where \(B C = \frac { 1 } { 2 } l\), and released from rest.
- Show that, while the string remains taut, \(P\) moves with simple harmonic motion about centre \(B\).
- Find the greatest magnitude of the acceleration of \(P\) while the string remains taut.
The point \(D\) is the midpoint of \(B C\). The time taken by \(P\) to move directly from \(D\) to the point where the string becomes slack for the first time is \(k \sqrt { \frac { l } { g } }\), where \(k\) is a constant.
- Find, to 2 significant figures, the value of \(k\).