4. A projectile \(P\) is fired vertically upwards from a point on the earth's surface. When \(P\) is at a distance \(x\) from the centre of the earth its speed is \(v\). Its acceleration is directed towards the centre of the earth and has magnitude \(\frac { k } { x ^ { 2 } }\), where \(k\) is a constant. The earth may be assumed to be a sphere of radius \(R\).
- Show that the motion of \(P\) may be modelled by the differential equation
$$v \frac { \mathrm {~d} v } { \mathrm {~d} x } = - \frac { g R ^ { 2 } } { x ^ { 2 } }$$
The initial speed of \(P\) is \(U\), where \(U ^ { 2 } < 2 g R\). The greatest distance of \(P\) from the centre of the earth is \(X\).
- Find \(X\) in terms of \(U , R\) and \(g\).
(6)
\section*{5.}
\begin{figure}[h]
\captionsetup{labelformat=empty}
\caption{Figure 3}
\includegraphics[alt={},max width=\textwidth]{c3026c4b-d499-4756-9e01-9b9929f2e04e-5_792_732_513_593}
\end{figure}
An ornament \(S\) is formed by removing a solid right circular cone, of radius \(r\) and height \(\frac { 1 } { 2 } h\), from a solid uniform cylinder, of radius \(r\) and height \(h\), as shown in Fig. 3. - Show that the distance of the centre of mass \(S\) from its plane face is \(\frac { 17 } { 40 } h\).
The ornament is suspended from a point on the circular rim of its open end. It hangs in equilibrium with its axis of symmetry inclined at an angle \(\alpha\) to the horizontal. Given that \(h = 4 r\),
- find, in degrees to one decimal place, the value of \(\alpha\).
(4)
\section*{6.}
\begin{figure}[h]
\captionsetup{labelformat=empty}
\caption{Figure 4}
\includegraphics[alt={},max width=\textwidth]{c3026c4b-d499-4756-9e01-9b9929f2e04e-6_841_942_507_605}
\end{figure}
A particle \(P\) of mass \(m\) is attached to two light inextensible strings. The other ends of the string are attached to fixed points \(A\) and \(B\). The point \(A\) is a distance \(h\) vertically above \(B\). The system rotates about the line \(A B\) with constant angular speed \(\omega\). Both strings are taut and inclined at \(60 ^ { \circ }\) to \(A B\), as shown in Fig. 4. The particle moves in a circle of radius \(r\). - Show that \(r = \frac { \sqrt { 3 } } { 2 } h\).
- Find, in terms of \(m , g , h\) and \(\omega\), the tension in \(A P\) and the tension in \(B P\).
The time taken for \(P\) to complete one circle is \(T\).
- Show that \(T < \pi \sqrt { \left( \frac { 2 h } { g } \right) }\).
(4)