The finite region bounded by the \(x\)-axis, the curve \(y = \frac { 1 } { x ^ { 2 } }\), the line \(x = \frac { 1 } { 4 }\) and the line \(x = 1\), is rotated through one complete revolution about the \(x\)-axis to form a uniform solid of revolution.
- Show that the volume of the solid is \(21 \pi\).
- Find the coordinates of the centre of mass of the solid.
- One end of a light inextensible string of length \(l\) is attached to a fixed point \(A\). The other end is attached to a particle \(P\) of mass \(m\), which is held at a point \(B\) with the string taut and \(A P\) making an angle \(\arccos \frac { 1 } { 4 }\) with the downward vertical. The particle is released from rest. When \(A P\) makes an angle \(\theta\) with the downward vertical, the string is taut and the tension in the string is \(T\).
- Show that
$$T = 3 m g \cos \theta - \frac { m g } { 2 }$$
(6)
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{5c9d14ac-0757-4cdd-9534-337e6b3acee0-08_678_629_815_653}
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\caption{Figure 3}
\end{figure}
At an instant when \(A P\) makes an angle of \(60 ^ { \circ }\) to the downward vertical, \(P\) is moving upwards, as shown in Figure 3. At this instant the string breaks. At the highest point reached in the subsequent motion, \(P\) is at a distance \(d\) below the horizontal through \(A\).