The diagram shows a uniform solid right circular cone of mass \(m \mathrm {~kg}\), height \(h \mathrm {~m}\) and base radius \(r \mathrm {~m}\) suspended by two vertical strings attached to the points \(P\) and \(Q\) on the circumference of the base. The vertex \(O\) of the cone is vertically below \(P\).
Show that the tension in the string attached at \(Q\) is \(\frac { 3 m g } { 8 } \mathrm {~N}\).
\includegraphics[max width=\textwidth, alt={}, center]{309da227-759c-475e-b12e-dcd9e338a417-2_296_277_269_1668}
Find, in terms of \(m\) and \(g\), the tension in the other string.
Two identical particles \(P\) and \(Q\) are connected by a light inextensible string passing through a small smooth-edged hole in a smooth table, as shown.
\(P\) moves on the table in a horizontal circle of radius 0.2 m and \(Q\) hangs at rest.
\includegraphics[max width=\textwidth, alt={}, center]{309da227-759c-475e-b12e-dcd9e338a417-2_309_430_859_1476}
Calculate the number of revolutions made per minute by \(P\).
(5 marks)
\(Q\) is now also made to move in a horizontal circle of radius 0.2 m below the table. The part of the string between \(Q\) and the table makes an angle of \(45 ^ { \circ }\) with the vertical.
Show that the numbers of revolutions per minute made by \(P\) and \(Q\) respectively are in the ratio \(2 ^ { 1 / 4 } : 1\).
\includegraphics[max width=\textwidth, alt={}, center]{309da227-759c-475e-b12e-dcd9e338a417-2_293_428_1213_1499}
A particle \(P\) of mass \(m \mathrm {~kg}\) is fixed to one end of a light elastic string of natural length \(l \mathrm {~m}\) and modulus of elasticity \(k m g \mathrm {~N}\). The other end of the string is fixed to a point \(X\) on a horizontal plane. \(P\) rests at \(O\), where \(O X = l \mathrm {~m}\), with the string just taut. It is then pulled away from \(X\) through a distance \(\frac { 3 l } { 4 } \mathrm {~m}\) and released from rest. On this side of \(O\), the plane is smooth.
Show that, as long as the string is taut, \(P\) performs simple harmonic motion.
Given that \(P\) first returns to \(O\) with speed \(\sqrt { } ( g l ) \mathrm { ms } ^ { - 1 }\), find the value of \(k\).
On the other side of \(O\) the plane is rough, the coefficient of friction between \(P\) and the plane being \(\mu\). If \(P\) does not reach \(X\) in the subsequent motion, show that \(\mu > \frac { 1 } { 2 }\). ( 4 marks)
If, further, \(\mu = \frac { 3 } { 4 }\), show that the time which elapses after \(P\) is released and before it comes to rest is \(\frac { 1 } { 24 } ( 9 \pi + 32 ) \sqrt { \frac { l } { g } }\) s.
(6 marks)