A light inelastic string of length \(l \mathrm {~m}\) passes through a small smooth ring which is fixed at a point \(O\) and is free to rotate about a vertical axis through \(O\). Particles \(P\) and \(Q\), of masses 0.06 kg and 0.04 kg respectively, are attached to the ends of the string.
\(Q\) describes a horizontal circle with centre \(P\), while \(P\) hangs at rest at a depth \(d \mathrm {~m}\) below \(O\). Show that \(d = \frac { 2 l } { 5 }\).
(6 marks)
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\(P\) and \(Q\) now both move in horizontal circles with the same angular velocity \(\omega \mathrm { rad } \mathrm { s } ^ { - 1 }\) about a vertical axis through \(O\). Show that \(O Q = \frac { 3 l } { 5 } \mathrm {~m}\). (7 marks)
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The figure show a wine glass consisting of a hemispherical cup of radius \(r\), a cylindrical solid stem of height \(r\) and a circular base of radius \(r\). The cup has mass \(M\) and the stem has mass \(m\). Modelling the cup as a thin, uniform hemispherical shell, the base as a uniform lamina made of the same thin
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material as the cup, and the stem as a uniform solid cylinder,
show that the mass of the circular base is \(\frac { 1 } { 2 } M\).
Given that the centre of mass of the glass is at a distance \(\frac { 13 r } { 14 }\) from the base along the vertical axis of symmetry,
express \(M\) in terms of \(m\).
If the cup is now filled with liquid whose mass is \(2 M\),
show that the position of the centre of mass rises through a distance \(\frac { 13 r } { 35 }\).
State an assumption that you have made about the liquid.