A particle \(P\) of mass \(m \mathrm {~kg}\) moves in a horizontal circle at one end of a light elastic string of natural length \(l \mathrm {~m}\) and modulus of elasticity \(m g \mathrm {~N}\). The other end of the string is attached to a fixed point \(O\). Given that the string makes an angle of \(60 ^ { \circ }\) with the vertical,
show that \(O P = 31 \mathrm {~m}\).
Find, in terms of \(l\) and \(g\), the angular speed of \(P\).
A particle \(P\) of mass \(m \mathrm {~kg}\) moves vertically upwards under gravity, starting from ground level. It is acted on by a resistive force of magnitude \(m \mathrm { f } ( x ) \mathrm { N }\), where \(\mathrm { f } ( x )\) is a function of the height \(x \mathrm {~m}\) of \(P\) above the ground. When \(P\) is at this height, its upward speed \(v \mathrm {~ms} ^ { - 1 }\) is given by \(v ^ { 2 } = 2 \mathrm { e } ^ { - 2 g x } - 1\).
Write down a differential equation for the motion of \(P\) and hence determine \(\mathrm { f } ( x )\) in terms of \(g\) and \(x\).
Show that the greatest height reached by \(P\) above the ground is \(\frac { 1 } { 2 g } \ln 2 \mathrm {~m}\).
Given that the work, in J , done by \(P\) against the resisting force as it moves from ground level to a point \(H \mathrm {~m}\) above the ground is equal to \(\int _ { 0 } ^ { H } m \mathrm { f } ( x ) \mathrm { d } x\),
show that the total work done by \(P\) against the resistance during its upward motion is \(\frac { 1 } { 2 } m ( 1 - \ln 2 ) \mathrm { J }\).