A particle of mass \(m \mathrm {~kg}\) is attached to one end of a light inextensible string of length \(l \mathrm {~m}\) whose other end is fixed to a point \(O\). The particle is made to move in a vertical circle with centre \(O\), with constant angular velocity \(\omega \mathrm { rad } \mathrm { s } ^ { - 1 }\). At a certain instant it is in the position shown, where the string makes an angle \(\theta\) radians with the downward vertical through \(O\).
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Find an expression, in terms of \(m , l\) and \(\omega\), for the kinetic energy of the particle at this instant.
Find an expression, in terms of \(m , g , l\) and \(\theta\), for the potential energy of the particle relative to the horizontal plane through the lowest point \(A\).
Determine the position of the particle when the rate of increase of its total energy, with respect to time, is a maximum.
A particle moves along a straight line in such a way that its displacement \(x \mathrm {~m}\) from a fixed point \(O\) on the line, at time \(t\) seconds after it leaves \(O\), is given by \(x = p \sin \omega t + q \cos \omega t\) where \(p , q\) and \(\omega\) are constants.
Show that the motion of the particle is simple harmonic.
If the particle leaves \(O\) with speed \(15 \mathrm {~ms} ^ { - 1 }\), and \(\omega = 3\), find the amplitude of the motion.
A particle \(P\) of mass 0.2 kg moves in a horizontal circle on one end of an elastic string whose other end is attached to a fixed point \(O\). The angular velocity of \(P\) is \(\pi \mathrm { rad } \mathrm { s } ^ { - 1 }\). The natural length of the string is 1 m and, while \(P\) is in motion, the distance \(O P = 1.15 \mathrm {~m}\).
Calculate, to 3 significant figures, the modulus of elasticity of the string.
The motion now ceases and \(P\) hangs at rest vertically below \(O\).
Show that the extension in the string in this position is about 13 cm .