A motorcyclist rides in a cylindrical well of radius 5 m . He maintains a horizontal circular path at a constant speed of \(10 \mathrm {~ms} ^ { - 1 }\). The coefficient of friction between the wall and the wheels of the cycle is \(\mu\).
Modelling the cyclist and his machine as a particle in contact
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with the wall, show that he will not slip downwards provided that \(\mu \geq 0.49\).
A particle \(P\) moves with simple harmonic motion in a straight line. The centre of oscillation is \(O\). When \(P\) is at a distance 1 m from \(O\), its speed is \(8 \mathrm {~ms} ^ { - 1 }\). When it is at a distance 2 m from \(O\), its speed is \(4 \mathrm {~ms} ^ { - 1 }\).
Find the amplitude of the motion.
Show that the period of motion is \(\frac { \pi } { 2 } \mathrm {~s}\).
A particle of mass \(m \mathrm {~kg}\) is attached to the end \(B\) of a light elastic string \(A B\). The string has natural length \(l \mathrm {~m}\) and modulus of elasticity \(\lambda . \mathrm { N }\).
The end \(A\) is attached to a fixed point on a smooth plane
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inclined at an angle \(\alpha\) to the horizontal, as shown, and the particle rests in equilibrium with the length \(A B = \frac { 5 l } { 4 } \mathrm {~m}\).