A light spring, of natural length 30 cm , is fixed in a vertical position. When a small ball of mass 0.4 kg rests on top of it, the spring is compressed by 10 cm . The ball is then held at a height of 15 cm vertically above the top of the spring and released from rest.
Calculate the maximum compression of the string in the resulting motion.
Aliya, whose mass is \(m \mathrm {~kg}\), is playing rounders. She rounds the first base at a speed of \(v \mathrm {~ms} ^ { - 1 }\), making the turn on a horizontal circular path of radius \(r \mathrm {~m}\).
Write down, in terms of \(m , v\) and \(r\), the magnitude of the horizontal force acting on her.
Show that if she continues on the same circular path, the reaction force exerted on her by the ground must act at an angle \(\theta\) to the vertical, where \(\tan \theta = \frac { v ^ { 2 } } { g r }\).
A particle \(P\) of mass 0.2 kg is suspended by two identical light inelastic strings, with one end of each string attached to \(P\) and the other ends fixed to points \(O\) and \(X\) on the same horizontal level. Both strings are inclined at \(30 ^ { \circ }\) to the horizontal.
Find the tension in the strings when \(P\) is at rest.
The string \(X P\) is suddenly cut, so that \(P\) begins to move in a vertical circle with centre \(O\).
Find the tension in the string \(O P\) when it makes an angle of \(60 ^ { \circ }\) with the horizontal.