Elastic string in circular motion

A question is this type if and only if the particle is attached by a light elastic string (not inextensible) and circular motion involves extension and Hooke's law.

1 questions · Challenging +1.2

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Pre-U Pre-U 9795/2 2010 June Q4
11 marks Challenging +1.2
4 One end of a light elastic string of natural length 0.2 m and modulus of elasticity 100 N is attached to a fixed point \(A\). The other end is attached to a particle of mass 5 kg . The particle moves with angular speed \(\omega\) radians per second in a horizontal circle with the centre vertically below \(A\). The string makes an angle \(\theta\) with the vertical.
  1. By considering the horizontal component of the tension in the string, show that the tension in the string is \(( 1 + 5 x ) \omega ^ { 2 } \mathrm {~N}\), where \(x\) is the extension, in metres, of the string.
  2. (a) By considering vertical forces and also Hooke's law, deduce that \(\cos \theta = \frac { 1 } { 10 x }\).
    (b) Show that \(\omega > \frac { 10 \sqrt { 3 } } { 3 }\).
  3. When the value of \(\omega\) is \(5 \sqrt { 2 }\), find the radius of the circular motion.