Vertical elastic string: projected vertically or mid-motion analysis

A particle attached to a vertical elastic string is projected with an initial speed, or you must find acceleration/speed at a specific instant during motion, using energy or Newton's second law.

4 questions · Standard +0.7

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CAIE M2 2016 June Q2
8 marks Standard +0.3
2 One end of a light elastic string of natural length 0.4 m is attached to a fixed point \(O\). The other end of the string is attached to a particle of weight 5 N which hangs in equilibrium 0.6 m vertically below \(O\).
  1. Find the modulus of elasticity of the string. The particle is projected vertically upwards from the equilibrium position and comes to instantaneous rest after travelling 0.3 m upwards.
  2. Calculate the speed of projection of the particle.
  3. Calculate the greatest extension of the string in the subsequent motion.
CAIE M2 2019 November Q3
6 marks Standard +0.8
3 A particle \(P\) of mass 0.5 kg is attached to one end of a light elastic string of natural length 0.6 m and modulus of elasticity 12 N . The other end of the string is attached to a fixed point \(O\). The particle \(P\) is projected vertically downwards with speed \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from the point 0.5 m vertically below \(O\). For an instant when the acceleration of \(P\) is \(4 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) downwards, find the extension of the string and the speed of \(P\).
CAIE M2 2015 June Q5
8 marks Standard +0.3
A particle \(P\) of mass \(0.3\) kg is attached to one end of a light elastic string of natural length \(0.9\) m and modulus of elasticity \(18\) N. The other end of the string is attached to a fixed point \(O\) which is \(3\) m above the ground.
  1. Find the extension of the string when \(P\) is in the equilibrium position. [2]
\(P\) is projected vertically downwards from the equilibrium position with initial speed \(6\) m s\(^{-1}\). At the instant when the tension in the string is \(12\) N the string breaks. \(P\) continues to descend vertically.
    1. Calculate the height of \(P\) above the ground at the instant when the string breaks. [2]
    2. Find the speed of \(P\) immediately before it strikes the ground. [4]
CAIE Further Paper 3 2022 November Q3
6 marks Challenging +1.2
One end of a light elastic string, of natural length \(a\) and modulus of elasticity \(\frac{16}{9}Mg\), is attached to a fixed point \(O\). A particle \(P\) of mass \(4M\) is attached to the other end of the string and hangs vertically in equilibrium. Another particle of mass \(2M\) is attached to \(P\) and the combined particle is then released from rest. The speed of the combined particle when it has descended a distance \(\frac{1}{4}a\) is \(v\). Find an expression for \(v\) in terms of \(g\) and \(a\). [6]