Vertical elastic string: released from rest at natural length or above (string initially slack)

A particle attached to a vertical elastic string is released from rest at or above the point where the string becomes taut (e.g. released from the fixed point or from a point where string is slack), so free fall occurs before the string becomes taut, then energy methods are applied.

30 questions · Standard +0.6

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Edexcel M3 Q4
10 marks Challenging +1.2
A small stone \(P\) of mass \(m\) kg is attached to one end of a light elastic string of modulus \(3mg\) N and natural length \(2l\) m. The other end of the string is fixed to a point \(O\) at a height \(3l\) m above a horizontal surface. \(P\) is released from rest at \(O\); it hits the surface and rebounds to a height of \(2l\) m. The coefficient of restitution between \(P\) and the surface is \(e\). Calculate the value of \(e\). [9 marks] State one assumption (other than the string being light) that you have used in your solution. [1 mark]
AQA Further Paper 3 Mechanics 2024 June Q6
10 marks Standard +0.3
In this question use \(g = 9.8\) m s\(^{-2}\) A light elastic string has natural length 3 metres and modulus of elasticity 18 newtons. One end of the elastic string is attached to a particle of mass 0.25 kg The other end of the elastic string is attached to a fixed point \(O\) The particle is released from rest at a point \(A\), which is 4.5 metres vertically below \(O\)
  1. Calculate the elastic potential energy of the string when the particle is at \(A\) [2 marks]
  2. The point \(B\) is 3 metres vertically below \(O\) Calculate the gravitational potential energy gained by the particle as it moves from \(A\) to \(B\) [2 marks]
  3. Find the speed of the particle at \(B\) [3 marks]
  4. The point \(C\) is 3.6 metres vertically below \(O\) Explain, showing any calculations that you make, why the speed of the particle is increasing the first time that the particle is at \(C\) [3 marks]
OCR MEI Further Mechanics Major 2020 November Q1
5 marks Standard +0.3
A particle P of mass \(0.5\) kg is attached to a fixed point O by a light elastic string of natural length \(3\) m and modulus of elasticity \(75\) N. P is released from rest at O and is allowed to fall freely. Determine the length of the string when P is at its lowest point in the subsequent motion. [5]
WJEC Further Unit 3 2022 June Q5
14 marks Challenging +1.2
One end of a light elastic string, of natural length 2.5 m and modulus of elasticity \(30g\) N, is fixed to a point O. A particle \(P\), of mass 2 kg, is attached to the other end of the string. Initially, \(P\) is held at rest at the point O. It is then released and allowed to fall under gravity.
  1. Show that, while the string is taut, $$v^2 = g(5 + 2x - 6x^2),$$ where \(v\text{ ms}^{-1}\) denotes the velocity of the particle when the extension in the string is \(x\) m. [6]
  2. Calculate the maximum extension of the string. [3]
    1. Find the extension of the string when \(P\) attains its maximum speed.
    2. Hence determine the maximum speed of \(P\). [5]
CAIE M2 2013 June Q2
Standard +0.8
2 A particle \(P\) of mass 0.3 kg is attached to one end of a light elastic string of natural length 0.6 m and modulus of elasticity 45 N . The other end of the string is attached to a fixed point \(O\). The particle \(P\) is released from rest at \(O\) and falls vertically. Find the extension of the string when \(P\) is at its lowest position.