Particle at midpoint of string between two horizontal fixed points: vertical motion

A particle attached to the midpoint of an elastic string with ends fixed at two points on the same horizontal level hangs or moves vertically; energy methods find equilibrium position, speed, or modulus.

27 questions · Standard +0.8

6.02h Elastic PE: 1/2 k x^26.02i Conservation of energy: mechanical energy principle
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Edexcel M3 Q4
10 marks Challenging +1.2
Two light elastic strings, each of length \(l\) m and modulus of elasticity \(\lambda\) N, are attached to a particle \(P\) of mass \(m\) kg. The other ends of the strings are attached to fixed points \(A\) and \(B\) on the same horizontal level, where \(AB = 2l\) m. \(P\) is held vertically below the mid-point of \(AB\), with each string taut and inclined at \(30°\) to the horizontal, and released from rest. Given that \(P\) comes to instantaneous rest when each string makes an angle of \(60°\) with the horizontal, show that \(\lambda = \frac{3mg}{6 - 2\sqrt{3}}\). \includegraphics{figure_1} [10 marks]
OCR M3 2009 June Q5
11 marks Challenging +1.2
\includegraphics{figure_5} Each of two identical strings has natural length \(1.5\) m and modulus of elasticity \(18\) N. One end of one of the strings is attached to \(A\) and one end of the other string is attached to \(B\), where \(A\) and \(B\) are fixed points which are \(3\) m apart and at the same horizontal level. \(M\) is the mid-point of \(AB\). A particle \(P\) of mass \(m\) kg is attached to the other end of each of the strings. \(P\) is held at rest at the point \(0.8\) m vertically above \(M\), and then released. The lowest point reached by \(P\) in the subsequent motion is \(2\) m below \(M\) (see diagram).
  1. Find the maximum tension in each of the strings during \(P\)'s motion. [3]
  2. By considering energy,
    1. show that the value of \(m\) is \(0.42\), correct to 2 significant figures, [5]
    2. find the speed of \(P\) at \(M\). [3]