Elastic string with friction

A question is this type if and only if it involves an elastic string or spring attached to a particle on a rough surface, requiring analysis of energy, extension, and friction forces.

4 questions · Standard +0.9

6.02i Conservation of energy: mechanical energy principle
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CAIE M2 2013 June Q5
8 marks Challenging +1.2
5 \includegraphics[max width=\textwidth, alt={}, center]{c85aa042-7b8c-44cc-b579-a5deef91e7e5-3_341_529_260_808} A block \(B\) of mass 3 kg is attached to one end of a light elastic string of modulus of elasticity 70 N and natural length 1.4 m . The other end of the string is attached to a particle \(P\) of mass 0.3 kg . \(B\) is at rest 0.9 m from the edge of a horizontal table and the string passes over a small smooth pulley at the edge of the table. \(P\) is released from rest at a point next to the pulley and falls vertically. At the first instant when \(P\) is 0.8 m below the pulley and descending, \(B\) is in limiting equilibrium with the part of the string attached to \(B\) horizontal (see diagram).
  1. Calculate the speed of \(P\) when \(B\) is first in limiting equilibrium.
  2. Find the coefficient of friction between \(B\) and the table.
Edexcel M3 2003 June Q1
6 marks Standard +0.8
A particle \(P\) of mass \(m\) is held at a point \(A\) on a rough horizontal plane. The coefficient of friction between \(P\) and the plane is \(\frac{2}{3}\). The particle is attached to one end of a light elastic string, of natural length \(a\) and modulus of elasticity \(4mg\). The other end of the string is attached to a fixed point \(O\) on the plane, where \(OA = \frac{3}{4}a\). The particle \(P\) is released from rest and comes to rest at a point \(B\), where \(OB < a\). Using the work-energy principle, or otherwise, calculate the distance \(AB\). [6]
Edexcel M3 2014 June Q3
11 marks Challenging +1.2
One end \(A\) of a light elastic string \(AB\), of modulus of elasticity \(mg\) and natural length \(a\), is fixed to a point on a rough plane inclined at an angle \(\theta\) to the horizontal. The other end \(B\) of the string is attached to a particle of mass \(m\) which is held at rest on the plane. The string \(AB\) lies along a line of greatest slope of the plane, with \(B\) lower than \(A\) and \(AB = a\). The coefficient of friction between the particle and the plane is \(\mu\), where \(\mu < \tan \theta\). The particle is released from rest.
  1. Show that when the particle comes to rest it has moved a distance \(2a(\sin \theta - \mu \cos \theta)\) down the plane. [6]
  2. Given that there is no further motion, show that \(\mu \geqslant \frac{1}{3} \tan \theta\). [5]
Edexcel M3 Q2
8 marks Standard +0.3
The diagram shows a particle of mass \(0.7\) kg resting on a rough horizontal table. The coefficient of friction between the particle and the table is \(0.25\). A light elastic string, of natural length \(50\) cm and modulus of elasticity \(6.86\) N, is attached to the particle. The string is kept at an angle of \(60°\) to the horizontal and is gradually extended by pulling on it until the particle moves. Show that the particle starts to move when the extension in the string is \(17\) cm. \includegraphics{figure_2} [8 marks]