Questions — Edexcel M4 (178 questions)

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Edexcel M4 Specimen Q5
12 marks Challenging +1.3
An elastic string spring of modulus \(2mg\) and natural length \(l\) is fixed at one end. To the other end is attached a mass \(m\) which is allowed to hang in equilibrium. The mass is then pulled vertically downwards through a distance \(l\) and released from rest. The air resistance is modelled as having magnitude \(2m\omega v\), where \(v\) is the speed of the particle and \(\omega = \sqrt{\frac{g}{l}}\). The particle is at distance \(x\) from its equilibrium position at time \(t\).
  1. Show that \(\frac{\mathrm{d}^2 x}{\mathrm{d} t^2} + 2\omega \frac{\mathrm{d} x}{\mathrm{d} t} + 2\omega^2 x = 0\). [7]
  2. Find the general solution of this differential equation. [4]
  3. Hence find the period of the damped harmonic motion. [1]
Edexcel M4 Specimen Q6
14 marks Standard +0.3
Two horizontal roads cross at right angles. One is directed from south to north, and the other from east to west. A tractor travels north on the first road at a constant speed of 6 m s\(^{-1}\) and at noon is 200 m south of the junction. A car heads west on the second road at a constant speed of 24 m s\(^{-1}\) and at noon is 960 m east of the junction.
  1. Find the magnitude and direction of the velocity of the car relative to the tractor. [6]
  2. Find the shortest distance between the car and the tractor. [8]
Edexcel M4 Specimen Q7
16 marks Challenging +1.8
\includegraphics{figure_3} A uniform rod \(AB\) has mass \(m\) and length \(2a\). The end \(A\) is smoothly hinged at a fixed point on a fixed straight horizontal wire. A smooth light ring \(R\) is threaded on the wire. The ring \(R\) is attached by a light elastic string, of natural length \(a\) and modulus of elasticity \(mg\), to the end \(B\) of the rod. The end \(B\) is always vertically below \(R\) and angle \(\angle RAB = \theta\), as shown in Fig. 3.
  1. Show that the potential energy of the system is $$mga(2\sin^2\theta - 3\sin\theta) + \text{constant}.$$ [6]
  2. Hence determine the value of \(\theta\), \(0 < \frac{\pi}{2}\), for which the system is in equilibrium. [5]
  3. Determine whether this position of equilibrium is stable or unstable. [5]