Edexcel M4 (Mechanics 4) 2006 January

1. A particle \(P\) of mass 0.5 kg is released from rest at time \(t = 0\) and falls vertically through a liquid. The motion of \(P\) is resisted by a force of magnitude \(2 v \mathrm {~N}\), where \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is the speed of \(v\) at time \(t\) seconds.
   1. Show that \(5 \frac { \mathrm {~d} v } { \mathrm {~d} t } = 49 - 20 v\).
   2. Find the speed of \(P\) when \(t = 1\).
      (5)
   3. A small smooth sphere \(S\) of mass \(m\) is attached to one end of a light inextensible string of length \(2 a\). The other end of the string is attached to a fixed point \(A\) which is at a distance \(a \sqrt { } 3\) from a smooth vertical wall. The sphere \(S\) hangs at rest in equilibrium. It is then projected horizontally towards the wall with a speed \(\sqrt { \left( \frac { 37 g a } { 5 } \right) }\).
   4. Show that \(S\) strikes the wall with speed \(\sqrt { \left( \frac { 27 g a } { 5 } \right) }\).
      (4)
   Given that the loss in kinetic energy due to the impact with the wall is \(\frac { 3 m g a } { 5 }\),
2. find the coefficient of restitution between \(S\) and the wall.
   (7)

3. Two ships \(P\) and \(Q\) are moving with constant velocity. At 3 p.m., \(P\) is 20 km due north of \(Q\) and is moving at \(16 \mathrm {~km} \mathrm {~h} ^ { - 1 }\) due west. To an observer on ship \(P\), ship \(Q\) appears to be moving on a bearing of \(030 ^ { \circ }\) at \(10 \mathrm {~km} \mathrm {~h} ^ { - 1 }\). Find

1. 1. the speed of \(Q\),
   2. the direction in which \(Q\) is moving, giving your answer as a bearing to the nearest degree,
2. the shortest distance between the ships,
3. the time at which the two ships are closest together.
   (3)

4. A particle \(P\) of mass \(m\) is suspended from a fixed point by a light elastic spring. The spring has natural length \(a\) and modulus of elasticity \(2 m \omega ^ { 2 } a\), where \(\omega\) is a positive constant. At time \(t = 0\) the particle is projected vertically downwards with speed \(U\) from its equilibrium position. The motion of the particle is resisted by a force of magnitude \(2 m \omega v\), where \(v\) is the speed of the particle. At time \(t\), the displacement of \(P\) downwards from its equilibrium position is \(x\).

1. Show that \(\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } + 2 \omega \frac { \mathrm {~d} x } { \mathrm {~d} t } + 2 \omega ^ { 2 } x = 0\). Given that the solution of this differential equation is \(x = \mathrm { e } ^ { - \omega t } ( A \cos \omega t + B \sin \omega t )\), where \(A\) and \(B\) are constants,
2. find \(A\) and \(B\).
3. Find an expression for the time at which \(P\) first comes to rest.
   (3)

5. Two smooth uniform spheres \(A\) and \(B\) have equal radii. Sphere \(A\) has mass \(m\) and sphere \(B\) has mass \(k m\). The spheres are at rest on a smooth horizontal table. Sphere \(A\) is then projected along the table with speed \(u\) and collides with \(B\). Immediately before the collision, the direction of motion of \(A\) makes an angle of \(60 ^ { \circ }\) with the line joining the centres of the two spheres. The coefficient of restitution between the spheres is \(\frac { 1 } { 2 }\).

1. Show that the speed of \(B\) immediately after the collision is \(\frac { 3 u } { 4 ( k + 1 ) }\).
   (6) Immediately after the collision the direction of motion of \(A\) makes an angle arctan \(( 2 \sqrt { 3 } )\) with the direction of motion of \(B\).
2. Show that \(k = \frac { 1 } { 2 }\).
3. Find the loss of kinetic energy due to the collision.
   (4) \section*{6.} \begin{figure}[h]
   \captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{fe647e21-9c4f-4035-b28f-b12a00087692-4_515_1077_301_502}
   \end{figure} A smooth wire with ends \(A\) and \(B\) is in the shape of a semi-circle of radius \(a\). The mid-point of \(A B\) is \(O\). The wire is fixed in a vertical plane and hangs below \(A B\) which is horizontal. A small ring \(R\), of mass \(m \sqrt { 2 }\), is threaded on the wire and is attached to two light inextensible strings. The other end of each string is attached to a particle of mass \(\frac { 3 m } { 2 }\). The particles hang vertically under gravity, as shown in Figure 1.
4. Show that, when the radius \(O R\) makes an angle \(2 \theta\) with the vertical, the potential energy, \(V\), of the system is given by $$V = \sqrt { } 2 m g a ( 3 \cos \theta - \cos 2 \theta ) + \text { constant }$$
5. Find the values of \(\theta\) for which the system is in equilibrium.
6. Determine the stability of the position of equilibrium for which \(\theta > 0\).