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Edexcel M4 2004 June Q3
11 marks Standard +0.8
3. At noon, two boats \(A\) and \(B\) are 6 km apart with \(A\) due east of \(B\). Boat \(B\) is moving due north at a constant speed of \(13 \mathrm {~km} \mathrm {~h} ^ { - 1 }\). Boat \(A\) is moving with constant speed \(12 \mathrm {~km} \mathrm {~h} ^ { - 1 }\) and sets a course so as to pass as close as possible to boat \(B\). Find
  1. the direction of motion of \(A\), giving your answer as a bearing,
  2. the time when the boats are closest,
  3. the shortest distance between the boats.
Edexcel M4 2004 June Q4
15 marks Challenging +1.2
4. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{4eb9c38d-66f4-40ba-b7cf-2c2bd19ad087-3_506_967_339_608}
\end{figure} A uniform rod \(P Q\), of length \(2 a\) and mass \(m\), is free to rotate in a vertical plane about a fixed smooth horizontal axis through the end \(P\). The end \(Q\) is attached to one end of a light elastic string, of natural length \(a\) and modulus of elasticity \(\frac { m g } { 2 \sqrt { 3 } }\). The other end of the string is attached to a fixed point \(O\), where \(O P\) is horizontal and \(O P = 2 a\), as shown in Fig. 2. \(\angle O P Q\) is denoted by \(2 \theta\).
  1. Show that, when the string is taut, the potential energy of the system is $$- \frac { m g a } { \sqrt { 3 } } ( 2 \cos 2 \theta + \sqrt { 3 } \sin 2 \theta + 2 \sin \theta ) + \text { constant } .$$
  2. Verify that there is a position of equilibrium at \(\theta = \frac { \pi } { 6 }\).
  3. Determine whether this is a position of stable equilibrium.
Edexcel M4 2004 June Q5
16 marks Standard +0.8
5. A particle \(P\) of mass \(m\) is attached to one end of a light elastic string, of natural length \(a\) and modulus of elasticity \(2 m a k ^ { 2 }\), where \(k\) is a positive constant. The other end of the string is attached to a fixed point \(A\). At time \(t = 0 , P\) is released from rest from a point which is a distance \(2 a\) vertically below \(A\). When \(P\) is moving with speed \(v\), the air resistance has magnitude \(2 m k v\). At time \(t\), the extension of the string is \(x\).
  1. Show that, while the string is taut, $$\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } + 2 k \frac { \mathrm {~d} x } { \mathrm {~d} t } + 2 k ^ { 2 } x = g$$ You are given that the general solution of this differential equation is $$x = \mathrm { e } ^ { - k t } ( C \sin k t + D \cos k t ) + \frac { g } { 2 k ^ { 2 } } , \quad \text { where } C \text { and } D \text { are constants. }$$
  2. Find the value of \(C\) and the value of \(D\). Assuming that the string remains taut,
  3. find the value of \(t\) when \(P\) first comes to rest,
  4. show that \(2 k ^ { 2 } a < g \left( 1 + \mathrm { e } ^ { \pi } \right)\).
Edexcel M4 2004 June Q6
16 marks Challenging +1.8
6. A particle \(P\) of mass \(m\) is attached to one end of a light inextensible string and hangs at rest at time \(t = 0\). The other end of the string is then raised vertically by an engine which is working at a constant rate \(k m g\), where \(k > 0\). At time \(t\), the distance of \(P\) above its initial position is \(x\), and \(P\) is moving upwards with speed \(v\).
  1. Show that \(v ^ { 2 } \frac { \mathrm {~d} v } { \mathrm {~d} x } = ( k - v ) g\).
  2. Show that \(g x = k ^ { 2 } \ln \left( \frac { k } { k - v } \right) - k v - \frac { 1 } { 2 } v ^ { 2 }\).
  3. Hence, or otherwise, find \(t\) in terms of \(k , v\) and \(g\).
Edexcel M4 2005 June Q1
7 marks Standard +0.3
  1. A small smooth ball of mass \(\frac { 1 } { 2 } \mathrm {~kg}\) is falling vertically. The ball strikes a smooth plane which is inclined at an angle \(\alpha\) to the horizontal, where tan \(\alpha = \frac { 3 } { 4 }\). Immediately before striking the plane the ball has speed \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The coefficient of restitution between ball and plane is \(\frac { 1 } { 2 }\). Find
    1. the speed, to 3 significant figures, of the ball immediately after the impact,
    2. the magnitude of the impulse received by the ball as it strikes the plane.
    3. A cyclist \(P\) is cycling due north at a constant speed of \(20 \mathrm {~km} \mathrm {~h} ^ { - 1 }\). At 12 noon another cyclist \(Q\) is due west of \(P\). The speed of \(Q\) is constant at \(10 \mathrm {~km} \mathrm {~h} ^ { - 1 }\). Find the course which \(Q\) should set in order to pass as close to \(P\) as possible, giving your answer as a bearing.
      (5)
    \begin{figure}[h]
    \captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{6895ccda-84b8-45a5-9524-e5bfc37a2fee-2_437_1232_1174_443}
    \end{figure} A smooth sphere \(P\) lies at rest on a smooth horizontal plane. A second identical sphere \(Q\), moving on the plane, collides with the sphere \(P\). Immediately before the collision the direction of motion of \(Q\) makes an angle \(\alpha\) with the line joining the centres of the spheres. Immediately after the collision the direction of motion of \(Q\) makes an angle \(\beta\) with the line joining the centres of spheres, as shown in Figure 1. The coefficient of restitution between the spheres is \(e\). Show that \(( 1 - e ) \tan \beta = 2 \tan \alpha\).
Edexcel M4 2005 June Q4
11 marks Standard +0.3
4. A lorry of mass \(M\) is moving along a straight horizontal road. The engine produces a constant driving force of magnitude \(F\). The total resistance to motion is modelled as having magnitude \(k v ^ { 2 }\), where \(k\) is a constant, and \(v\) is the speed of the lorry. Given the lorry moves with constant speed \(V\),
  1. show that \(V = \sqrt { \frac { F } { k } }\). Given instead that the lorry starts from rest,
  2. show that the distance travelled by the lorry in attaining a speed of \(\frac { 1 } { 2 } V\) is $$\frac { M } { 2 k } \ln \left( \frac { 4 } { 3 } \right)$$
Edexcel M4 2005 June Q5
12 marks Challenging +1.3
  1. A non-uniform rod \(B C\) has mass \(m\) and length \(3 l\). The centre of mass of the rod is at distance \(l\) from \(B\). The rod can turn freely about a fixed smooth horizontal axis through \(B\). One end of a light elastic string, of natural length \(l\) and modulus of elasticity \(\frac { m g } { 6 }\), is attached to \(C\). The other end of the string is attached to a point \(P\) which is at a height \(3 l\) vertically above \(B\).
    1. Show that, while the string is stretched, the potential energy of the system is
    $$m g l \left( \cos ^ { 2 } \theta - \cos \theta \right) + \text { constant, }$$ where \(\theta\) is the angle between the string and the downward vertical and \(- \frac { \pi } { 2 } < \theta < \frac { \pi } { 2 }\).
  2. Find the values of \(\theta\) for which the system is in equilibrium with the string stretched.
Edexcel M4 2005 June Q6
12 marks Challenging +1.2
6. A ship \(A\) has maximum speed \(30 \mathrm {~km} \mathrm {~h} ^ { - 1 }\). At time \(t = 0 , A\) is 70 km due west of \(B\) which is moving at a constant speed of \(36 \mathrm {~km} \mathrm {~h} ^ { - 1 }\) on a bearing of \(300 ^ { \circ }\). Ship \(A\) moves on a straight course at a constant speed and intercepts \(B\). The course of \(A\) makes an angle \(\theta\) with due north.
  1. Show that \(- \arctan \frac { 4 } { 3 } \leq \theta \leq \arctan \frac { 4 } { 3 }\).
  2. Find the least time for \(A\) to intercept \(B\).
    (5)
Edexcel M4 2005 June Q7
17 marks Challenging +1.3
7. A light elastic string, of natural length \(a\) and modulus of elasticity \(5 m a \omega ^ { 2 }\), lies unstretched along a straight line on a smooth horizontal plane. A particle of mass \(m\) is attached to one end of the spring. At time \(t = 0\), the other end of the spring starts to move with constant speed \(U\) along the line of the spring and away from the particle. As the particle moves along the plane it is subject to a resistance of magnitude \(2 m \omega v\), where \(v\) is its speed. At time \(t\), the extension of the spring is \(x\) and the displacement of the particle from its initial position is \(y\). Show that
  1. \(x + y = U t\),
  2. \(\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } + 2 \omega \frac { \mathrm {~d} x } { \mathrm {~d} t } + 5 \omega ^ { 2 } x = 2 \omega U\).
  3. Find \(x\) in terms of \(\omega , U\) and \(t\).
Edexcel M4 2006 June Q1
5 marks Standard +0.8
  1. At noon, a boat \(P\) is on a bearing of \(120 ^ { \circ }\) from boat \(Q\). Boat \(P\) is moving due east at a constant speed of \(12 \mathrm {~km} \mathrm {~h} ^ { - 1 }\). Boat \(Q\) is moving in a straight line with a constant speed of \(15 \mathrm {~km} \mathrm {~h} ^ { - 1 }\) on a course to intercept \(P\). Find the direction of motion of \(Q\), giving your answer as a bearing.
  2. A smooth uniform sphere \(S\) of mass \(m\) is moving on a smooth horizontal plane when it collides with a fixed smooth vertical wall. Immediately before the collision, the speed of \(S\) is \(U\) and its direction of motion makes an angle \(\alpha\) with the wall. The coefficient of restitution between \(S\) and the wall is \(e\). Find the kinetic energy of \(S\) immediately after the collision.
    (6)
  3. A cyclist \(C\) is moving with a constant speed of \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) due south. Cyclist \(D\) is moving with a constant speed of \(16 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) on a bearing of \(240 ^ { \circ }\).
    1. Show that the magnitude of the velocity of \(C\) relative to \(D\) is \(14 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
    At \(2 \mathrm { pm } , D\) is 4 km due east of \(C\).
  4. Find
    1. the shortest distance between \(C\) and \(D\) during the subsequent motion,
    2. the time, to the nearest minute, at which this shortest distance occurs.
Edexcel M4 2006 June Q4
12 marks Challenging +1.8
4. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{fc091589-cb39-47a4-a8b3-06f5fd5ce06a-3_606_1312_260_372}
\end{figure} A uniform rod \(P Q\) has mass \(m\) and length \(2 l\). A small smooth light ring is fixed to the end \(P\) of the rod. This ring is threaded on to a fixed horizontal smooth straight wire. A second small smooth light ring \(R\) is threaded on to the wire and is attached by a light elastic string, of natural length \(l\) and modulus of elasticity \(k m g\), to the end \(Q\) of the rod, where \(k\) is a constant.
  1. Show that, when the rod \(P Q\) makes an angle \(\theta\) with the vertical, where \(0 < \theta \leq \frac { \pi } { 3 }\), and \(Q\) is vertically below \(R\), as shown in Figure 1, the potential energy of the system is $$m g l \left[ 2 k \cos ^ { 2 } \theta - ( 2 k + 1 ) \cos \theta \right] + \text { constant. }$$ Given that there is a position of equilibrium with \(\theta > 0\),
  2. show that \(k > \frac { 1 } { 2 }\).
Edexcel M4 2006 June Q5
11 marks Challenging +1.2
5. A train of mass \(m\) is moving along a straight horizontal railway line. A time \(t\), the train is moving with speed \(v\) and the resistance to motion has magnitude \(k v\), where \(k\) is a constant. The engine of the train is working at a constant rate \(P\).
  1. Show that, when \(v > 0 , \quad m v \frac { \mathrm {~d} v } { \mathrm {~d} t } + k v ^ { 2 } = P\). When \(t = 0\), the speed of the train is \(\frac { 1 } { 3 } \sqrt { \left( \frac { P } { k } \right) }\).
  2. Find, in terms of \(m\) and \(k\), the time taken for the train to double its initial speed.
    (8) \section*{6.} \begin{figure}[h]
    \captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{fc091589-cb39-47a4-a8b3-06f5fd5ce06a-4_638_285_315_897}
    \end{figure} Two small smooth spheres \(A\) and \(B\), of equal size and of mass \(m\) and \(2 m\) respectively, are moving initially with the same speed \(U\) on a smooth horizontal floor. The spheres collide when their centres are on a line \(L\). Before the collision the spheres are moving towards each other, with their directions of motion perpendicular to each other and each inclined at an angle of \(45 ^ { \circ }\) to the line \(L\), as shown in Figure 2. The coefficient of restitution between the spheres is \(\frac { 1 } { 2 }\).
  3. Find the magnitude of the impulse which acts on \(A\) in the collision. \begin{figure}[h]
    \captionsetup{labelformat=empty} \caption{Figure 3} \includegraphics[alt={},max width=\textwidth]{fc091589-cb39-47a4-a8b3-06f5fd5ce06a-4_481_737_1610_792}
    \end{figure} The line \(L\) is parallel to and a distance \(d\) from a smooth vertical wall, as shown in Figure 3.
  4. Find, in terms of \(d\), the distance between the points at which the spheres first strike the wall.
    (5)
Edexcel M4 2006 June Q7
17 marks Challenging +1.2
7. \section*{Figure 4}
\includegraphics[max width=\textwidth, alt={}]{fc091589-cb39-47a4-a8b3-06f5fd5ce06a-5_346_787_328_628}
A light elastic spring has natural length \(l\) and modulus of elasticity \(4 m g\). One end of the spring is attached to a point \(A\) on a plane that is inclined to the horizontal at an angle \(\alpha\), where \(\tan \alpha = \frac { 3 } { 4 }\). The other end of the spring is attached to a particle \(P\) of mass \(m\). The plane is rough and the coefficient of friction between \(P\) and the plane is \(\frac { 1 } { 2 }\). The particle \(P\) is held at a point \(B\) on the plane where \(B\) is below \(A\) and \(A B = l\), with the spring lying along a line of greatest slope of the plane, as shown in Figure 4. At time \(t = 0\), the particle is projected up the plane towards \(A\) with speed \(\frac { 1 } { 2 } \sqrt { } ( g l )\). At time \(t\), the compression of the spring is \(x\).
  1. Show that $$\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } + 4 \omega ^ { 2 } x = - g , \text { where } \omega = \sqrt { \left( \frac { g } { l } \right) }$$
  2. Find \(x\) in terms of \(l , \omega\) and \(t\).
  3. Find the distance that \(P\) travels up the plane before first coming to rest.
Edexcel M4 2007 June Q1
10 marks Standard +0.3
  1. A small ball is moving on a horizontal plane when it strikes a smooth vertical wall. The coefficient of restitution between the ball and the wall is \(e\). Immediately before the impact the direction of motion of the ball makes an angle of \(60 ^ { \circ }\) with the wall. Immediately after the impact the direction of motion of the ball makes an angle of \(30 ^ { \circ }\) with the wall.
    1. Find the fraction of the kinetic energy of the ball which is lost in the impact.
      (6)
    2. Find the value of \(e\).
      (4)
    3. A lorry of mass \(M\) moves along a straight horizontal road against a constant resistance of magnitude \(R\). The engine of the lorry works at a constant rate \(R U\), where \(U\) is a constant. At time \(t\), the lorry is moving with speed \(v\).
    4. Show that \(M v \frac { \mathrm {~d} v } { \mathrm {~d} t } = R ( U - v )\).
      (3)
    At time \(t = 0\), the lorry has speed \(\frac { 1 } { 4 } U\) and the time taken by the lorry to attain a speed of \(\frac { 1 } { 3 } U\) is \(\frac { k M U } { R }\), where \(k\) is a constant.
  2. Find the exact value of \(k\).
    (7)
Edexcel M4 2007 June Q3
12 marks Challenging +1.2
3. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{0760e668-c87a-4159-a59c-f880640d14e7-3_479_444_214_815} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} A framework consists of two uniform rods \(A B\) and \(B C\), each of mass \(m\) and length \(2 a\), joined at \(B\). The mid-points of the rods are joined by a light rod of length \(a \sqrt { } 2\), so that angle \(A B C\) is a right angle. The framework is free to rotate in a vertical plane about a fixed smooth horizontal axis. This axis passes through the point \(A\) and is perpendicular to the plane of the framework. The angle between the \(\operatorname { rod } A B\) and the downward vertical is denoted by \(\theta\), as shown in Fig. 1.
  1. Show that the potential energy of the framework is $$- m g a ( 3 \cos \theta + \sin \theta ) + \text { constant } .$$
  2. Find the value of \(\theta\) when the framework is in equilibrium, with \(B\) below the level of \(A\).
  3. Determine the stability of this position of equilibrium.
Edexcel M4 2007 June Q4
13 marks Challenging +1.2
4. At 12 noon, \(\operatorname { ship } A\) is 20 km from ship \(B\), on a bearing of \(300 ^ { \circ }\). Ship \(A\) is moving at a constant speed of \(15 \mathrm {~km} \mathrm {~h} ^ { - 1 }\) on a bearing of \(070 ^ { \circ }\). Ship \(B\) moves in a straight line with constant speed \(V \mathrm {~km} \mathrm {~h} ^ { - 1 }\) and intercepts \(A\).
  1. Find, giving your answer to 3 significant figures, the minimum possible for \(V\). It is now given that \(V = 13\).
  2. Explain why there are two possible times at which ship \(B\) can intercept ship \(A\).
  3. Find, giving your answer to the nearest minute, the earlier time at which ship \(B\) can intercept ship \(A\).
    (8)
Edexcel M4 2007 June Q5
13 marks Challenging +1.2
5. A smooth uniform sphere \(A\) has mass \(2 m \mathrm {~kg}\) and another smooth uniform sphere \(B\), with the same radius as \(A\), has mass \(m \mathrm {~kg}\). The spheres are moving on a smooth horizontal plane when they collide. At the instant of collision the line joining the centres of the spheres is parallel to j. Immediately after the collision, the velocity of \(A\) is ( \(3 \mathbf { i } - \mathbf { j }\) ) \(\mathrm { m } \mathrm { s } ^ { - 1 }\) and the velocity of \(B\) is \(( 2 \mathbf { i } + \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\). The coefficient of restitution between the spheres is \(\frac { 1 } { 2 }\).
  1. Find the velocities of the two spheres immediately before the collision.
  2. Find the magnitude of the impulse in the collision.
  3. Find, to the nearest degree, the angle through which the direction of motion of \(A\) is deflected by the collision.
Edexcel M4 2007 June Q6
17 marks Challenging +1.2
6. A small ball is attached to one end of a spring. The ball is modelled as a particle of mass 0.1 kg and the spring is modelled as a light elastic spring \(A B\), of natural length 0.5 m and modulus of elasticity 2.45 N . The particle is attached to the end \(B\) of the spring. Initially, at time \(t = 0\), the end \(A\) is held at rest and the particle hangs at rest in equilibrium below \(A\) at the point \(E\). The end \(A\) then begins to move along the line of the spring in such a way that, at time \(t\) seconds, \(t \leq 1\), the downward displacement of \(A\) from its initial position is 2 sin \(2 t\) metres. At time \(t\) seconds, the extension of the spring is \(x\) metres and the displacement of the particle below \(E\) is \(y\) metres.
  1. Show, by referring to a simple diagram, that \(y + 0.2 = x + 2 \sin 2 t\).
  2. Hence show that \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} t ^ { 2 } } + 49 y = 98 \sin 2 t\). Given that \(y = \frac { 98 } { 45 } \sin 2 t\) is a particular integral of this differential equation,
  3. find \(y\) in terms of \(t\).
  4. Find the time at which the particle first comes to instantaneous rest.
Edexcel M4 2008 June Q1
5 marks Standard +0.3
  1. \hspace{0pt} [In this question \(\mathbf { i }\) and \(\mathbf { j }\) are unit vectors due east and due north respectively.]
A ship \(P\) is moving with velocity ( \(5 \mathbf { i } - 4 \mathbf { j }\) ) \(\mathrm { km } \mathrm { h } ^ { - 1 }\) and a ship \(Q\) is moving with velocity \(( 3 \mathbf { i } + 7 \mathbf { j } ) \mathrm { km } \mathrm { h } ^ { - 1 }\). Find the direction that ship \(Q\) appears to be moving in, to an observer on ship \(P\), giving your answer as a bearing.
Edexcel M4 2008 June Q2
5 marks Standard +0.3
2. Two small smooth spheres \(A\) and \(B\) have equal radii. The mass of \(A\) is \(2 m \mathrm {~kg}\) and the mass of \(B\) is \(m \mathrm {~kg}\). The spheres are moving on a smooth horizontal plane and they collide. Immediately before the collision the velocity of \(A\) is \(( 2 \mathbf { i } - 2 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) and the velocity of \(B\) is \(( - 3 \mathbf { i } - \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\). Immediately after the collision the velocity of \(A\) is \(( \mathbf { i } - 3 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\). Find the speed of \(B\) immediately after the collision.
(5)
Edexcel M4 2008 June Q3
8 marks Challenging +1.2
3. At time \(t = 0\), a particle of mass \(m\) is projected vertically downwards with speed \(U\) from a point above the ground. At time \(t\) the speed of the particle is \(v\) and the magnitude of the air resistance is modelled as being \(m k v\), where \(k\) is a constant. Given that \(U < \frac { \boldsymbol { g } } { \mathbf { 2 } \boldsymbol { k } }\), find, in terms of \(k , U\) and \(g\), the time taken for the particle to double its speed.
(8)
Edexcel M4 2008 June Q4
8 marks Standard +0.8
4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{376d12ab-022c-4070-a1e0-88eacc2fe48e-2_451_357_1672_852} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} A small smooth ball \(B\), moving on a horizontal plane, collides with a fixed vertical wall. Immediately before the collision the angle between the direction of motion of \(B\) and the wall is \(2 \theta\), where \(0 ^ { \circ } < \theta < 45 ^ { \circ }\). Immediately after the collision the angle between the direction of motion of \(B\) and the wall is \(\theta\), as shown in Figure 1. Given that the coefficient of restitution between \(B\) and the wall is \(\frac { 3 } { 8 }\), find the value of \(\tan \theta\).
(8)
Edexcel M4 2008 June Q5
15 marks Challenging +1.2
5. A light elastic spring has natural length \(l\) and modulus of elasticity \(m g\). One end of the spring is fixed to a point \(O\) on a rough horizontal table. The other end is attached to a particle \(P\) of mass \(m\) which is at rest on the table with \(O P = l\). At time \(t = 0\) the particle is projected with speed \(\sqrt { } ( g l )\) along the table in the direction \(O P\). At time \(t\) the displacement of \(P\) from its initial position is \(x\) and its speed is \(v\). The motion of \(P\) is subject to air resistance of magnitude \(2 m v \omega\), where \(\omega = \sqrt { \frac { g } { l } }\). The coefficient of friction between \(P\) and the table is 0.5 .
  1. Show that, until \(P\) first comes to rest, $$\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } + 2 \omega \frac { \mathrm {~d} x } { \mathrm {~d} t } + \omega ^ { 2 } x = - 0.5 g$$
  2. Find \(x\) in terms of \(t , l\) and \(\omega\).
  3. Hence find, in terms of \(\omega\), the time taken for \(P\) to first come to instantaneous rest.
    (3)
Edexcel M4 2008 June Q6
16 marks Challenging +1.8
6. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{376d12ab-022c-4070-a1e0-88eacc2fe48e-4_448_803_242_630} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} A river is 30 m wide and flows between two straight parallel banks. At each point of the river, the direction of flow is parallel to the banks. At time \(t = 0\), a boat leaves a point \(O\) on one bank and moves in a straight line across the river to a point \(P\) on the opposite bank. Its path \(O P\) is perpendicular to both banks and \(O P = 30 \mathrm {~m}\), as shown in Figure 2. The speed of flow of the river, \(r \mathrm {~m} \mathrm {~s} ^ { - 1 }\), at a point on \(O P\) which is at a distance \(x \mathrm {~m}\) from \(O\), is modelled as $$r = \frac { 1 } { 10 } x , \quad 0 \leq x \leq 30$$ The speed of the boat relative to the water is constant at \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). At time \(t\) seconds the boat is at a distance \(x \mathrm {~m}\) from \(O\) and is moving with speed \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in the direction \(O P\).
  1. Show that $$100 v ^ { 2 } = 2500 - x ^ { 2 }$$
  2. Hence show that $$\frac { \mathbf { d } ^ { 2 } x } { \mathbf { d } t ^ { 2 } } + \frac { x } { 100 } = 0$$
  3. Find the total time taken for the boat to cross the river from \(O\) to \(P\).
    (9)
Edexcel M4 2008 June Q7
18 marks Challenging +1.2
7. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{376d12ab-022c-4070-a1e0-88eacc2fe48e-5_917_814_303_587} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} A uniform rod \(A B\), of length \(2 a\) and mass \(k M\) where \(k\) is a constant, is free to rotate in a vertical plane about the fixed point \(A\). One end of a light inextensible string of length \(6 a\) is attached to the end \(B\) of the rod and passes over a small smooth pulley which is fixed at the point \(P\). The line \(A P\) is horizontal and of length \(2 a\). The other end of the string is attached to a particle of mass \(M\) which hangs vertically below the point \(P\), as shown in Figure 3. The angle \(P A B\) is \(2 \theta\), where \(0 ^ { \circ } \leq \theta \leq 180 ^ { \circ }\).
  1. Show that the potential energy of the system is $$M g a ( 4 \sin \theta - k \sin 2 \theta ) + \text { constant. }$$ The system has a position of equilibrium when \(\cos \theta = \frac { 3 } { 4 }\).
  2. Find the value of \(k\).
  3. Hence find the value of \(\cos \theta\) at the other position of equilibrium.
  4. Determine the stability of each of the two positions of equilibrium.