Questions M4 (303 questions)

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Edexcel M4 2014 June Q1
  1. A particle \(A\) has constant velocity \(( 3 \mathbf { i } + \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) and a particle \(B\) has constant velocity \(( \mathbf { i } - \mathbf { k } ) \mathrm { m } \mathrm { s } ^ { - 1 }\). At time \(t = 0\) seconds, the position vectors of the particles \(A\) and \(B\) with respect to a fixed origin \(O\) are \(( - 6 \mathbf { i } + 4 \mathbf { j } - 3 \mathbf { k } ) \mathrm { m }\) and \(( - 2 \mathbf { i } + 2 \mathbf { j } + 3 \mathbf { k } ) \mathrm { m }\) respectively.
    1. Show that, in the subsequent motion, the minimum distance between \(A\) and \(B\) is \(4 \sqrt { } 2 \mathrm {~m}\).
    2. Find the position vector of \(A\) at the instant when the distance between \(A\) and \(B\) is a minimum.
    3. A car of mass 1000 kg is moving along a straight horizontal road. The engine of the car is working at a constant rate of 25 kW . When the speed of the car is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), the resistance to motion has magnitude 10 v newtons.
    4. Show that, at the instant when \(v = 20\), the acceleration of the car is \(1.05 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
    5. Find the distance travelled by the car as it accelerates from a speed of \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) to a speed of \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
    6. A small ball is moving on a smooth horizontal plane when it collides obliquely with a smooth plane vertical wall. The coefficient of restitution between the ball and the wall is \(\frac { 1 } { 3 }\). The speed of the ball immediately after the collision is half the speed of the ball immediately before the collision.
    Find the angle through which the path of the ball is deflected by the collision.
Edexcel M4 2014 June Q4
4. At noon two ships \(A\) and \(B\) are 20 km apart with \(A\) on a bearing of \(230 ^ { \circ }\) from \(B\). Ship \(B\) is moving at \(6 \mathrm {~km} \mathrm {~h} ^ { - 1 }\) on a bearing of \(015 ^ { \circ }\). The maximum speed of \(A\) is \(12 \mathrm {~km} \mathrm {~h} ^ { - 1 }\). Ship \(A\) sets a course to intercept \(B\) as soon as possible.
  1. Find the course set by \(A\), giving your answer as a bearing to the nearest degree.
  2. Find the time at which \(A\) intercepts \(B\).
Edexcel M4 2014 June Q5
5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{904c44f8-bd97-4a1d-8eb1-73cb52ddc8c5-07_478_1185_267_376} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Two smooth uniform spheres \(A\) and \(B\) have equal radii. The mass of \(A\) is \(m\) and the mass of \(B\) is \(3 m\). The spheres are moving on a smooth horizontal plane when they collide obliquely. Immediately before the collision, \(A\) is moving with speed \(3 u\) at angle \(\alpha\) to the line of centres and \(B\) is moving with speed \(u\) at angle \(\beta\) to the line of centres, as shown in Figure 1. The coefficient of restitution between the two spheres is \(\frac { 1 } { 5 }\). It is given that \(\cos \alpha = \frac { 1 } { 3 }\) and \(\cos \beta = \frac { 2 } { 3 }\) and that \(\alpha\) and \(\beta\) are both acute angles.
  1. Find the magnitude of the impulse on \(A\) due to the collision in terms of \(m\) and \(u\).
  2. Express the kinetic energy lost by \(A\) in the collision as a fraction of its initial kinetic energy.
Edexcel M4 2014 June Q6
6. A particle of mass \(m \mathrm {~kg}\) is attached to one end of a light elastic string of natural length a metres and modulus of elasticity 5ma newtons. The other end of the string is attached to a fixed point \(O\) on a smooth horizontal plane. The particle is held at rest on the plane with the string stretched to a length \(2 a\) metres and then released at time \(t = 0\). During the subsequent motion, when the particle is moving with speed \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), the particle experiences a resistance of magnitude \(4 m v\) newtons. At time \(t\) seconds after the particle is released, the length of the string is ( \(a + x\) ) metres, where \(0 \leqslant x \leqslant a\).
  1. Show that, from \(t = 0\) until the string becomes slack, $$\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } + 4 \frac { \mathrm {~d} x } { \mathrm {~d} t } + 5 x = 0$$
  2. Hence express \(x\) in terms of \(a\) and \(t\).
  3. Find the speed of the particle at the instant when the string first becomes slack, giving your answer in the form \(k a\), where \(k\) is a constant to be found correct to 2 significant figures.
Edexcel M4 2014 June Q7
7. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{904c44f8-bd97-4a1d-8eb1-73cb52ddc8c5-11_595_552_260_712} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} A bead \(B\) of mass \(m\) is threaded on a smooth circular wire of radius \(r\), which is fixed in a vertical plane. The centre of the circle is \(O\), and the highest point of the circle is \(A\). A light elastic string of natural length \(r\) and modulus of elasticity \(k m g\) has one end attached to the bead and the other end attached to \(A\). The angle between the string and the downward vertical is \(\theta\), and the extension in the string is \(x\), as shown in Figure 2. Given that the string is taut,
  1. show that the potential energy of the system is $$2 m g r \left\{ ( k - 1 ) \cos ^ { 2 } \theta - k \cos \theta \right\} + \text { constant }$$ Given also that \(k = 3\),
  2. find the positions of equilibrium and determine their stability.
    \includegraphics[max width=\textwidth, alt={}, center]{904c44f8-bd97-4a1d-8eb1-73cb52ddc8c5-12_109_127_2480_1818}
Edexcel M4 2015 June Q1
  1. Particles \(P\) and \(Q\) move in a plane with constant velocities. At time \(t = 0\) the position vectors of \(P\) and \(Q\), relative to a fixed point \(O\) in the plane, are \(( 16 \mathbf { i } - 12 \mathbf { j } ) \mathrm { m }\) and \(( - 5 \mathbf { i } + 4 \mathbf { j } ) \mathrm { m }\) respectively. The velocity of \(P\) is \(( \mathbf { i } + 2 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) and the velocity of \(Q\) is \(( 2 \mathbf { i } + \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\)
Find the shortest distance between \(P\) and \(Q\) in the subsequent motion.
Edexcel M4 2015 June Q2
  1. When a woman walks due North at a constant speed of \(4 \mathrm {~km} \mathrm {~h} ^ { - 1 }\), the wind appears to be blowing from due East. When she runs due South at a constant speed of \(8 \mathrm {~km} \mathrm {~h} ^ { - 1 }\), the speed of the wind appears to be \(20 \mathrm {~km} \mathrm {~h} ^ { - 1 }\).
Assuming that the velocity of the wind relative to the earth is constant, find
  1. the speed of the wind,
  2. the direction from which the wind is blowing.
Edexcel M4 2015 June Q3
3. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{44066c44-e366-4f87-b1b2-c5a894e407fa-08_350_1123_258_408} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Two smooth uniform spheres \(A\) and \(B\) with equal radii have masses \(m\) and \(2 m\) respectively. The spheres are moving in opposite directions on a smooth horizontal surface and collide obliquely. Immediately before the collision, \(A\) has speed \(3 u\) with its direction of motion at an angle \(\theta\) to the line of centres, and \(B\) has speed \(u\) with its direction of motion at an angle \(\theta\) to the line of centres, as shown in Figure 1. The coefficient of restitution between the spheres is \(\frac { 1 } { 8 }\) Immediately after the collision, the speed of \(A\) is twice the speed of \(B\).
Find the size of the angle \(\theta\).
Edexcel M4 2015 June Q4
4. A car of mass 900 kg is moving along a straight horizontal road with the engine of the car working at a constant rate of 22.5 kW . At time \(t\) seconds, the speed of the car is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 } ( 0 < v < 30 )\) and the total resistance to the motion of the car has magnitude \(25 v\) newtons.
  1. Show that when the speed of the car is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), the acceleration of the car is $$\frac { 900 - v ^ { 2 } } { 36 v } \mathrm {~m} \mathrm {~s} ^ { - 2 }$$ The time taken for the car to accelerate from \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) to \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is \(T\) seconds.
  2. Show that $$T = 18 \ln \frac { 8 } { 5 }$$
  3. Find the distance travelled by the car as it accelerates from \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) to \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\)
Edexcel M4 2015 June Q5
5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{44066c44-e366-4f87-b1b2-c5a894e407fa-16_193_1367_274_287} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} A particle \(P\) of mass 1.5 kg is attached to the midpoint of a light elastic spring \(A B\), of natural length 2 m and modulus of elasticity 12 N . The end \(A\) of the spring is attached to a fixed point on a smooth horizontal floor. The end \(B\) is held at a point on the floor where \(A B = 6 \mathrm {~m}\). At time \(t = 0 , P\) is at rest on the floor at the point \(O\), where \(A O = 3 \mathrm {~m}\), as shown in Figure 2. The end \(B\) is now moved along the floor in such a way that \(A O B\) remains a straight line and at time \(t\) seconds, \(t \geqslant 0\), $$A B = \left( 6 + \frac { 1 } { 4 } \sin 2 t \right) \mathrm { m }$$ At time \(t\) seconds, \(A P = ( 3 + x ) \mathrm { m }\).
  1. Show that, for \(t \geqslant 0\), $$\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } + 16 x = 2 \sin 2 t$$ The general solution of this differential equation is $$x = C \cos 4 t + D \sin 4 t + \frac { 1 } { 6 } \sin 2 t$$ where \(C\) and \(D\) are constants.
  2. Find the time at which \(P\) first comes to instantaneous rest. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{44066c44-e366-4f87-b1b2-c5a894e407fa-20_705_1104_116_420} \captionsetup{labelformat=empty} \caption{Figure 3}
    \end{figure}
Edexcel M4 2015 June Q6
6. A smooth wire, with ends \(A\) and \(B\), is in the shape of a semicircle of radius \(r\). The line \(A B\) is horizontal and the midpoint of \(A B\) is \(O\). The wire is fixed in a vertical plane. A small ring \(R\) of mass \(2 m\) is threaded on the wire and is attached to two light inextensible strings. One string passes through a small smooth ring fixed at \(A\) and is attached to a particle of mass \(\sqrt { 6 } m\). The other string passes through a small smooth ring fixed at \(B\) and is attached to a second particle of mass \(\sqrt { 6 } \mathrm {~m}\). The particles hang freely under gravity, as shown in Figure 3. The angle between the radius \(O R\) and the downward vertical is \(2 \theta\), where \(- \frac { \pi } { 4 } < \theta < \frac { \pi } { 4 }\)
  1. Show that the potential energy of the system is $$2 m g r ( 2 \sqrt { 3 } \cos \theta - \cos 2 \theta ) + \text { constant }$$
  2. Find the values of \(\theta\) for which the system is in equilibrium.
  3. Determine the stability of the position of equilibrium for which \(\theta > 0\)
Edexcel M4 2015 June Q7
7. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{44066c44-e366-4f87-b1b2-c5a894e407fa-24_494_936_260_536} \captionsetup{labelformat=empty} \caption{Figure 4}
\end{figure} Figure 4 represents the plan view of part of a smooth horizontal floor, where \(A B\) and \(B C\) are smooth vertical walls. The angle between \(A B\) and \(B C\) is \(120 ^ { \circ }\). A ball is projected along the floor towards \(A B\) with speed \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\) on a path at an angle of \(60 ^ { \circ }\) to \(A B\). The ball hits \(A B\) and then hits \(B C\). The ball is modelled as a particle. The coefficient of restitution between the ball and each wall is \(\frac { 1 } { 2 }\)
  1. Show that the speed of the ball immediately after it has hit \(A B\) is \(\frac { \sqrt { 7 } } { 4 } u\). The speed of the ball immediately after it has hit \(B C\) is \(w \mathrm {~m} \mathrm {~s} ^ { - 1 }\)
  2. Find \(w\) in terms of \(u\).
Edexcel M4 2016 June Q1
1. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{b01b3a41-3ed1-4104-b20d-4cfb845df4a1-02_476_835_121_552} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} A smooth uniform sphere \(A\) of mass \(m\) is moving on a smooth horizontal plane when it collides with a second smooth uniform sphere \(B\), which is at rest on the plane. The sphere \(B\) has mass \(4 m\) and the same radius as \(A\). Immediately before the collision the direction of motion of \(A\) makes an angle \(\alpha\) with the line of centres of the spheres, as shown in Figure 1. The direction of motion of \(A\) is turned through an angle of \(90 ^ { \circ }\) by the collision and the coefficient of restitution between the spheres is \(\frac { 1 } { 2 }\) Find the value of \(\tan \alpha\).
1.
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Edexcel M4 2016 June Q2
2. Figure 2 A small spherical ball \(P\) is at rest at the point \(A\) on a smooth horizontal floor. The ball is struck and travels along the floor until it hits a fixed smooth vertical wall at the point \(X\). The angle between \(A X\) and this wall is \(\alpha\), where \(\alpha\) is acute. A second fixed smooth vertical wall is perpendicular to the first wall and meets it in a vertical line through the point \(C\) on the floor. The ball rebounds from the first wall and hits the second wall at the point \(Y\). After \(P\) rebounds from the second wall, \(P\) is travelling in a direction parallel to \(X A\), as shown in Figure 2. The coefficient of restitution between the ball and the first wall is \(e\). The coefficient of restitution between the ball and the second wall is ke. Find the value of \(k\).
2.
\includegraphics[max width=\textwidth, alt={}, center]{b01b3a41-3ed1-4104-b20d-4cfb845df4a1-03_582_645_118_648}
Edexcel M4 2016 June Q3
3. Two straight horizontal roads cross at right angles at the point \(X\). A girl is running, with constant speed \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), due north towards \(X\) on one road. A car is travelling, with constant speed \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), due west towards \(X\) on the other road.
  1. Find the magnitude and direction of the velocity of the car relative to the girl, giving the direction as a bearing.
    (6) At noon the girl is 150 m south of \(X\) and the car is 800 m east of \(X\).
  2. Find the shortest distance between the car and the girl during the subsequent motion.
Edexcel M4 2016 June Q4
4. A particle \(P\) of mass 9 kg moves along the horizontal positive \(x\)-axis under the action of a force directed towards the origin. At time \(t\) seconds, the displacement of \(P\) from \(O\) is \(x\) metres, \(P\) is moving with speed \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the force has magnitude \(16 x\) newtons. The particle \(P\) is also subject to a resistive force of magnitude \(24 v\) newtons.
  1. Show that the equation of motion of \(P\) is $$9 \frac { \mathrm {~d} ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } + 24 \frac { \mathrm {~d} x } { \mathrm {~d} t } + 16 x = 0$$ It is given that the general solution of this differential equation is $$x = \mathrm { e } ^ { - \frac { 4 } { 3 } t } ( A t + B )$$ where \(A\) and \(B\) are arbitrary constants.
    When \(t = \frac { 3 } { 4 } , P\) is travelling towards \(O\) with its maximum speed of \(8 \mathrm { e } ^ { - 1 } \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(x = d\).
  2. Find the value of \(d\).
  3. Find the value of \(x\) when \(t = 0\)
Edexcel M4 2016 June Q5
5. A toy car of mass 0.5 kg is attached to one end \(A\) of a light elastic string \(A B\), of natural length 1.5 m and modulus of elasticity 27 N . Initially the car is at rest on a smooth horizontal floor and the string lies in a straight line with \(A B = 1.5 \mathrm {~m}\). The end \(B\) is moved in a straight horizontal line directly away from the car, with constant speed \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\). At time \(t\) seconds after \(B\) starts to move, the extension of the string is \(x\) metres and the car has moved a distance \(y\) metres. The effect of air resistance on the car can be ignored. By modelling the car as a particle, show that, while the string remains taut,
    1. \(x + y = u t\)
    2. \(\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } + 36 x = 0\)
  2. Hence show that the string becomes slack when \(t = \frac { \pi } { 6 }\)
  3. Find, in terms of \(u\), the speed of the car when \(t = \frac { \pi } { 12 }\)
  4. Find, in terms of \(u\), the distance the car has travelled when it first reaches end \(B\) of the string.
Edexcel M4 2016 June Q6
6. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{b01b3a41-3ed1-4104-b20d-4cfb845df4a1-11_664_786_221_587} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} Figure 3 shows a uniform rod \(A B\), of length \(2 l\) and mass \(4 m\). A particle of mass \(2 m\) is attached to the rod at \(B\). The rod can turn freely in a vertical plane about a fixed smooth horizontal axis through \(A\). One end of a light elastic spring, of natural length \(2 l\) and modulus of elasticity \(k m g\), where \(k > 4\), is attached to the rod at \(B\). The other end of the spring is attached to a fixed point \(C\) which is vertically above \(A\), where \(A C = 2 l\). The angle \(B A C\) is \(2 \theta\), where \(\frac { \pi } { 6 } < \theta \leqslant \frac { \pi } { 2 }\)
  1. Show that the potential energy of the system is $$4 m g l \left\{ ( k - 4 ) \sin ^ { 2 } \theta - k \sin \theta \right\} + \text { constant }$$ Given that there is a position of equilibrium with \(\theta \neq \frac { \pi } { 2 }\)
  2. show that \(k > 8\) Given that \(k = 10\)
  3. determine the stability of this position of equilibrium.
Edexcel M4 2017 June Q1
  1. \hspace{0pt} [In this question the horizontal unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are due east and due north respectively.]
A ship \(A\) has constant velocity \(( 4 \mathbf { i } + 2 \mathbf { j } ) \mathrm { kmh } ^ { - 1 }\) and a ship \(B\) has constant velocity \(( - \mathbf { i } + 3 \mathbf { j } ) \mathrm { km } \mathrm { h } ^ { - 1 }\). At noon, the position vectors of the ships \(A\) and \(B\) with respect to a fixed origin \(O\) are \(( - 2 \mathbf { i } + \mathbf { j } ) \mathrm { km }\) and \(( 5 \mathbf { i } - 2 \mathbf { j } ) \mathrm { km }\) respectively. Find
  1. the time at which the two ships are closest together,
  2. the length of time for which ship \(A\) is within 2 km of ship \(B\).
Edexcel M4 2017 June Q2
2. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{23e6a9ae-bf00-45a3-b462-e18760d9af45-04_912_988_260_470} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Two smooth uniform spheres \(A\) and \(B\) have masses \(3 m \mathrm {~kg}\) and \(m \mathrm {~kg}\) respectively and equal radii. The spheres are moving on a smooth horizontal surface. Initially, sphere \(A\) has velocity \(( 5 \mathbf { i } - 2 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) and sphere \(B\) has velocity \(( 3 \mathbf { i } + 4 \mathbf { j } ) \mathrm { ms } ^ { - 1 }\). When the spheres collide, the line joining their centres is parallel to \(\mathbf { j }\), as shown in Figure 1.
The coefficient of restitution between the two spheres is \(e\).
The kinetic energy of sphere \(B\) immediately after the collision is \(85 \%\) of its kinetic energy immediately before the collision. Find
  1. the velocity of each sphere immediately after the collision,
  2. the value of \(e\).
Edexcel M4 2017 June Q3
3. A cyclist and her bicycle have a combined mass of 75 kg . The cyclist travels along a straight horizontal road. The cyclist produces a constant driving force of magnitude 150 N . At time \(t\) seconds, the speed of the cyclist is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), where \(v < \sqrt { 50 }\). As the cyclist moves, the total resistance to motion of the cyclist and her bicycle has magnitude \(3 v ^ { 2 }\) newtons. The cyclist starts from rest. At time \(t\) seconds, she has travelled a distance \(x\) metres from her starting point. Find
  1. \(v\) in terms of \(x\),
  2. \(t\) in terms of \(v\).
Edexcel M4 2017 June Q4
4. [In this question, the unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are in a vertical plane, \(\mathbf { i }\) being horizontal and \(\mathbf { j }\) being vertically upwards.] A line of greatest slope of a fixed smooth plane is parallel to the vector \(( - 4 \mathbf { i } - 3 \mathbf { j } )\). A particle \(P\) falls vertically and strikes the plane. Immediately before the impact, \(P\) has velocity \(- 7 \mathbf { j } \mathrm {~ms} ^ { - 1 }\). Immediately after the impact, \(P\) has velocity \(( - a \mathbf { i } + \mathbf { j } ) \mathrm { ms } ^ { - 1 }\), where \(a\) is a positive constant.
  1. Show that \(a = 6\)
  2. Find the coefficient of restitution between \(P\) and the plane.
Edexcel M4 2017 June Q5
5. A cyclist riding due north at a steady speed of \(12 \mathrm {~km} \mathrm {~h} ^ { - 1 }\) notices that the wind appears to come from the north-west. At the same time, another cyclist, moving on a bearing of \(120 ^ { \circ }\) and also riding at a steady speed of \(12 \mathrm {~km} \mathrm {~h} ^ { - 1 }\), notices that the wind appears to come from due south. The velocity of the wind is assumed to be constant. Find
  1. the wind speed,
  2. the direction from which the wind is blowing, giving your answer as a bearing.
Edexcel M4 2017 June Q6
6. A particle \(P\) of mass 0.2 kg is suspended from a fixed point by a light elastic spring. The spring has natural length 0.8 m and modulus of elasticity 7 N . At time \(t = 0\) the particle is released from rest from a point 0.2 metres vertically below its equilibrium position. The motion of \(P\) is resisted by a force of magnitude \(2 v\) newtons, where \(v \mathrm {~ms} ^ { - 1 }\) is the speed of \(P\). At time \(t\) seconds, \(P\) is \(x\) metres below its equilibrium position.
  1. Show that \(\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } + 10 \frac { \mathrm {~d} x } { \mathrm {~d} t } + 43.75 x = 0\)
  2. Find \(x\) in terms of \(t\).
  3. Find the value of \(t\) when \(P\) first comes to instantaneous rest.
Edexcel M4 2017 June Q7
7. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{23e6a9ae-bf00-45a3-b462-e18760d9af45-24_655_890_239_529} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Figure 2 shows four uniform rods, each of mass \(m\) and length \(2 a\). The rods are freely hinged at their ends to form a rhombus \(A B C D\). Point \(A\) is attached to a fixed point on a ceiling and the rhombus hangs freely with \(C\) vertically below \(A\). A light elastic spring of natural length \(2 a\) and modulus of elasticity \(7 m g\) connects the points \(A\) and \(C\). A particle of mass \(3 m\) is attached to point \(C\).
  1. Show that, when \(A D\) is at an angle \(\theta\) to the downward vertical, the potential energy \(V\) of the system is given by $$V = 28 m g a \cos ^ { 2 } \theta - 48 m g a \cos \theta + \text { constant }$$ Given that \(\theta > 0\)
  2. find the value of \(\theta\) for which the system is in equilibrium,
  3. determine the stability of this position of equilibrium.