Questions — Edexcel M2 (623 questions)

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Edexcel M2 2023 October Q2
14 marks Standard +0.3
2. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{f180f5f0-43c5-4365-b0d8-7284220b481e-04_784_814_260_646} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows a template where
  • PQUY is a uniform square lamina with sides of length \(4 a\)
  • RSTU is a uniform square lamina with sides of length \(2 a\)
  • VWXY is a uniform square lamina with sides of length \(2 a\)
  • the three squares all lie in the same plane
  • the mass per unit area of \(V W X Y\) is double the mass per unit area of \(P Q U Y\)
  • the mass per unit area of \(R S T U\) is double the mass per unit area of \(P Q U Y\)
  • the distance of the centre of mass of the template from \(P X\) is \(d\)
    1. Show that \(d = \frac { 5 } { 2 } a\)
The template is freely pivoted about \(Q\) and hangs in equilibrium with \(P Q\) at an angle of \(\theta\) to the downward vertical.
  • Find the value of \(\tan \theta\) The mass of the template is \(M\) The template is still freely pivoted about \(Q\), but it is now held in equilibrium, with \(P Q\) vertical, by a horizontal force of magnitude \(F\) which acts on the template at \(X\). The line of action of the force lies in the same plane as the template.
  • Find \(F\) in terms of \(M\) and \(g\)
    •
    Edexcel M2 2023 October Q3
    6 marks Standard +0.3
    3. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{f180f5f0-43c5-4365-b0d8-7284220b481e-08_424_752_246_667} \captionsetup{labelformat=empty} \caption{Figure 2}
    \end{figure} A particle \(Q\) of mass 0.25 kg is moving in a straight line on a smooth horizontal surface with speed \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when it receives an impulse of magnitude \(I \mathrm { Ns }\). The impulse acts parallel to the horizontal surface and at \(60 ^ { \circ }\) to the original direction of motion of \(Q\). Immediately after receiving the impulse, the speed of \(Q\) is \(12 \mathrm {~ms} ^ { - 1 }\) As a result of receiving the impulse, the direction of motion of \(Q\) is turned through \(\alpha ^ { \circ }\), as shown in Figure 2. Find the value of \(I\)
    Edexcel M2 2023 October Q4
    12 marks Standard +0.3
    1. \hspace{0pt} [In this question \(\mathbf { i }\) and \(\mathbf { j }\) are unit vectors, with \(\mathbf { i }\) horizontal and \(\mathbf { j }\) vertical.]
    \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{f180f5f0-43c5-4365-b0d8-7284220b481e-12_278_891_294_587} \captionsetup{labelformat=empty} \caption{Figure 3}
    \end{figure} The fixed points \(A\) and \(B\) lie on horizontal ground.
    At time \(t = 0\), a particle \(P\) is projected from \(A\) with velocity ( \(4 \mathbf { i } + 4 \mathbf { j }\) ) \(\mathrm { ms } ^ { - 1 }\) Particle \(P\) moves freely under gravity and hits the ground at \(B\), as shown in Figure 3 .
    At time \(T _ { 1 }\) seconds, \(P\) is at its highest point above the ground.
    1. Find the value of \(T _ { 1 }\) At time \(t = 0\), a particle \(Q\) is also projected from \(A\) but with velocity \(( 5 \mathbf { i } + 7 \mathbf { j } ) \mathrm { ms } ^ { - 1 }\) Particle \(Q\) moves freely under gravity.
    2. Find the vertical distance between \(Q\) and \(P\) at time \(T _ { 1 }\) seconds, giving your answer to 2 significant figures. At the instant when particle \(P\) reaches \(B\), particle \(Q\) is moving at \(\alpha ^ { \circ }\) below the horizontal.
    3. Find the value of \(\alpha\). At time \(T _ { 2 }\) seconds, the direction of motion of \(Q\) is perpendicular to the initial direction of motion of \(Q\).
    4. Find the value of \(T _ { 2 }\)
    Edexcel M2 2023 October Q5
    13 marks Standard +0.3
    1. A cyclist is travelling on a straight horizontal road and working at a constant rate of 500 W .
    The total mass of the cyclist and her cycle is 80 kg .
    The total resistance to the motion of the cyclist is modelled as a constant force of magnitude 60 N .
    1. Using this model, find the acceleration of the cyclist at the instant when her speed is \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) On the following day, the cyclist travels up a straight road from a point \(A\) to a point \(B\).
      The distance from \(A\) to \(B\) is 20 km .
      Point \(A\) is 500 m above sea level and point \(B\) is 800 m above sea level.
      The cyclist starts from rest at \(A\).
      At the instant she reaches \(B\) her speed is \(8 \mathrm {~ms} ^ { - 1 }\) The total resistance to the motion of the cyclist from non-gravitational forces is modelled as a constant force of magnitude 60 N .
    2. Using this model, find the total work done by the cyclist in the journey from \(A\) to \(B\). Later on, the cyclist is travelling up a straight road which is inclined at an angle \(\alpha\) to the horizontal, where \(\sin \alpha = \frac { 1 } { 20 }\) The cyclist is now working at a constant rate of \(P\) watts and has a constant speed of \(7 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) The total resistance to the motion of the cyclist from non-gravitational forces is again modelled as a constant force of magnitude 60 N .
    3. Using this model, find the value of \(P\)
    Edexcel M2 2023 October Q6
    9 marks Standard +0.3
    6. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{f180f5f0-43c5-4365-b0d8-7284220b481e-20_593_745_246_667} \captionsetup{labelformat=empty} \caption{Figure 4}
    \end{figure} A uniform \(\operatorname { rod } A B\) has length \(8 a\) and weight \(W\).
    The end \(A\) of the rod is freely hinged to a fixed point on a vertical wall.
    A particle of weight \(\frac { 1 } { 4 } W\) is attached to the rod at \(B\).
    A light inelastic string of length \(5 a\) has one end attached to the rod at the point \(C\), where \(A C = 5 a\). The other end of the string is attached to the wall at the point \(D\), where \(D\) is above \(A\) and \(A D = 5 a\), as shown in Figure 4. The rod rests in equilibrium.
    The tension in the string is \(T\).
    1. Show that \(T = \frac { 6 } { 5 } \mathrm {~W}\)
    2. Find, in terms of \(W\), the magnitude of the force exerted on the rod by the hinge at \(A\).
    Edexcel M2 2023 October Q7
    14 marks Standard +0.3
    1. Particle \(P\) has mass \(4 m\) and particle \(Q\) has mass \(2 m\).
    The particles are moving in opposite directions along the same straight line on a smooth horizontal surface. Particle \(P\) collides directly with particle \(Q\).
    Immediately before the collision, the speed of \(P\) is \(2 u\) and the speed of \(Q\) is \(3 u\).
    Immediately after the collision, the speed of \(P\) is \(x\) and the speed of \(Q\) is \(y\).
    The direction of motion of each particle is reversed as a result of the collision.
    The total kinetic energy of \(P\) and \(Q\) after the collision is half of the total kinetic energy of \(P\) and \(Q\) before the collision.
    1. Show that \(y = \frac { 8 } { 3 } u\) The coefficient of restitution between \(P\) and \(Q\) is \(e\).
    2. Find the value of \(e\). After the collision, \(Q\) hits a smooth fixed vertical wall that is perpendicular to the direction of motion of \(Q\). Particle \(Q\) rebounds.
      The coefficient of restitution between \(Q\) and the wall is \(f\).
      Given that there is no second collision between \(P\) and \(Q\),
    3. find the range of possible values of \(f\). Given that \(f = \frac { 1 } { 4 }\)
    4. find, in terms of \(m\) and \(u\), the magnitude of the impulse received by \(Q\) as a result of its impact with the wall.
    Edexcel M2 2018 Specimen Q1
    8 marks Standard +0.2
    1. A car of mass 900 kg is travelling up a straight road inclined at an angle \(\theta\) to the horizontal, where \(\sin \theta = \frac { 1 } { 25 }\). The car is travelling at a constant speed of \(14 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the resistance to motion from non-gravitational forces has a constant magnitude of 800 N . The car takes 10 seconds to travel from \(A\) to \(B\), where \(A\) and \(B\) are two points on the road.
      1. Find the work done by the engine of the car as the car travels from \(A\) to \(B\).
      When the car is at \(B\) and travelling at a speed of \(14 \mathrm {~ms} ^ { - 1 }\) the rate of working of the engine of the car is suddenly increased to \(P \mathrm {~kW}\), resulting in an initial acceleration of the car of \(0.7 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). The resistance to motion from non-gravitational forces still has a constant magnitude of 800 N .
    2. Find the value of \(P\).
      □
    Edexcel M2 2018 Specimen Q2
    10 marks Moderate -0.3
    2. A particle \(P\) of mass 0.7 kg is moving in a straight line on a smooth horizontal surface. The particle \(P\) collides with a particle \(Q\) of mass 1.2 kg which is at rest on the surface. Immediately before the collision the speed of \(P\) is \(6 \mathrm {~ms} ^ { - 1 }\). Immediately after the collision both particles are moving in the same direction. The coefficient of restitution between the particles is \(e\).
    1. Show that \(e < \frac { 7 } { 12 }\) Given that \(e = \frac { 1 } { 4 }\)
    2. find the magnitude of the impulse exerted on \(Q\) in the collision.
    Edexcel M2 2018 Specimen Q3
    11 marks Standard +0.3
    3. At time \(t\) seconds \(( t \geqslant 0 )\) a particle \(P\) has velocity \(\mathbf { v } \mathrm { m } \mathrm { s } ^ { - 1 }\), where $$\mathbf { v } = \left( 6 t ^ { 2 } + 6 t \right) \mathbf { i } + \left( 3 t ^ { 2 } + 24 \right) \mathbf { j }$$ When \(t = 0\) the particle \(P\) is at the origin \(O\). At time \(T\) seconds, \(P\) is at the point \(A\) and \(\mathbf { v } = \lambda ( \mathbf { i } + \mathbf { j } )\), where \(\lambda\) is a constant. Find
    1. the value of \(T\),
    2. the acceleration of \(P\) as it passes through the point \(A\),
    3. the distance \(O A\).
    Edexcel M2 2018 Specimen Q4
    11 marks Standard +0.3
    4. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{f30ed5b8-880e-42de-860e-d1538fa68f11-12_540_1116_251_342} \captionsetup{labelformat=empty} \caption{Figure 1}
    \end{figure} Two particles \(P\) and \(Q\), of mass 2 kg and 4 kg respectively, are connected by a light inextensible string. Initially \(P\) is held at rest at the point \(A\) on a rough fixed plane inclined at \(\alpha\) to the horizontal ground, where \(\sin \alpha = \frac { 3 } { 5 }\). The string passes over a small smooth pulley fixed at the top of the plane. The particle \(Q\) hangs freely below the pulley and 2.5 m above the ground, as shown in Figure 1. The part of the string from \(P\) to the pulley lies along a line of greatest slope of the plane. The system is released from rest with the string taut. At the instant when \(Q\) hits the ground, \(P\) is at the point \(B\) on the plane. The coefficient of friction between \(P\) and the plane is \(\frac { 1 } { 4 }\).
    1. Find the work done against friction as \(P\) moves from \(A\) to \(B\).
    2. Find the total potential energy lost by the system as \(P\) moves from \(A\) to \(B\).
    3. Find, using the work-energy principle, the speed of \(P\) as it passes through \(B\).
    Edexcel M2 2018 Specimen Q5
    10 marks Standard +0.3
    5. Figure 2 The uniform lamina \(A B C D E F\), shown in Figure 2, consists of two identical rectangles with sides of length \(a\) and \(3 a\). The mass of the lamina is \(M\). A particle of mass \(k M\) is attached to the lamina at \(E\). The lamina, with the attached particle, is freely suspended from \(A\) and hangs in equilibrium with \(A F\) at an angle \(\theta\) to the downward vertical. Given that \(\tan \theta = \frac { 4 } { 7 }\), find the value of \(k\). \includegraphics[max width=\textwidth, alt={}, center]{f30ed5b8-880e-42de-860e-d1538fa68f11-16_677_677_244_580}
    VIIIV SIHI NI JIIIM ION OCVIIV SIHI NI JAHAM ION OOVI4V SIHIL NI JIIIM ION OC
    Edexcel M2 2018 Specimen Q6
    11 marks Standard +0.8
    6. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{f30ed5b8-880e-42de-860e-d1538fa68f11-20_757_1264_233_333} \captionsetup{labelformat=empty} \caption{Figure 3}
    \end{figure} A uniform rod \(A B\), of mass \(3 m\) and length \(2 a\), is freely hinged at \(A\) to a fixed point on horizontal ground. A particle of mass \(m\) is attached to the rod at the end \(B\). The system is held in equilibrium by a force \(\mathbf { F }\) acting at the point \(C\), where \(A C = b\). The rod makes an acute angle \(\theta\) with the ground, as shown in Figure 3. The line of action of \(\mathbf { F }\) is perpendicular to the rod and in the same vertical plane as the rod.
    1. Show that the magnitude of \(\mathbf { F }\) is \(\frac { 5 m g a } { b } \cos \theta\) The force exerted on the rod by the hinge at \(A\) is \(\mathbf { \mathbf { R } }\), which acts upwards at an angle \(\phi\) above the horizontal, where \(\phi > \theta\).
    2. Find
      1. the component of \(\mathbf { R }\) parallel to the rod, in terms of \(m , g\) and \(\theta\),
      2. the component of \(\mathbf { R }\) perpendicular to the rod, in terms of \(a , b , m , g\) and \(\theta\).
    3. Hence, or otherwise, find the range of possible values of \(b\), giving your answer in terms of \(a\).
    Edexcel M2 2018 Specimen Q7
    14 marks Standard +0.8
    7. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{f30ed5b8-880e-42de-860e-d1538fa68f11-24_549_1284_258_322} \captionsetup{labelformat=empty} \caption{Figure 4}
    \end{figure} At time \(t = 0\), a particle \(P\) of mass 0.7 kg is projected with speed \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from a fixed point \(O\) at an angle \(\theta ^ { \circ }\) to the horizontal. The particle moves freely under gravity. At time \(t = 2\) seconds, \(P\) passes through the point \(A\) with speed \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and is moving downwards at \(45 ^ { \circ }\) to the horizontal, as shown in Figure 4. Find
    1. the value of \(\theta\),
    2. the kinetic energy of \(P\) as it reaches the highest point of its path. For an interval of \(T\) seconds, the speed, \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), of \(P\) is such that \(v \leqslant 6\)
    3. Find the value of \(T\).
      VIIV STHI NI JINM ION OCVIAV SIHI NI JMAM/ION OCVIAV SIHI NI JIIYM ION OO
    Edexcel M2 Specimen Q1
    6 marks Moderate -0.3
    1 A particle P moves on the x-axis. The acceleration of P at time t seconds, \(\mathrm { t } \geqslant 0\), is \(( 3 \mathrm { t } + 5 ) \mathrm { ms } ^ { - 2 }\) in the positive x -direction. When \(\mathrm { t } = 0\), the velocity of P is \(2 \mathrm {~ms} ^ { - 1 }\) in the positive x -direction. When \(\mathrm { t } = \mathrm { T }\), the velocity of P is \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in the positive x -direction. Find the value of T .
    (6)
    Edexcel M2 Specimen Q2
    8 marks Standard +0.3
    2 A particle \(P\) of mass 0.6 kg is released from rest and slides down a line of greatest slope of a rough plane. The plane is inclined at \(30 ^ { \circ }\) to the horizontal. When P has moved 12 m , its speed is \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Given that friction is the only non-gravitational resistive force acting on P , find
    1. the work done against friction as the speed of \(P\) increases from \(0 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) to \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\),
    2. the coefficient of friction between the particle and the plane.
    Edexcel M2 Specimen Q3
    9 marks Standard +0.3
    3. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{0a4e4cdd-bec4-4059-b9f7-9ce00bc34b71-08_613_629_125_660} \captionsetup{labelformat=empty} \caption{Figure 1}
    \end{figure} A triangular frame is formed by cutting a uniform rod into 3 pieces which are then joined to form a triangle ABC , where \(\mathrm { AB } = \mathrm { AC } = 10 \mathrm {~cm}\) and \(\mathrm { BC } = 12 \mathrm {~cm}\), as shown in Figure 1.
    1. Find the distance of the centre of mass of the frame from \(B C\). The frame has total mass M . A particle of mass M is attached to the frame at the mid-point of BC . The frame is then freely suspended from B and hangs in equilibrium.
    2. Find the size of the angle between BC and the vertical.
    Edexcel M2 Specimen Q4
    8 marks Moderate -0.3
    4. A car of mass 750 kg is moving up a straight road inclined at an angle \(\theta\) to the horizontal, where \(\sin \theta = \frac { 1 } { 15 }\). The resistance to motion of the car from non-gravitational forces has constant magnitude R newtons. The power developed by the car's engine is 15 kW and the car is moving at a constant speed of \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
    1. Show that \(\mathrm { R } = 260\). The power developed by the car's engine is now increased to 18 kW . The magnitude of the resistance to motion from non-gravitational forces remains at 260 N . At the instant when the car is moving up the road at \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) the car's acceleration is a \(\mathrm { m } \mathrm { s } ^ { - 2 }\).
    2. Find the value of a.
    Edexcel M2 Specimen Q5
    9 marks Moderate -0.3
    5. [In this question \(\mathbf { i }\) and \(\mathbf { j }\) are perpendi cular unit vectors in a horizontal plane.] A ball of mass 0.5 kg is moving with velocity \(( 10 \mathbf { i } + 24 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) when it is struck by a bat. Immediately after the impact the ball is moving with velocity \(20 \mathbf { i } \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find
    1. the magnitude of the impulse of the bat on the ball,
    2. the size of the angle between the vector \(\mathbf { i }\) and the impulse exerted by the bat on the ball,
    3. the kinetic energy lost by the ball in the impact.
    Edexcel M2 Specimen Q6
    8 marks Standard +0.3
    6. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{0a4e4cdd-bec4-4059-b9f7-9ce00bc34b71-20_721_958_127_495} \captionsetup{labelformat=empty} \caption{Figure2}
    \end{figure} Figure 2 shows a uniform rod \(A B\) of mass \(m\) and length 4a. The end \(A\) of the rod is freely hinged to a point on a vertical wall. A particle of mass \(m\) is attached to the rod at \(B\). One end of a light inextensible string is attached to the rod at C , where \(\mathrm { AC } = 3 \mathrm { a }\). The other end of the string is attached to the wall at D , where \(\mathrm { AD } = 2 \mathrm { a }\) and D is vertically above A . The rod rests horizontally in equilibrium in a vertical plane perpendicular to the wall and the tension in the string is T .
    1. Show that \(\mathrm { T } = \mathrm { mg } \sqrt { } 13\).
      (5) The particle of mass \(m\) at \(B\) is removed from the rod and replaced by a particle of mass \(M\) which is attached to the rod at B . The string breaks if the tension exceeds \(2 \mathrm { mg } \sqrt { } 13\). Given that the string does not break,
    2. show that \(M \leqslant \frac { 5 } { 2 } m\).
    Edexcel M2 Specimen Q7
    12 marks Standard +0.3
    7. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{0a4e4cdd-bec4-4059-b9f7-9ce00bc34b71-24_629_1029_251_461} \captionsetup{labelformat=empty} \caption{Figure 3}
    \end{figure} A ball is projected with speed \(40 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from a point \(P\) on a cliff above horizontal ground. The point O on the ground is vertically below P and OP is 36 m . The ball is projected at an angle \(\theta ^ { \circ }\) to the horizontal. The point Q is the highest point of the path of the ball and is 12 m above the level of P. The ball moves freely under gravity and hits the ground at the point R , as shown in Figure 3. Find
    1. the value of \(\theta\),
    2. the distance OR ,
    3. the speed of the ball as it hits the ground at R.
    Edexcel M2 Specimen Q8
    15 marks Standard +0.3
    8. A small ball A of mass 3 m is moving with speed u in a straight line on a smooth horizontal table. The ball collides directly with another small ball B of mass m moving with speed \(u\) towards \(A\) along the same straight line. The coefficient of restitution between \(A\) and \(B\) is \(\frac { 1 } { 2 }\). The balls have the same radius and can be modelled as particles.
    1. Find
      1. the speed of A immediately after the collision,
      2. the speed of B immediately after the collision. A fter the collision \(B\) hits a smooth vertical wall which is perpendicular to the direction of motion of \(B\). The coefficient of restitution between \(B\) and the wall is \(\frac { 2 } { 5 }\).
    2. Find the speed of B immediately after hitting the wall.
      (2) The first collision between A and B occurred at a distance 4a from the wall. The balls collide again \(T\) seconds after the first collision.
    3. Show that \(T = \frac { 112 a } { 15 u }\).
    Edexcel M2 2004 January Q1
    5 marks Moderate -0.3
    1. A car of mass 400 kg is moving up a straight road inclined at an angle \(\theta\) to the horizontal, where \(\sin \theta = \frac { 1 } { 14 }\). The resistance to motion of the car from non-gravitational forces is modelled as a constant force of magnitude \(R\) newtons. When the car is moving at a constant speed of \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), the power developed by the car's engine is 10 kW .
    Find the value of \(R\).
    Edexcel M2 2004 January Q2
    9 marks Standard +0.3
    2. A particle \(P\) of mass 0.75 kg is moving under the action of a single force \(\mathbf { F }\) newtons. At time \(t\) seconds, the velocity \(\mathbf { v } \mathrm { m } \mathrm { s } ^ { - 1 }\) of \(P\) is given by $$\mathbf { v } = \left( t ^ { 2 } + 2 \right) \mathbf { i } - 6 t \mathbf { j }$$
    1. Find the magnitude of \(\mathbf { F }\) when \(t = 4\).
      (5) When \(t = 5\), the particle \(P\) receives an impulse of magnitude \(9 \sqrt { } 2 \mathrm { Ns }\) in the direction of the vector \(\mathbf { i } - \mathbf { j }\).
    2. Find the velocity of \(P\) immediately after the impulse.
    Edexcel M2 2004 January Q3
    9 marks Standard +0.3
    3. \begin{figure}[h]
    \captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{fe64e6f1-e36b-465d-a41c-ac834439623b-3_435_832_379_571}
    \end{figure} A particle \(P\) of mass 2 kg is projected from a point \(A\) up a line of greatest slope \(A B\) of a fixed plane. The plane is inclined at an angle of \(30 ^ { \circ }\) to the horizontal and \(A B = 3 \mathrm {~m}\) with \(B\) above \(A\), as shown in Fig. 1. The speed of \(P\) at \(A\) is \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Assuming the plane is smooth,
    1. find the speed of \(P\) at \(B\). The plane is now assumed to be rough. At \(A\) the speed of \(P\) is \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and at \(B\) the speed of \(P\) is \(7 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). By using the work-energy principle, or otherwise,
    2. find the coefficient of friction between \(P\) and the plane.
    Edexcel M2 2004 January Q4
    10 marks Standard +0.8
    4. \begin{figure}[h]
    \captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{fe64e6f1-e36b-465d-a41c-ac834439623b-4_889_741_370_639}
    \end{figure} A uniform ladder, of weight \(W\) and length \(2 a\), rests in equilibrium with one end \(A\) on a smooth horizontal floor and the other end \(B\) on a rough vertical wall. The ladder is in a vertical plane perpendicular to the wall. The coefficient of friction between the wall and the ladder is \(\mu\). The ladder makes an angle \(\theta\) with the floor, where \(\tan \theta = 2\). A horizontal light inextensible string \(C D\) is attached to the ladder at the point \(C\), where \(A C = \frac { 1 } { 2 } a\). The string is attached to the wall at the point \(D\), with \(B D\) vertical, as shown in Fig. 2. The tension in the string is \(\frac { 1 } { 4 } W\). By modelling the ladder as a rod,
    1. find the magnitude of the force of the floor on the ladder,
    2. show that \(\mu \geqslant \frac { 1 } { 2 }\).
    3. State how you have used the modelling assumption that the ladder is a rod.