Questions — Edexcel M2 (551 questions)

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Edexcel M2 2021 October Q6
6. Two particles, \(A\) and \(B\), are moving in opposite directions along the same straight line on a smooth horizontal surface when they collide directly.
The mass of \(A\) is \(2 m\) and the mass of \(B\) is \(3 m\).
Immediately after the collision, \(A\) and \(B\) are moving in opposite directions with the same speed \(v\).
In the collision, \(A\) receives an impulse of magnitude \(5 m v\).
  1. Find the coefficient of restitution between \(A\) and \(B\).
    (6) After the collision with \(A\), particle \(B\) strikes a smooth fixed vertical wall and rebounds. The wall is perpendicular to the direction of motion of the particles.
    The coefficient of restitution between \(B\) and the wall is \(f\).
    As a result of its collision with \(A\) and with the wall, the total kinetic energy lost by \(B\) is \(E\). As a result of its collision with \(B\), the kinetic energy lost by \(A\) is \(2 E\).
  2. Find the value of \(f\). \includegraphics[max width=\textwidth, alt={}, center]{80dceee7-2eea-4082-ad20-7b3fe4e8bb25-19_2664_107_106_6}
    "
    ,
    \includegraphics[max width=\textwidth, alt={}, center]{80dceee7-2eea-4082-ad20-7b3fe4e8bb25-19_108_67_2613_1884}
Edexcel M2 2021 October Q7
7. In this question you may use, without proof, the formula for the centre of mass of a uniform sector of a circle, as given in the formulae book. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{80dceee7-2eea-4082-ad20-7b3fe4e8bb25-20_444_625_354_662} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} The uniform lamina \(A B C D E\), shown shaded in Figure 3, is formed by joining a rectangle to a sector of a circle.
  • The rectangle \(A B C E\) has \(A B = E C = a\) and \(A E = B C = d\)
  • The sector \(C D E\) has centre \(C\) and radius \(a\)
  • Angle \(E C D = \frac { \pi } { 3 }\) radians
The centre of mass of the lamina lies on EC.
  1. Show that \(a = \sqrt { 3 } d\) The lamina is freely suspended from \(B\) and hangs in equilibrium with \(B C\) at an angle \(\beta\) radians to the downward vertical.
  2. Find the value of \(\beta\)
Edexcel M2 2021 October Q8
8. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{80dceee7-2eea-4082-ad20-7b3fe4e8bb25-24_470_824_214_561} \captionsetup{labelformat=empty} \caption{Figure 4}
\end{figure} The fixed point \(A\) is \(h\) metres vertically above the point \(O\) that is on horizontal ground. At time \(t = 0\), a particle \(P\) is projected from \(A\) with speed \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The particle moves freely under gravity. At time \(t = 2.5\) seconds, \(P\) strikes the ground at the point \(B\). At the instant when \(P\) strikes the ground, the speed of \(P\) is \(18 \mathrm {~ms} ^ { - 1 }\), as shown in Figure 4.
  1. By considering energy, find the value of \(h\).
  2. Find the distance \(O B\). As \(P\) moves from \(A\) to \(B\), the speed of \(P\) is less than or equal to \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) for \(T\) seconds.
  3. Find the value of \(T\)
Edexcel M2 2022 October Q1
  1. Three particles of masses \(2 m , 3 m\) and \(4 m\) are placed at the points with coordinates \(( - 2,5 ) , ( 2 , - 3 )\) and \(( 3 k , k )\) respectively, where \(k\) is a constant. The centre of mass of the three particles is at the point \(( \bar { x } , \bar { y } )\).
    1. Show that \(\bar { x } = \frac { 2 + 12 k } { 9 }\)
    The centre of mass of the three particles lies at a point on the straight line with equation \(x + 2 y = 3\)
  2. Find the value of \(k\).
Edexcel M2 2022 October Q2
2. A car of mass 900 kg is moving down a straight road which is inclined at an angle \(\alpha\) to the horizontal, where \(\sin \alpha = \frac { 1 } { 12 }\) The engine of the car is working at a constant rate of 15 kW .
The resistance to the motion of the car is modelled as a constant force of magnitude 400 N . Find the acceleration of the car at the instant when it is moving at \(16 \mathrm {~ms} ^ { - 1 }\)
VIAV SIHI NI IIIIIM ION OCVIIIV SIHI NI III IM I O N OCVIIV SIHI NI IIIIM I I ON OC
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Edexcel M2 2022 October Q3
  1. A particle \(P\) of mass 0.2 kg is moving with velocity \(( 4 \mathbf { i } - 3 \mathbf { j } ) \mathrm { ms } ^ { - 1 }\)
The particle receives an impulse \(\lambda ( \mathbf { i } + \mathbf { j } ) \mathrm { Ns }\), where \(\lambda\) is a constant.
Immediately after receiving the impulse, the speed of \(P\) is \(7 \mathrm {~m} \mathrm {~s} ^ { - 1 }\)
Find the possible values of \(\lambda\)
Edexcel M2 2022 October Q4
4. At time \(t\) seconds \(( 0 \leqslant t < 5 )\), a particle \(P\) has velocity \(\mathbf { v m s } ^ { - 1 }\), where $$\mathbf { v } = ( \sqrt { 5 - t } ) \mathbf { i } + \left( t ^ { 2 } + 2 t - 3 \right) \mathbf { j }$$ When \(t = \lambda\), particle \(P\) is moving in a direction parallel to the vector \(\mathbf { i }\).
  1. Find the acceleration of \(P\) when \(t = \lambda\) The position vector of \(P\) is measured relative to the fixed point \(O\) When \(t = 1\), the position vector of \(P\) is \(( - 2 \mathbf { i } + \mathbf { j } ) \mathrm { m }\). Given that \(1 \leqslant T < 5\)
  2. find, in terms of \(T\), the position vector of \(P\) when \(t = T\)
Edexcel M2 2022 October Q5
5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{1732eb73-8c16-4a45-8d3b-a88e659e47ea-12_424_1118_221_420} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} A uniform rod \(A B\) has length \(8 a\) and weight \(W\).
The end \(A\) of the rod is freely hinged to horizontal ground.
The rod rests in equilibrium against a block which is also fixed to the ground.
The block is modelled as a smooth solid hemisphere with radius \(2 a\) and centre \(D\).
The point of contact between the rod and the block is \(C\), where \(A C = 5 a\)
The rod is at an angle \(\theta\) to the ground, as shown in Figure 1.
Points \(A , B , C\) and \(D\) all lie in the same vertical plane.
  1. Show that \(A D = \sqrt { 29 } a\)
  2. Show that the magnitude of the normal reaction at \(C\) between the rod and the block is \(\frac { 4 } { \sqrt { 29 } } W\) The resultant force acting on the rod at \(A\) has magnitude \(k W\) and acts at an angle \(\alpha\) to the ground.
  3. Find (i) the exact value of \(k\)
    (ii) the exact value of \(\tan \alpha\)
    \includegraphics[max width=\textwidth, alt={}, center]{1732eb73-8c16-4a45-8d3b-a88e659e47ea-15_72_1819_2709_114}
Edexcel M2 2022 October Q6
6. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{1732eb73-8c16-4a45-8d3b-a88e659e47ea-16_588_871_219_539} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} The uniform lamina \(P Q R S T U V\) shown in Figure 2 is formed from two identical rectangles, \(P Q U V\) and \(Q R S T U\).
The rectangles have sides \(P Q = R S = 2 a\) and \(P V = Q R = k a\).
  1. Show that the centre of mass of the lamina is \(\left( \frac { 6 + k } { 4 } \right) a\) from \(P V\) The lamina is freely suspended from \(P\) and hangs in equilibrium with \(P R\) at an angle of \(\alpha\) to the downward vertical. Given that \(\tan \alpha = \frac { 7 } { 15 }\)
  2. find the value of \(k\).
Edexcel M2 2022 October Q7
7. Particle \(A\) has mass \(m\) and particle \(B\) has mass \(2 m\). The particles are moving in the same direction along the same straight line on a smooth horizontal surface.
Particle \(A\) collides directly with particle \(B\).
Immediately before the collision, the speed of \(A\) is \(3 u\) and the speed of \(B\) is \(u\).
The coefficient of restitution between \(A\) and \(B\) is \(e\).
    1. Show that the speed of \(B\) immediately after the collision is \(\frac { 5 + 2 e } { 3 } u\)
    2. Find the speed of \(A\) immediately after the collision. After the collision, \(B\) hits a smooth fixed vertical wall that is perpendicular to the direction of motion of \(B\).
      The coefficient of restitution between \(B\) and the wall is \(\frac { 1 } { 3 }\)
      Particle \(B\) rebounds and there is a second collision between \(A\) and \(B\).
      The first collision between \(A\) and \(B\) occurs at a distance \(d\) from the wall.
      The time between the two collisions is \(T\).
      Given that \(e = \frac { 1 } { 2 }\)
  2. find \(T\) in terms of \(d\) and \(u\).
Edexcel M2 2022 October Q8
8. [In this question, the unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are in a vertical plane, with \(\mathbf { i }\) being horizontal and \(\mathbf { j }\) being vertically upwards.] \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{1732eb73-8c16-4a45-8d3b-a88e659e47ea-24_378_1219_347_349} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} A rough ramp is fixed to horizontal ground.
The ramp is inclined to the ground at an angle \(\alpha\), where \(\tan \alpha = \frac { 7 } { 24 }\)
The point \(A\) is at the bottom of the ramp and the point \(B\) is at the top of the ramp. The line \(A B\) is a line of greatest slope of the ramp and \(A B = 15 \mathrm {~m}\), as shown in Figure 3. A particle \(P\) of mass 0.3 kg is projected with speed \(U \mathrm {~ms} ^ { - 1 }\) from \(A\) directly towards \(B\). At the instant \(P\) reaches the point \(B\), the velocity of \(P\) is \(( 24 \mathbf { i } + 7 \mathbf { j } ) \mathrm { ms } ^ { - 1 }\) The particle leaves the ramp at \(B\), and moves freely under gravity until it hits the horizontal ground at the point \(C\).
The coefficient of friction between \(P\) and the ramp is \(\frac { 1 } { 5 }\)
  1. Find the work done against friction as \(P\) moves from \(A\) to \(B\).
  2. Use the work-energy principle to find the value of \(U\).
  3. Find the time taken by \(P\) to move from \(B\) to \(C\). At the instant immediately before \(P\) hits the ground at \(C\), the particle is moving downwards at \(\theta ^ { \circ }\) to the horizontal.
  4. Find the value of \(\theta\)
Edexcel M2 2023 October Q1
  1. At time \(t\) seconds, \(t > 0\), a particle \(P\) is at the point with position vector \(\mathbf { r } \mathrm { m }\), where
$$\mathbf { r } = \left( t ^ { 4 } - 8 t ^ { 2 } \right) \mathbf { i } + \left( 6 t ^ { 2 } - 2 t ^ { \frac { 3 } { 2 } } \right) \mathbf { j }$$
  1. Find the velocity of \(P\) when \(P\) is moving in a direction parallel to the vector \(\mathbf { j }\)
  2. Find the acceleration of \(P\) when \(t = 4\)
Edexcel M2 2023 October Q2
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
    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
    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
    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
    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
    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
    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
    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
    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
    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
    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
    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
    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