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Edexcel M5 2015 June Q2
8 marks Standard +0.8
2. A particle \(P\) moves so that its position vector, \(\mathbf { r }\) metres, at time \(t\) seconds, where \(0 \leqslant t < \frac { \pi } { 2 }\), satisfies the differential equation $$\frac { \mathrm { d } \mathbf { r } } { \mathrm {~d} t } - ( \tan t ) \mathbf { r } = ( \sin t ) \mathbf { i }$$ When \(t = 0 , \mathbf { r } = - \frac { 1 } { 2 } \mathbf { i }\).
Find \(\mathbf { r }\) in terms of \(t\).
Edexcel M5 2015 June Q3
12 marks Challenging +1.2
A rigid body is in equilibrium under the action of three forces \(\mathbf { F } _ { 1 } , \mathbf { F } _ { 2 }\) and \(\mathbf { F } _ { 3 } \mathbf { F } _ { 1 }\) and \(\mathbf { F } _ { 2 }\) act at the points with position vectors \(\mathbf { r } _ { 1 }\) and \(\mathbf { r } _ { 2 }\) respectively, where \(\mathbf { F } _ { 1 } = ( 2 \mathbf { j } + \mathbf { k } ) \mathrm { N } \quad \mathbf { r } _ { 1 } = ( \mathbf { i } + 2 \mathbf { j } + 2 \mathbf { k } ) \mathrm { m } \mathbf { F } _ { 2 } = ( - 2 \mathbf { i } - \mathbf { j } ) \mathrm { N } \quad \mathbf { r } _ { 2 } = ( - \mathbf { i } - \mathbf { j } + \mathbf { k } ) \mathrm { m }\)
  1. Find the magnitude of \(\mathbf { F } _ { 3 }\)
  2. Find a vector equation of the line of action of \(\mathbf { F } _ { 3 }\), giving your answer in the form \(\mathbf { r } = \mathbf { a } + t \mathbf { b }\), where \(\mathbf { a }\) and \(\mathbf { b }\) are constant vectors and \(t\) is a scalar parameter. \includegraphics[max width=\textwidth, alt={}, center]{cac4dd38-796c-414b-9b80-fe39ab12d41b-11_62_49_2643_1886}
Edexcel M5 2015 June Q4
12 marks Challenging +1.8
4. A particle \(P\), whose initial mass is \(m _ { 0 }\), is projected vertically upwards from the ground at time \(t = 0\) with speed \(\frac { g } { k }\), where \(k\) is a constant. As the particle moves upwards it gains mass by picking up small droplets of moisture from the atmosphere. The droplets are at rest before they are picked up. At time \(t\) the speed of \(P\) is \(v\) and its mass has increased to \(m _ { 0 } \mathrm { e } ^ { k t }\). Assuming that, during the motion, the acceleration due to gravity is constant,
  1. show that, while \(P\) is moving upwards, $$k v + \frac { \mathrm { d } v } { \mathrm {~d} t } = - g$$
  2. find, in terms of \(m _ { 0 }\), the mass of \(P\) when it reaches its greatest height above the ground.
    (6)
Edexcel M5 2015 June Q5
9 marks Challenging +1.8
  1. A uniform circular disc, of mass \(m\) and radius \(a\), is free to rotate about a fixed smooth horizontal axis \(L\). The axis \(L\) is a tangent to the disc at the point \(A\). The centre \(O\) of the disc moves in a vertical plane that is perpendicular to \(L\).
The disc is held at rest with its plane horizontal and released.
  1. Find the angular acceleration of the disc when it has turned through an angle of \(\frac { \pi } { 3 }\)
  2. Find the magnitude of the component, in a direction perpendicular to the disc, of the force of the axis \(L\) acting on the disc at \(A\), when the disc has turned through an angle of \(\frac { \pi } { 3 }\)
Edexcel M5 2015 June Q6
16 marks Challenging +1.3
A pendulum is modelled as a uniform rod \(A B\), of mass \(3 m\) and length \(2 a\), which has a particle of mass \(2 m\) attached at \(B\). The pendulum is free to rotate in a vertical plane about a fixed smooth horizontal axis \(L\) which passes through \(A\). The vertical plane is perpendicular to the axis \(L\).
  1. Find the period of small oscillations of the pendulum about its position of stable equilibrium. The pendulum is hanging at rest in a vertical position, with \(B\) below \(A\), when it is given a horizontal impulse of magnitude \(J\). The impulse acts at \(B\) in a vertical plane which is perpendicular to the axis \(L\). Given that the pendulum turns through an angle of \(60 ^ { \circ }\) before first coming to instantaneous rest,
  2. find \(J\).
Edexcel M5 2015 June Q7
12 marks Hard +2.3
7.
  1. Find, using integration, the moment of inertia of a uniform solid hemisphere, of mass \(m\) and radius \(a\), about a diameter of its plane face.
    [0pt] [You may assume, without proof, that the moment of inertia of a uniform circular disc, of mass \(m\) and radius \(r\), about a diameter is \(\frac { 1 } { 4 } m r ^ { 2 }\).]
  2. Hence find the moment of inertia of a uniform solid sphere, of mass \(M\) and radius \(a\), about a diameter.
Edexcel M5 2016 June Q1
7 marks Standard +0.8
  1. \hspace{0pt} [In this question, \(\mathbf { i }\) and \(\mathbf { j }\) are perpendicular unit vectors in a horizontal plane.]
A bead \(P\) of mass 0.4 kg is threaded on a smooth straight horizontal wire. The wire lies along the line with vector equation \(\mathbf { r } = ( \mathbf { i } + 2 \mathbf { j } ) + \lambda ( - 2 \mathbf { i } + 3 \mathbf { j } )\). The bead is initially at rest at the point \(A\) with position vector \(( - \mathbf { i } + 5 \mathbf { j } ) \mathrm { m }\). A constant horizontal force \(( 0.5 \mathbf { i } + \mathbf { j } ) \mathrm { N }\) acts on \(P\) and moves it along the wire to the point \(B\). At \(B\) the speed of \(P\) is \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find the position vector of \(B\).
Edexcel M5 2016 June Q2
13 marks Challenging +1.3
  1. A particle \(P\) is moving in a plane. At time \(t\) seconds the position vector of \(P\) is \(\mathbf { r }\) metres and the velocity of \(P\) is \(\mathbf { v } \mathrm { m } \mathrm { s } ^ { - 1 }\). When \(t = \frac { \pi } { 2 } , P\) is instantaneously at rest at the point with position vector \(( \mathbf { i } - \mathbf { j } ) \mathrm { m }\).
Given that \(\mathbf { r }\) satisfies the differential equation $$\frac { \mathrm { d } ^ { 2 } \mathbf { r } } { \mathrm {~d} t ^ { 2 } } + 4 \mathbf { r } = ( 3 \sin t ) \mathbf { i }$$ find \(\mathbf { v }\) in terms of \(t\).
(13)
Edexcel M5 2016 June Q3
11 marks Standard +0.8
  1. Three forces \(\mathbf { F } _ { 1 } , \mathbf { F } _ { 2 }\) and \(\mathbf { F } _ { 3 }\) act on a rigid body at the points with position vectors \(\mathbf { r } _ { 1 } , \mathbf { r } _ { 2 }\) and \(\mathbf { r } _ { 3 }\) respectively, where \(\mathbf { F } _ { 1 } = ( 2 \mathbf { j } - \mathbf { k } ) \mathrm { N }\) \(\mathbf { F } _ { 3 } = ( \mathbf { i } + \mathbf { j } ) \mathrm { N }\) \(\mathbf { r } _ { 1 } = ( 4 \mathbf { j } - \mathbf { k } ) \mathrm { m }\) \(\mathbf { r } _ { 3 } = ( 3 \mathbf { i } + \mathbf { j } + \mathbf { k } ) \mathrm { m }\) \(\mathbf { F } _ { 1 } = ( 2 \mathbf { j } - \mathbf { k } ) \mathrm { N }\) \(\mathbf { r } _ { 1 } = ( 4 \mathbf { j } - \mathbf { k } ) \mathrm { m }\)
$$\begin{aligned} & \mathbf { F } _ { 2 } = ( \mathbf { i } + \mathbf { k } ) \mathrm { N } \\ & \mathbf { r } _ { 2 } = ( 2 \mathbf { i } + \mathbf { k } ) \mathrm { m } \end{aligned}$$ j The system of the three forces is equivalent to a single force \(\mathbf { R }\) acting through the point with position vector \(( \mathbf { i } - \mathbf { j } + \mathbf { k } ) \mathrm { m }\), together with a couple of moment \(\mathbf { G }\).
  1. Find \(\mathbf { R }\).
  2. Find \(\mathbf { G }\). respectively, where The
Edexcel M5 2016 June Q4
10 marks Challenging +1.8
4. Find, using integration, the moment of inertia of a uniform cylindrical shell of radius \(r\), height \(h\) and mass \(M\), about a diameter of one end.
(10)
Edexcel M5 2016 June Q5
11 marks Challenging +1.2
5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{f932d7cb-1299-41d1-8248-cfbf639795ed-08_613_649_221_644} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} A uniform piece of wire \(A B C\), of mass \(2 m\) and length \(4 a\), is bent into two straight equal portions, \(A B\) and \(B C\), which are at right angles to each other, as shown in Figure 1. The wire rotates freely in a vertical plane about a fixed smooth horizontal axis \(L\) which passes through \(A\) and is perpendicular to the plane of the wire.
  1. Show that the moment of inertia of the wire about \(L\) is \(\frac { 20 m a ^ { 2 } } { 3 }\)
  2. By writing down an equation of rotational motion for the wire as it rotates about \(L\), find the period of small oscillations of the wire about its position of stable equilibrium.
Edexcel M5 2016 June Q6
12 marks Challenging +1.8
6. A firework rocket, excluding its fuel, has mass \(m _ { 0 } \mathrm {~kg}\). The rocket moves vertically upwards by ejecting burnt fuel vertically downwards with constant speed \(u \mathrm {~m} \mathrm {~s} ^ { - 1 } , u > 24.5\), relative to the rocket. The rocket starts from rest on the ground at time \(t = 0\). At time \(t\) seconds, \(t \leqslant 2\), the speed of the rocket is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the mass of the rocket including its fuel is \(m _ { 0 } ( 5 - 2 t ) \mathrm { kg }\). It is assumed that air resistance is negligible and the acceleration due to gravity is constant.
  1. Show that, for \(t \leqslant 2\) $$\frac { \mathrm { d } v } { \mathrm {~d} t } = \frac { 2 u } { 5 - 2 t } - 9.8$$
  2. Find the speed of the rocket at the instant when all of its fuel has been burnt.
Edexcel M5 2016 June Q7
11 marks Challenging +1.8
7. A uniform square lamina \(P Q R S\), of mass \(m\) and side \(2 a\), is free to rotate about a fixed smooth horizontal axis which passes through \(P\) and \(Q\). The lamina hangs at rest in a vertical plane with \(S R\) below \(P Q\) and is given a horizontal impulse of magnitude \(J\) at the midpoint of \(S R\). The impulse is perpendicular to \(S R\).
  1. Find the initial angular speed of the lamina.
  2. Find the magnitude of the angular deceleration of the lamina at the instant when the lamina has turned through \(\frac { \pi } { 6 }\) radians.
  3. Find the magnitude of the component of the force exerted on the lamina by the axis, in a direction perpendicular to the lamina, at the instant when the lamina has turned through \(\frac { \pi } { 6 }\) radians. \includegraphics[max width=\textwidth, alt={}, center]{f932d7cb-1299-41d1-8248-cfbf639795ed-12_2255_50_315_1978}
Edexcel M5 2017 June Q1
7 marks Standard +0.3
  1. \hspace{0pt} [In this question, \(\mathbf { i }\) and \(\mathbf { j }\) are perpendicular unit vectors in a horizontal, \(x - y\) plane.]
A bead \(P\) of mass 0.08 kg is threaded on a smooth straight horizontal wire which lies along the line with equation \(y = 2 x - 1\). The unit of length on both axes is the metre. Initially the bead is at rest at the point \(( a , b )\). A force \(( 6 \mathbf { i } - 2 \mathbf { j } ) \mathrm { N }\) acts on \(P\) and moves it along the wire so that \(P\) passes through the point \(( 5,9 )\) with speed \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find the value of \(a\) and the value of \(b\).
Edexcel M5 2017 June Q2
10 marks Standard +0.3
2. [In this question, \(\mathbf { i }\) and \(\mathbf { j }\) are perpendicular unit vectors in a horizontal plane and \(\mathbf { k }\) is a unit vector vertically upwards.] A particle of mass 2 kg moves under the action of a constant gravitational force \(- 19.6 \mathbf { k } \mathrm {~N}\). The particle is subject to a resistive force \(- \mathbf { v }\) newtons, where \(\mathbf { v } \mathrm { m } \mathrm { s } ^ { - 1 }\) is the velocity of the particle at time \(t\) seconds.
  1. By writing down an equation of motion of the particle, show that \(\mathbf { v }\) satisfies the differential equation $$\frac { \mathrm { d } \mathbf { v } } { \mathrm {~d} t } + 0.5 \mathbf { v } = - 9.8 \mathbf { k }$$ When \(t = 0 , \mathbf { v } = ( 4 \mathbf { i } - 6 \mathbf { j } + 11.6 \mathbf { k } )\)
  2. Find \(\mathbf { v }\) when \(t = \ln 4\)
Edexcel M5 2017 June Q3
15 marks Challenging +1.8
The position vectors of the points \(P\) and \(Q\) on a rigid body are \(( \mathbf { i } - 2 \mathbf { j } + 3 \mathbf { k } ) \mathrm { m }\) and \(( \mathbf { i } - \mathbf { j } + \mathbf { k } ) \mathrm { m }\) respectively, relative to a fixed origin \(O\). A force \(\mathbf { F } _ { 1 }\) of magnitude 6 N acts at \(P\) in the direction \(( \mathbf { i } - 2 \mathbf { j } + 2 \mathbf { k } )\). A force \(\mathbf { F } _ { 2 }\) of magnitude 14 N acts at \(Q\) in the direction \(( 3 \mathbf { i } - 6 \mathbf { j } + 2 \mathbf { k } )\). When a force \(\mathbf { F } _ { 3 }\) acts at \(O\), which is also a point on the rigid body, the system of three forces is equivalent to a couple of moment \(\mathbf { G }\)
  1. Find \(\mathbf { F } _ { 3 }\)
  2. Find G When an additional force \(\mathbf { F } _ { 4 } = ( \mathbf { i } + 3 \mathbf { j } + 4 \mathbf { k } ) \mathrm { N }\) also acts at \(O\), the system of four forces is equivalent to a single force \(\mathbf { R }\).
  3. Write down \(\mathbf { R }\).
  4. Find an equation of the line of action of \(\mathbf { R }\) in the form \(\mathbf { r } = \mathbf { a } + t \mathbf { b }\), where \(\mathbf { a }\) and \(\mathbf { b }\) are constant vectors and \(t\) is a parameter.
Edexcel M5 2017 June Q4
16 marks Challenging +1.8
A uniform lamina \(P Q R\) of mass \(m\) is in the shape of an isosceles triangle, with \(P Q = P R = 5 a\) and \(Q R = 6 a\). The midpoint of \(Q R\) is \(T\).
  1. Show, using integration, that the moment of inertia of the lamina about an axis which passes through \(P\) and is parallel to \(Q R\), is \(8 m a ^ { 2 }\).
  2. Show, using integration, that the moment of inertia of the lamina about an axis which passes through \(P\) and \(T\), is \(1.5 m a ^ { 2 }\).
    [0pt] [You may assume without proof that the moment of inertia of a uniform rod, of mass \(m\) and length \(2 l\), about an axis perpendicular to the rod through its midpoint is \(\frac { 1 } { 3 } m l ^ { 2 }\) ]
    (4) The lamina is now free to rotate in a vertical plane about a fixed smooth horizontal axis \(A\) which passes through \(P\) and is perpendicular to the plane of the lamina. The lamina makes small oscillations about its position of stable equilibrium.
  3. By writing down an equation of rotational motion for the lamina as it rotates about \(A\), find the approximate period of these small oscillations.
Edexcel M5 2017 June Q5
15 marks Challenging +1.2
A uniform rod \(A B\), of mass \(M\) and length \(2 L\), is free to rotate in a vertical plane about a smooth fixed horizontal axis through \(A\). The rod is hanging vertically at rest, with \(B\) below \(A\), when it is struck at its midpoint by a particle of mass \(\frac { 1 } { 2 } M\). Immediately before this impact, the particle is moving with speed \(u\), in a direction which is horizontal and perpendicular to the axis. The particle is brought to rest by the impact and immediately after the impact the rod moves with angular speed \(\omega\).
  1. Show that \(\omega = \frac { 3 u } { 8 L }\) Immediately after the impact, the magnitude of the vertical component of the force exerted on the \(\operatorname { rod }\) at \(A\) by the axis is \(\frac { 3 M g } { 2 }\)
  2. Find \(u\) in terms of \(L\) and \(g\).
  3. Show that the magnitude of the horizontal component of the force exerted on the rod at \(A\) by the axis, immediately after the impact, is zero. The rod first comes to instantaneous rest after it has turned through an angle \(\alpha\).
  4. Find the size of \(\alpha\). \includegraphics[max width=\textwidth, alt={}, center]{3ce3d486-0c4d-4d30-be86-e175b303fda8-19_56_58_2631_1875}
Edexcel M5 2017 June Q6
12 marks Challenging +1.8
6. A small object \(P\), of mass \(m _ { 0 }\), is projected vertically upwards from the ground with speed \(U\). As \(P\) moves upwards it picks up droplets of moisture from the atmosphere. The droplets are at rest immediately before they are picked up. In a model of the motion, \(P\) is modelled as a particle, air resistance is assumed to be negligible and the acceleration due to gravity is assumed to have the constant value of \(g\). When \(P\) is at a height \(x\) above the ground, the combined mass of \(P\) and the moisture is \(m _ { 0 } ( 1 + k x )\), where \(k\) is a constant, and the speed of \(P\) is \(v\).
  1. Show that, while \(P\) is moving upwards $$\frac { \mathrm { d } } { \mathrm {~d} x } \left( v ^ { 2 } \right) + \frac { 2 k v ^ { 2 } } { ( 1 + k x ) } = - 2 g$$ The general solution of this differential equation is given by \(v ^ { 2 } = \frac { A } { ( 1 + k x ) ^ { 2 } } - \frac { 2 g } { 3 k } ( 1 + k x )\),
    where \(A\) is an arbitrary constant. Given that \(U = \sqrt { 2 g h }\) and \(k = \frac { 7 } { 3 h }\)
  2. find, in terms of \(h\), the height of \(P\) above the ground when \(P\) first comes to rest.
Edexcel M5 2018 June Q1
5 marks Standard +0.3
  1. A small bead is threaded on a smooth straight horizontal wire. The wire is modelled as a line with vector equation \(\mathbf { r } = ( 2 + \lambda ) \mathbf { i } + ( 2 \lambda - 1 ) \mathbf { j }\), where the unit of length is the metre. The bead is moved a distance of \(\sqrt { 80 } \mathrm {~m}\) along the wire by a force \(\mathbf { F } = ( 4 \mathbf { i } - 3 \mathbf { j } ) \mathrm { N }\). Find the magnitude of the work done by \(\mathbf { F }\).
    (5)
Edexcel M5 2018 June Q2
11 marks Challenging +1.2
2. Three forces \(\mathbf { F } _ { 1 } = ( a \mathbf { i } + b \mathbf { j } - 2 \mathbf { k } ) \mathrm { N } , \mathbf { F } _ { 2 } = ( - \mathbf { i } + \mathbf { j } - 2 \mathbf { k } ) \mathrm { N }\) and \(\mathbf { F } _ { 3 } = ( - \mathbf { i } - 3 \mathbf { j } + \mathbf { k } ) \mathrm { N }\), where \(a\) and \(b\) are constants, act on a rigid body. The force \(\mathbf { F } _ { 1 }\) acts through the point with position vector \(\mathbf { k } \mathrm { m }\), the force \(\mathbf { F } _ { 2 }\) acts through the point with position vector \(( 3 \mathbf { i } - \mathbf { j } + \mathbf { k } ) \mathrm { m }\) and the force \(\mathbf { F } _ { 3 }\) acts through the point with position vector \(( \mathbf { j } + 2 \mathbf { k } ) \mathrm { m }\). The system of three forces is equivalent to a single force \(\mathbf { R }\) acting through the origin together with a couple of moment \(\mathbf { G }\). The direction of \(\mathbf { R }\) is parallel to the direction of \(\mathbf { G }\). Find the value of \(a\) and the value of \(b\).
Edexcel M5 2018 June Q3
8 marks Challenging +1.2
3. A particle \(P\) moves in the \(x y\)-plane in such a way that its position vector \(\mathbf { r }\) metres at time \(t\) seconds, where \(0 \leqslant t < \pi\), satisfies the differential equation $$\sec ^ { 2 } \left( \frac { 1 } { 2 } t \right) \frac { \mathrm { d } \mathbf { r } } { \mathrm {~d} t } + \sec ^ { 3 } \left( \frac { 1 } { 2 } t \right) \sin \left( \frac { 1 } { 2 } t \right) \mathbf { r } = \sin \left( \frac { 1 } { 2 } t \right) \mathbf { i } + \sec ^ { 2 } \left( \frac { 1 } { 2 } t \right) \mathbf { j }$$ When \(t = 0\), the particle is at the point with position vector \(( - \mathbf { i } + \mathbf { j } ) \mathrm { m }\).
Find \(\mathbf { r }\) in terms of \(t\).
Edexcel M5 2018 June Q4
6 marks Challenging +1.2
4. A uniform lamina of mass \(M \mathrm {~kg}\) is modelled as the region which is bounded by the curve with equation \(y = x ^ { 2 }\), the positive \(x\)-axis and the line with equation \(x = 2\). The unit of length on both axes is the metre. Find the moment of inertia of the lamina about the \(x\)-axis.
(6)
Edexcel M5 2018 June Q5
14 marks Challenging +1.8
5. At time \(t = 0\) a rocket is launched. The rocket has initial mass \(M\), of which mass \(\lambda M\), \(0 < \lambda < 1\), is fuel. The rocket is launched vertically upwards, from rest, from the surface of the Earth. The rocket burns fuel and the burnt fuel is ejected vertically downwards with constant speed \(U\) relative to the rocket. At time \(t\), the rocket has mass \(m\) and velocity \(v\). Ignoring air resistance and any variation in \(g\),
  1. show, from first principles, that until all the fuel is used, $$m \frac { \mathrm {~d} v } { \mathrm {~d} t } + U \frac { \mathrm {~d} m } { \mathrm {~d} t } = - m g$$ The rocket accelerates vertically upwards with constant acceleration \(g\).
  2. Show that \(m = M \mathrm { e } ^ { \frac { - 2 g t } { U } }\)
  3. Find, in terms of \(M , U\) and \(\lambda\), an expression for the kinetic energy of the rocket at the instant when all of the fuel has been used.
Edexcel M5 2018 June Q6
15 marks Challenging +1.2
6. Three equal uniform rods, each of mass \(m\) and length \(2 a\), form the sides of a rigid equilateral triangular frame \(A B C\). The frame is free to rotate in a vertical plane about a fixed smooth horizontal axis \(L\) which passes through \(A\) and is perpendicular to the plane of the frame.
  1. Show that the moment of inertia of the frame about \(L\) is \(6 m a ^ { 2 }\). The frame is held with \(A B\) horizontal and \(C\) below \(A B\), and released from rest. Given that the centre of mass of the frame is two thirds of the way along a median from a vertex,
  2. find the magnitude of the force exerted by the axis on the frame at \(A\) at the instant when the frame is released.