Questions — Edexcel (9685 questions)

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Edexcel M5 2002 June Q7
17 marks Challenging +1.8
7. A uniform plane circular disc, of mass \(m\) and radius \(a\), hangs in equilibrium from a point \(B\) on its circumference. The disc is free to rotate about a fixed smooth horizontal axis which is in the plane of the disc and tangential to the disc at \(B\). A particle \(P\), of mass \(m\), is moving horizontally with speed \(u\) in a direction which is perpendicular to the plane of the disc. At time \(t = 0 , P\) strikes the disc at its centre and adheres to the disc.
  1. Show that the angular speed of the disc immediately after it has been struck by \(P\) is \(\frac { 4 u } { 9 a }\).
    (6) It is given that \(u ^ { 2 } = \frac { 1 } { 10 } a g\), and that air resistance is negligible.
  2. Find the angle through which the disc turns before it first comes to instantaneous rest. The disc first returns to its initial position at time \(t = T\).
    1. Write down an equation of motion for the disc.
    2. Hence find \(T\) in terms of \(a , g\) and \(m\), using a suitable approximation which should be justified.
Edexcel M5 2003 June Q1
6 marks Challenging +1.2
  1. 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 small smooth ring of mass 0.1 kg is threaded onto a smooth horizontal wire which is parallel to \(( \mathbf { i } + 2 \mathbf { j } )\). The only forces acting on the ring are its weight, the normal reaction from the wire and a constant force \(( \mathbf { i } + 2 \mathbf { j } - 2 \mathbf { k } )\) N. The ring starts from rest at the point \(A\) on the wire, whose position vector relative to a fixed origin is \(( 2 \mathbf { i } - 2 \mathbf { j } - 3 \mathbf { k } ) \mathrm { m }\), and passes through the point \(B\) with speed \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find the position vector of \(B\).
(6)
Edexcel M5 2003 June Q2
9 marks Standard +0.8
2. With respect to a fixed origin \(O\), the position vector, \(\mathbf { r }\) metres, of a particle \(P\) at time \(t\) seconds satisfies $$\frac { \mathrm { d } \mathbf { r } } { \mathrm {~d} t } + \mathbf { r } = ( \mathbf { i } - \mathbf { j } ) \mathrm { e } ^ { - 2 t } .$$ Given that \(P\) is at \(O\) when \(t = 0\), find
  1. \(\mathbf { r }\) in terms of \(t\),
  2. a cartesian equation of the path of \(P\).
Edexcel M5 2003 June Q3
13 marks Challenging +1.2
3. \section*{Figure 1}
\includegraphics[max width=\textwidth, alt={}]{1fa7336c-20aa-45c1-b685-d8e205367227-3_528_755_317_618}
Figure 1 shows a box in the shape of a cuboid \(P Q R S T U V W\) where \(\overrightarrow { P Q } = 3 \mathbf { i }\) metres, \(\overrightarrow { P S } = 4 \mathbf { j }\) metres and \(\overrightarrow { P T } = 3 \mathbf { k }\) metres. A force \(( 4 \mathbf { i } - 2 \mathbf { j } ) \mathrm { N }\) acts at \(Q\), a force \(( 4 \mathbf { i } + 2 \mathbf { j } ) \mathrm { N }\) acts at \(R\), a force \(( - 2 \mathbf { j } + \mathbf { k } ) \mathrm { N }\) acts at \(T\), and a force \(( 2 \mathbf { j } + \mathbf { k } ) \mathrm { N }\) acts at \(W\). Given that these are the only forces acting on the box, find
  1. the resultant force acting on the box,
  2. the resultant vector moment about \(P\) of the four forces acting on the box. When an additional force \(\mathbf { F }\) acts on the box at a point \(X\) on the edge \(P S\), the box is in equilibrium.
  3. Find \(\mathbf { F }\).
  4. Find the length of \(P X\).
Edexcel M5 2003 June Q4
13 marks Challenging +1.8
4. A rocket-driven car propels itself forwards in a straight line on a horizontal track by ejecting burnt fuel backwards at a constant rate \(\lambda \mathrm { kg } \mathrm { s } ^ { - 1 }\) and at a constant speed \(U \mathrm {~m} \mathrm {~s} ^ { - 1 }\) relative to the car. At time \(t\) seconds, the speed of the car is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the total resistance to the motion of the car has magnitude \(k v \mathrm {~N}\), where \(k\) is a positive constant. When \(t = 0\) the total mass of the car, including fuel, is \(M \mathrm {~kg}\). Assuming that at time \(t\) seconds some fuel remains in the car,
  1. show that $$\frac { \mathrm { d } v } { \mathrm {~d} t } = \frac { \lambda U - k v } { M - \lambda t }$$
  2. find the speed of the car at time \(t\) seconds, given that it starts from rest when \(t = 0\) and that \(\lambda = k = 10\).
Edexcel M5 2003 June Q5
16 marks Challenging +1.2
5. A uniform rod \(A B\), of mass \(m\) and length \(2 a\), is free to rotate in a vertical plane about a fixed smooth horizontal axis through \(A\). The rod is hanging in equilibrium with \(B\) below \(A\) when it is hit by a particle of mass \(m\) moving horizontally with speed \(v\) in a vertical plane perpendicular to the axis. The particle strikes the rod at \(B\) and immediately adheres to it.
  1. Show that the angular speed of the rod immediately after the impact is \(\frac { 3 v } { 8 a }\). Given that the rod rotates through \(120 ^ { \circ }\) before first coming to instantaneous rest,
  2. find \(v\) in terms of \(a\) and \(g\).
  3. find, in terms of \(m\) and \(g\), the magnitude of the vertical component of the force acting on the \(\operatorname { rod }\) at \(A\) immediately after the impact.
    (5)
Edexcel M5 2003 June Q6
18 marks Challenging +1.8
6. (a) Prove, using integration, that the moment of inertia of a uniform circular disc, of mass \(m\) and radius \(a\), about an axis through its centre \(O\) perpendicular to the plane of the disc is \(\frac { 1 } { 2 } m a ^ { 2 }\). The line \(A B\) is a diameter of the disc and \(P\) is the mid-point of \(O A\). The disc is free to rotate about a fixed smooth horizontal axis \(L\). The axis lies in the plane of the disc, passes through \(P\) and is perpendicular to \(O A\). A particle of mass \(m\) is attached to the disc at \(A\) and a particle of mass \(2 m\) is attached to the disc at \(B\).
(b) Show that the moment of inertia of the loaded disc about \(L\) is \(\frac { 21 } { 4 } m a ^ { 2 }\). At time \(t = 0 , P B\) makes a small angle with the downward vertical through \(P\) and the loaded disc is released from rest. By obtaining an equation of motion for the disc and using a suitable approximation,
(c) find the time when the loaded disc first comes to instantaneous rest. END
Edexcel M5 2004 June Q1
7 marks Challenging +1.2
  1. Three forces \(\mathbf { F } _ { 1 } , \mathbf { F } _ { 2 }\) and \(\mathbf { F } _ { 3 }\) act on a rigid body. \(\mathbf { F } _ { 1 } = ( 12 \mathbf { i } - 4 \mathbf { j } + 6 \mathbf { k } ) \mathrm { N }\) and acts at the point with position vector \(( 2 \mathbf { i } - 3 \mathbf { j } ) \mathrm { m } , \mathbf { F } _ { 2 } = ( - 3 \mathbf { j } + 2 \mathbf { k } ) \mathrm { N }\) and acts at the point with position vector \(( \mathbf { i } + \mathbf { j } + \mathbf { k } ) \mathrm { m }\). The force \(\mathbf { F } _ { 3 }\) acts at the point with position vector \(( 2 \mathbf { i } - \mathbf { k } ) \mathrm { m }\).
Given that this set of forces is equivalent to a couple, find
  1. \(\mathbf { F } _ { 3 }\),
  2. the magnitude of the couple.
Edexcel M5 2004 June Q2
8 marks Standard +0.3
2. Two constant forces \(\mathbf { F } _ { 1 }\) and \(\mathbf { F } _ { 2 }\) are the only forces acting on a particle \(P\) of mass 2 kg . The particle is initially at rest at the point \(A\) with position vector \(( - 2 \mathbf { i } - \mathbf { j } - 4 \mathbf { k } ) \mathrm { m }\). Four seconds later, \(P\) is at the point \(B\) with position vector \(( 6 \mathbf { i } - 5 \mathbf { j } + 8 \mathbf { k } ) \mathrm { m }\). Given that \(\mathbf { F } _ { 1 } = ( 12 \mathbf { i } - 4 \mathbf { j } + 6 \mathbf { k } ) \mathrm { N }\), find
  1. \(\mathbf { F } _ { 2 }\),
  2. the work done on \(P\) as it moves from \(A\) to \(B\).
Edexcel M5 2004 June Q3
9 marks Standard +0.8
3. A uniform lamina of mass \(m\) is in the shape of a rectangle \(P Q R S\), where \(P Q = 8 a\) and \(Q R = 6 a\).
  1. Find the moment of inertia of the lamina about the edge \(P Q\). \begin{figure}[h]
    \captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{26fef791-e6fb-45a8-89e9-16c4b4a1a4e4-3_336_772_528_642}
    \end{figure} The flap on a letterbox is modelled as such a lamina. The flap is free to rotate about an axis along its horizontal edge \(P Q\), as shown in Fig. 1. The flap is released from rest in a horizontal position. It then swings down into a vertical position.
  2. Show that the angular speed of the flap as it reaches the vertical position is \(\sqrt { \left( \frac { g } { 2 a } \right) }\).
  3. Find the magnitude of the vertical component of the resultant force of the axis \(P Q\) on the flap, as it reaches the vertical position.
Edexcel M5 2004 June Q4
10 marks Challenging +1.2
4. A uniform circular disc, of mass \(m\) and radius \(r\), has a diameter \(A B\). The point \(C\) on \(A B\) is such that \(A C = \frac { 1 } { 2 } r\). The disc can rotate freely in a vertical plane about a horizontal axis through \(C\), perpendicular to the plane of the disc. The disc makes small oscillations in a vertical plane about the position of equilibrium in which \(B\) is below \(A\).
  1. Show that the motion is approximately simple harmonic.
  2. Show that the period of this approximate simple harmonic motion is \(\pi \sqrt { \left( \frac { 6 r } { g } \right) }\). The speed of \(B\) when it is vertically below \(A\) is \(\sqrt { \left( \frac { g r } { 54 } \right) }\). The disc comes to rest when \(C B\) makes an angle \(\alpha\) with the downward vertical.
  3. Find an approximate value of \(\alpha\).
    (3)
Edexcel M5 2004 June Q5
10 marks Challenging +1.8
5. A rocket is launched vertically upwards under gravity from rest at time \(t = 0\). The rocket propels itself upward by ejecting burnt fuel vertically downwards at a constant speed \(u\) relative to the rocket. The initial mass of the rocket, including fuel, is \(M\). At time \(t\), before all the fuel has been used up, the mass of the rocket, including fuel, is \(M ( 1 - k t )\) and the speed of the rocket is \(v\).
  1. Show that \(\frac { \mathrm { d } v } { \mathrm {~d} t } = \frac { k u } { 1 - k t } - g\).
  2. Hence find the speed of the rocket when \(t = \frac { 1 } { 3 k }\).
Edexcel M5 2004 June Q6
15 marks Challenging +1.3
6. A particle \(P\) of mass 2 kg moves in the \(x - y\) plane. At time \(t\) seconds its position vector is \(\mathbf { r }\) metres. When \(t = 0\), the position vector of \(P\) is \(\mathbf { i }\) metres and the velocity of \(P\) is ( \(- \mathbf { i } + \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\). The vector \(\mathbf { r }\) satisfies the differential equation $$\frac { \mathrm { d } ^ { 2 } \mathbf { r } } { \mathrm {~d} t ^ { 2 } } + 2 \frac { \mathrm {~d} \mathbf { r } } { \mathrm {~d} t } + 2 \mathbf { r } = \mathbf { 0 }$$
  1. Find \(\mathbf { r }\) in terms of \(t\).
  2. Show that the speed of \(P\) at time \(t\) is \(\mathrm { e } ^ { - t } \sqrt { 2 } \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  3. Find, in terms of e, the loss of kinetic energy of \(P\) in the interval \(t = 0\) to \(t = 1\).
Edexcel M5 2004 June Q7
16 marks Challenging +1.8
7. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{26fef791-e6fb-45a8-89e9-16c4b4a1a4e4-5_313_1443_317_356}
\end{figure} A body consists of two uniform circular discs, each of mass \(m\) and radius \(a\), with a uniform rod. The centres of the discs are fixed to the ends \(A\) and \(B\) of the rod, which has mass \(3 m\) and length 8a. The discs and the rod are coplanar, as shown in Fig. 2. The body is free to rotate in a vertical plane about a smooth fixed horizontal axis. The axis is perpendicular to the plane of the discs and passes through the point \(O\) of the rod, where \(A O = 3 a\).
  1. Show that the moment of inertia of the body about the axis is \(54 m a ^ { 2 }\). The body is held at rest with \(A B\) horizontal and is then released. When the body has turned through an angle of \(30 ^ { \circ }\), the rod \(A B\) strikes a small fixed smooth peg \(P\) where \(O P = 3 a\). Given that the body rebounds from the peg with its angular speed halved by the impact,
  2. show that the magnitude of the impulse exerted on the body by the peg at the impact is $$9 m \sqrt { \left( \frac { 5 g a } { 6 } \right) } .$$ END
Edexcel M5 2005 June Q1
6 marks Standard +0.3
  1. Two constant forces \(\mathbf { F } _ { 1 }\) and \(\mathbf { F } _ { 2 }\) are the only forces acting on a particle. \(\mathbf { F } _ { 1 }\) has magnitude 9 N and acts in the direction of \(2 \mathbf { i } + \mathbf { j } + 2 \mathbf { k } . \mathbf { F } _ { 2 }\) has magnitude 18 N and acts in the direction of \(\mathbf { i } + 8 \mathbf { j } - 4 \mathbf { k }\).
Find the total work done by the two forces in moving the particle from the point with position vector \(( \mathbf { i } + \mathbf { j } + \mathbf { k } ) \mathrm { m }\) to the point with position vector \(( 3 \mathbf { i } + 2 \mathbf { j } - \mathbf { k } ) \mathrm { m }\).
(Total 6 marks)
Edexcel M5 2005 June Q2
7 marks Standard +0.3
2. At time \(t\) seconds the position vector of a particle \(P\), relative to a fixed origin \(O\), is \(\mathbf { r }\) metres, where \(\mathbf { r }\) satisfies the differential equation $$\frac { \mathrm { d } \mathbf { r } } { \mathrm {~d} t } + 2 \mathbf { r } = 3 \mathrm { e } ^ { - t } \mathbf { j }$$ Given that \(\mathbf { r } = 2 \mathbf { i } - \mathbf { j }\) when \(t = 0\), find \(\mathbf { r }\) in terms of \(t\).
(Total 7 marks)
Edexcel M5 2005 June Q3
9 marks Standard +0.3
3. A system of forces acting on a rigid body consists of two forces \(\mathbf { F } _ { 1 }\) and \(\mathbf { F } _ { 2 }\) acting at a point \(A\) of the body, together with a couple of moment \(\mathbf { G } . \mathbf { F } _ { 1 } = ( \mathbf { i } + 2 \mathbf { j } - \mathbf { k } ) \mathrm { N }\) and \(\mathbf { F } _ { 2 } = ( - 2 \mathbf { i } + \mathbf { j } + 3 \mathbf { k } ) N\). The position vector of the point \(A\) is \(( \mathbf { i } + \mathbf { j } + \mathbf { k } ) \mathrm { m }\) and \(\mathbf { G } = ( 7 \mathbf { i } - 3 \mathbf { j } + 8 \mathbf { k } ) \mathrm { Nm }\). Given that the system is equivalent to a single force \(\mathbf { R }\),
  1. find \(\mathbf { R }\),
  2. find a vector equation for the line of action of \(\mathbf { R }\).
    (Total 9 marks) \section*{4.} \section*{Figure 1}
    \includegraphics[max width=\textwidth, alt={}]{43ce237f-c8e4-428a-b8cd-04673e62abb9-3_896_515_276_772}
    A thin uniform rod \(P Q\) has mass \(m\) and length \(3 a\). A thin uniform circular disc, of mass \(m\) and radius \(a\), is attached to the rod at \(Q\) in such a way that the rod and the diameter \(Q R\) of the disc are in a straight line with \(P R = 5 a\). The rod together with the disc form a composite body, as shown in Figure 1. The body is free to rotate about a fixed smooth horizontal axis \(L\) through \(P\), perpendicular to \(P Q\) and in the plane of the disc.
Edexcel M5 2005 June Q5
12 marks Challenging +1.8
5. A uniform square lamina \(A B C D\), of mass \(m\) and side \(2 a\), 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 lamina. The moment of inertia of the lamina about \(L\) is \(\frac { 8 m a ^ { 2 } } { 3 }\). Given that the lamina is released from rest when the line \(A C\) makes an angle of \(\frac { \pi } { 3 }\) with the downward vertical,
  1. find the magnitude of the vertical component of the force acting on the lamina at \(A\) when the line \(A C\) is vertical. Given instead that the lamina now makes small oscillations about its position of stable equilibrium,
  2. find the period of these oscillations.
    (5)
    (Total 12 marks)
Edexcel M5 2005 June Q6
13 marks Challenging +1.8
6. A rocket-driven car moves along a straight horizontal road. The car has total initial mass \(M\). It propels itself forwards by ejecting mass backwards at a constant rate \(\lambda\) per unit time at a constant speed \(U\) relative to the car. The car starts from rest at time \(t = 0\). At time \(t\) the speed of the car is \(v\). The total resistance to motion is modelled as having magnitude \(k v\), where \(k\) is a constant. Given that \(t < \frac { M } { \lambda }\), show that
  1. \(\frac { \mathrm { d } v } { \mathrm {~d} t } = \frac { \lambda U - k v } { M - \lambda t }\),
  2. \(v = \frac { \lambda U } { k } \left\{ 1 - \left( 1 - \frac { \lambda t } { M } \right) ^ { \frac { k } { \lambda } } \right\}\).
    (6)
    (Total 13 marks)
Edexcel M5 2005 June Q7
17 marks Challenging +1.8
7. A uniform lamina of mass \(m\) is in the shape of an equilateral triangle \(A B C\) of perpendicular height \(h\). The lamina is free to rotate in a vertical plane about a fixed smooth horizontal axis \(L\) through \(A\) and perpendicular to the lamina.
  1. Show, by integration, that the moment of inertia of the lamina about \(L\) is \(\frac { 5 m h ^ { 2 } } { 9 }\). The centre of mass of the lamina is \(G\). The lamina is in equilibrium, with \(G\) below \(A\), when it is given an angular speed \(\sqrt { \left( \frac { 6 g } { 5 h } \right) }\).
  2. Find the angle between \(A G\) and the downward vertical, when the lamina first comes to rest.
  3. Find the greatest magnitude of the angular acceleration during the motion.
    (Total 17 marks)
Edexcel M5 2006 June Q1
6 marks Standard +0.8
  1. (a) Prove, using integration, that the moment of inertia of a uniform rod, of mass \(m\) and length \(2 a\), about an axis perpendicular to the rod through one end is \(\frac { 4 } { 3 } m a ^ { 2 }\).
    (b) Hence, or otherwise, find the moment of inertia of a uniform square lamina, of mass \(M\) and side \(2 a\), about an axis through one corner and perpendicular to the plane of the lamina.
  2. A particle of mass 0.5 kg is at rest at the point with position vector ( \(2 \mathbf { i } + 3 \mathbf { j } - 4 \mathbf { k }\) ) m. The particle is then acted upon by two constant forces \(\mathbf { F } _ { 1 }\) and \(\mathbf { F } _ { 2 }\). These are the only two forces acting on the particle. Subsequently, the particle passes through the point with position vector \(( 4 \mathbf { i } + 5 \mathbf { j } - 5 \mathbf { k } ) \mathrm { m }\) with speed \(12 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Given that \(\mathbf { F } _ { 1 } = ( \mathbf { i } + 2 \mathbf { j } - \mathbf { k } ) \mathrm { N }\), find \(\mathbf { F } _ { 2 }\).
  3. A particle \(P\) moves in the \(x - y\) plane and has position vector \(\mathbf { r }\) metres at time \(t\) seconds. It is given that \(\mathbf { r }\) satisfies the differential equation
$$\frac { \mathrm { d } ^ { 2 } \mathbf { r } } { \mathrm {~d} t ^ { 2 } } = 2 \frac { \mathrm {~d} \mathbf { r } } { \mathrm {~d} t }$$ When \(t = 0 , P\) is at the point with position vector \(3 \mathbf { i }\) metres and is moving with velocity \(\mathbf { j } \mathrm { m } \mathrm { s } ^ { - 1 }\).
(a) Find \(\mathbf { r }\) in terms of \(t\).
(b) Describe the path of \(P\), giving its cartesian equation.
Edexcel M5 2006 June Q4
12 marks Challenging +1.2
4. A force system consists of three forces \(\mathbf { F } _ { 1 } , \mathbf { F } _ { 2 }\) and \(\mathbf { F } _ { 3 }\) acting on a rigid body. \(\mathbf { F } _ { 1 } = ( \mathbf { i } + 2 \mathbf { j } ) \mathrm { N }\) and acts at the point with position vector \(( - \mathbf { i } + 4 \mathbf { j } ) \mathrm { m }\). \(\mathbf { F } _ { 2 } = ( - \mathbf { j } + \mathbf { k } ) \mathrm { N }\) and acts at the point with position vector ( \(\left. 2 \mathbf { i } + \mathbf { j } + \mathbf { k } \right) \mathrm { m }\). \(\mathbf { F } _ { 3 } = ( 3 \mathbf { i } - \mathbf { j } + \mathbf { k } ) \mathrm { N }\) and acts at the point with position vector \(( \mathbf { i } - \mathbf { j } + 2 \mathbf { k } ) \mathrm { m }\).
It is given that this system can be reduced to a single force \(\mathbf { R }\).
  1. Find \(\mathbf { R }\).
    (2)
  2. Find a vector equation of the line of action of \(\mathbf { R }\), giving your answer in the form \(\mathbf { r } = \mathbf { a } + \lambda \mathbf { b }\), where \(\mathbf { a }\) and \(\mathbf { b }\) are constant vectors and \(\lambda\) is a parameter.
    (10)
Edexcel M5 2006 June Q5
12 marks Challenging +1.2
5. A space-ship is moving in a straight line in deep space and needs to reduce its speed from \(U\) to \(V\). This is done by ejecting fuel from the front of the space-ship at a constant speed \(k\) relative to the space-ship. When the speed of the space-ship is \(v\), its mass is \(m\).
  1. Show that, while the space-ship is ejecting fuel, \(\frac { \mathrm { d } m } { \mathrm {~d} v } = \frac { m } { k }\). The initial mass of the space-ship is \(M\).
  2. Find, in terms of \(U , V , k\) and \(M\), the amount of fuel which needs to be used to reduce the speed of the space-ship from \(U\) to \(V\).
    (6)
Edexcel M5 2006 June Q6
12 marks Challenging +1.3
6. A uniform circular disc, of mass \(m\), radius \(a\) and centre \(O\), is free to rotate in a vertical plane about a fixed smooth horizontal axis. The axis passes through the mid-point \(A\) of a radius of the disc.
  1. Find an equation of motion for the disc when the line \(A O\) makes an angle \(\theta\) with the downward vertical through \(A\).
    (5)
  2. Hence find the period of small oscillations of the disc about its position of stable equilibrium. When the line \(A O\) makes an angle \(\theta\) with the downward vertical through \(A\), the force acting on the disc at \(A\) is \(\mathbf { F }\).
  3. Find the magnitude of the component of \(\mathbf { F }\) perpendicular to \(A O\).
    (5)
Edexcel M5 2006 June Q7
14 marks Challenging +1.8
7. Particles \(P\) and \(Q\) have mass \(3 m\) and \(m\) respectively. Particle \(P\) is attached to one end of a light inextensible string and \(Q\) is attached to the other end. The string passes over a circular pulley which can freely rotate in a vertical plane about a fixed horizontal axis through its centre \(O\). The pulley is modelled as a uniform circular disc of mass \(2 m\) and radius \(a\). The pulley is sufficiently rough to prevent the string slipping. The system is at rest with the string taut. A third particle \(R\) of mass \(m\) falls freely under gravity from rest for a distance \(a\) before striking and adhering to \(Q\). Immediately before \(R\) strikes \(Q\), particles \(P\) and \(Q\) are at rest with the string taut.
  1. Show that, immediately after \(R\) strikes \(Q\), the angular speed of the pulley is \(\frac { 1 } { 3 } \sqrt { \left( \frac { g } { 2 a } \right) }\). When \(R\) strikes \(Q\), there is an impulse in the string attached to \(Q\).
  2. Find the magnitude of this impulse. Given that \(P\) does not hit the pulley,
  3. find the distance that \(P\) moves upwards before first coming to instantaneous rest.