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Edexcel M4 Specimen Q3
10 marks Challenging +1.2
3. A ball of mass \(m\) is thrown vertically upwards from the ground. When its speed is \(v\) the magnitude of the air resistance is modelled as being \(m k v ^ { 2 }\), where \(k\) is a positive constant. The ball is projected with speed \(\sqrt { \frac { g } { k } }\). By modelling the ball as a particle,
  1. find the greatest height reached by the ball.
  2. State one physical factor which is ignored in this model.
Edexcel M4 Specimen Q4
11 marks Challenging +1.2
4. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{4737e682-1e1d-4c1a-91c2-7d051cb43aac-3_417_986_303_534}
\end{figure} Two smooth uniform spheres \(A\) and \(B\), of equal radius, are moving on a smooth horizontal plane. Sphere \(A\) has mass 3 kg and velocity \(( 2 \mathbf { i } + \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\), and sphere \(B\) has mass 5 kg and velocity \(( - \mathbf { i } + \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\). When the spheres collide the line joining their centres is parallel to \(\mathbf { i }\), as shown in Fig. 2. Given that the direction of \(A\) is deflected through a right angle by the collision, find
  1. the velocity of \(A\) after the collision,
  2. the coefficient of restitution between the spheres.
Edexcel M4 Specimen Q5
12 marks Challenging +1.2
5. An elastic string spring of modulus \(2 m g\) and natural length \(l\) is fixed at one end. To the other end is attached a mass \(m\) which is allowed to hang in equilibrium. The mass is then pulled vertically downwards through a distance \(l\) and released from rest. The air resistance is modelled as having magnitude \(2 m \omega v\), where \(v\) is the speed of the particle and \(\omega = \sqrt { \frac { g } { l } }\). The particle is at distance \(x\) from its equilibrium position at time \(t\).
  1. Show that \(\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } + 2 \omega \frac { \mathrm {~d} x } { \mathrm {~d} t } + 2 \omega ^ { 2 } x = 0\).
  2. Find the general solution of this differential equation.
  3. Hence find the period of the damped harmonic motion.
Edexcel M4 Specimen Q6
14 marks Standard +0.3
6. Two horizontal roads cross at right angles. One is directed from south to north, and the other from east to west. A tractor travels north on the first road at a constant speed of \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and at noon is 200 m south of the junction. A car heads west on the second road at a constant speed of \(24 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and at noon is 960 m east of the junction.
  1. Find the magnitude and direction of the velocity of the car relative to the tractor.
  2. Find the shortest distance between the car and the tractor.
Edexcel M4 Specimen Q7
16 marks Challenging +1.2
7. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 3} \includegraphics[alt={},max width=\textwidth]{4737e682-1e1d-4c1a-91c2-7d051cb43aac-4_558_1180_845_440}
\end{figure} A uniform rod \(A B\) has mass \(m\) and length \(2 a\). The end \(A\) is smoothly hinged at a fixed point on a fixed straight horizontal wire. A smooth light ring \(R\) is threaded on the wire. The ring \(R\) is attached by a light elastic string, of natural length \(a\) and modulus of elasticity \(m g\), to the end \(B\) of the rod. The end \(B\) is always vertically below \(R\) and angle \(\angle R A B = \theta\), as shown in Fig. 3.
  1. Show that the potential energy of the system is $$m g a \left( 2 \sin ^ { 2 } \theta - 3 \sin \theta \right) + \text { constant }$$ (6)
  2. Hence determine the value of \(\theta , \theta < \frac { \pi } { 2 }\), for which the system is in equilibrium.
  3. Determine whether this position of equilibrium is stable or unstable. END
Edexcel M5 Q1
5 marks Standard +0.8
  1. \hspace{0pt} [In this question \(\mathbf { i }\) and \(\mathbf { j }\) are horizontal unit vectors.]
A small smooth ring of mass 0.5 kg moves along a smooth horizontal wire. The only forces acting on the ring are its weight, the normal reaction from the wire, and a constant force \(( 5 \mathbf { i } + \mathbf { j } - 3 \mathbf { k } ) \mathrm { N }\). The ring is initially at rest at the point with position vector \(( \mathbf { i } + \mathbf { j } + \mathbf { k } ) \mathrm { m }\), relative to a fixed origin. Find the speed of the ring as it passes through the point with position vector \(( 3 \mathbf { i } + \mathbf { k } ) \mathrm { m }\).
Edexcel M5 Q2
6 marks Standard +0.8
2. Three forces, \(\mathbf { F } _ { 1 } , \mathbf { F } _ { 2 }\) and \(\mathbf { F } _ { 3 }\) act on a rigid body. \(\mathbf { F } _ { 1 } = ( 2 \mathbf { i } - \mathbf { j } + 3 \mathbf { k } ) \mathrm { N } , \mathbf { F } _ { 2 } = ( \mathbf { i } + \mathbf { j } - 4 \mathbf { k } )\) N and \(\mathbf { F } _ { 3 } = ( p \mathbf { i } + q \mathbf { j } + r \mathbf { k } ) \mathrm { N }\), where \(p , q\) and \(r\) are constants. All three forces act through the point with position vector \(( 3 \mathbf { i } - 2 \mathbf { j } + \mathbf { k } ) \mathrm { m }\), relative to a fixed origin. The three forces \(\mathbf { F } _ { 1 } , \mathbf { F } _ { 2 }\) and \(\mathbf { F } _ { 3 }\) are equivalent to a single force ( \(5 \mathbf { i } - 4 \mathbf { j } + 2 \mathbf { k }\) ) N , acting at the origin, together with a couple \(\mathbf { G }\).
  1. Find the values of \(p , q\) and \(r\).
  2. Find \(\mathbf { G }\).
Edexcel M5 Q3
7 marks Standard +0.3
3. At time \(t\) seconds, the position vector of a particle \(P\) is \(\mathbf { r }\) metres, relative to a fixed origin. The particle moves in such a way that $$\frac { \mathrm { d } ^ { 2 } \mathbf { r } } { \mathrm {~d} t ^ { 2 } } - 4 \frac { \mathrm {~d} \mathbf { r } } { \mathrm {~d} t } = \mathbf { 0 }$$ At \(t = 0 , P\) is moving with velocity ( \(8 \mathbf { i } - 6 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\).
Find the speed of \(P\) when \(t = \frac { 1 } { 2 } \ln 2\).
Edexcel M5 Q4
9 marks Challenging +1.8
4. A uniform plane lamina of mass \(m\) is in the shape of an equilateral triangle of side \(2 a\). Find, using integration, the moment of inertia of the lamina about one of its edges.
(9)
Edexcel M5 Q5
14 marks Challenging +1.8
5. A rocket is launched vertically upwards from rest. Initially, the total mass of the rocket and its fuel is 1000 kg . The rocket burns fuel at a rate of \(10 \mathrm {~kg} \mathrm {~s} ^ { - 1 }\). The burnt fuel is ejected vertically downwards with a speed of \(2000 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) relative to the rocket, and burning stops after one minute. At time \(t\) seconds, \(t \leq 60\), after the launch, the speed of the rocket is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Air resistance is assumed to be negligible.
  1. Show that $$- 9.8 ( 100 - t ) = ( 100 - t ) \frac { \mathrm { d } v } { \mathrm {~d} t } - 2000 .$$
  2. Find the speed of the rocket when burning stops.
Edexcel M5 Q6
17 marks Challenging +1.8
6. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{9e3d76a7-b997-4e46-a5dd-aeeaa5abfa4e-02_211_611_388_1845}
\end{figure} A rough uniform rod, of mass \(m\) and length \(4 a\), is rod is held on a rough horizontal table. The rod is perpendicular to the edge of the table and a length \(3 a\) projects horizontally over the edge, as shown in Fig. 1.
  1. Show that the moment of inertia of the rod about the edge of the table is \(\frac { 7 } { 3 } m a ^ { 2 }\). The rod is released from rest and rotates about the edge of the table. When the rod has turned through an angle \(\theta\), its angular speed is \(\dot { \theta }\). Assuming that the rod has not started to slip,
  2. show that \(\dot { \theta } ^ { 2 } = \frac { 6 g \sin \theta } { 7 a }\),
  3. find the angular acceleration of the rod,
  4. find the normal reaction of the table on the rod. The coefficient of friction between the rod and the edge of the table is \(\mu\).
  5. Show that the rod starts to slip when \(\tan \theta = \frac { 4 } { 13 } \mu\)
Edexcel M5 Q8
17 marks Challenging +1.8
8. A pendulum consists of a uniform rod \(P Q\), of mass \(3 m\) and length \(2 a\), which is rigidly fixed at its end \(Q\) to the centre of a uniform circular disc of mass \(m\) and radius \(a\). The rod is perpendicular to the plane of the disc. The pendulum is free to rotate about a fixed smooth horizontal axis \(L\) which passes through the end \(P\) of the rod and is perpendicular to the rod.
  1. Show that the moment of inertia of the pendulum about \(L\) is \(\frac { 33 } { 4 } m a ^ { 2 }\). The pendulum is released from rest in the position where \(P Q\) makes an angle \(\alpha\) with the downward vertical. At time \(t , P Q\) makes an angle \(\theta\) with the downward vertical.
  2. Show that the angular speed, \(\dot { \theta }\), of the pendulum satisfies $$\dot { \theta } ^ { 2 } = \frac { 40 g ( \cos \theta - \cos \alpha ) } { 33 a } .$$
  3. Hence, or otherwise, find the angular acceleration of the pendulum. Given that \(\alpha = \frac { \pi } { 20 }\) and that \(P Q\) has length \(\frac { 8 } { 33 } \mathrm {~m}\),
  4. find, to 3 significant figures, an approximate value for the angular speed of the pendulum 0.2 s after it has been released from rest. \section*{Advanced Level} \section*{Monday 25 June 2012 - Afternoon} \section*{Materials required for examination
    Mathematical Formulae (Pink)} Items included with question papers
    Nil Candidates may use any calculator allowed by the regulations of the Joint Council for Qualifications. Calculators must not have the facility for symbolic algebra manipulation, differentiation and integration, or have retrievable mathematical formulae stored in them.
Edexcel M5 2006 January Q1
4 marks Moderate -0.3
  1. A bead is threaded on a straight wire. The vector equation of the wire is
$$\mathbf { r } = \mathbf { i } - 3 \mathbf { j } + \mathbf { k } + t ( 2 \mathbf { i } - \mathbf { j } + 2 \mathbf { k } )$$ where the unit of length is the metre. The bead is moved from a point \(A\) on the wire through a distance of 6 m along the wire to a point \(B\) by a force \(\mathbf { F } = ( 7 \mathbf { i } + 4 \mathbf { j } - 2 \mathbf { k } ) \mathrm { N }\). Find the magnitude of the work done by \(\mathbf { F }\) in moving the bead from \(A\) to \(B\).
(Total 4 marks)
Edexcel M5 2006 January Q2
5 marks Standard +0.8
2. A uniform circular disc has radius \(a\) and mass \(m\). Prove, using integration, that the moment of inertia of the disc about an axis through its centre and perpendicular to the plane of the disc is \(\frac { 1 } { 2 } m a ^ { 2 }\).
(Total 5 marks)
Edexcel M5 2006 January Q3
6 marks Standard +0.3
3. The position vector \(\mathbf { r }\) of a particle \(P\) at time \(t\) satisfies the vector differential equation $$\frac { \mathrm { d } \mathbf { r } } { \mathrm {~d} t } + 2 \mathbf { r } = 4 \mathbf { i }$$ Given that the position vector of \(P\) at time \(t = 0\) is \(2 \mathbf { j }\), find the position vector of \(P\) at time \(t\).
(Total 6 marks)
Edexcel M5 2006 January Q4
6 marks Standard +0.8
4. A uniform \(\operatorname { rod } A B\), of mass \(m\) and length \(2 a\), is free to rotate in a vertical plane about a fixed smooth horizontal axis through \(A\) and perpendicular to the plane. The rod hangs in equilibrium with \(B\) below \(A\). The rod is rotated through a small angle and released from rest at time \(t = 0\).
  1. Show that the motion of the rod is approximately simple harmonic.
  2. Using this approximation, find the time \(t\) when the rod is first vertical after being released.
    (Total 6 marks)
Edexcel M5 2006 January Q5
10 marks Challenging +1.8
5. A uniform circular disc has mass \(m\) and radius \(a\). The disc can rotate freely about an axis that is in the same plane as the disc and tangential to the disc at a point \(A\) on its circumference. The disc hangs at rest in equilibrium with its centre \(O\) vertically below \(A\). A particle \(P\) of mass \(m\) is moving horizontally and perpendicular to the disc with speed \(\sqrt { } ( k g a )\), where \(k\) is a constant. The particle then strikes the disc at \(O\) and adheres to it at \(O\). Given that the disc rotates through an angle of \(90 ^ { \circ }\) before first coming to instantaneous rest, find the value of \(k\).
(Total 10 marks)
Edexcel M5 2006 January Q6
12 marks Challenging +1.3
6. The vertices of a tetrahedron \(P Q R S\) have position vectors \(\mathbf { p } , \mathbf { q } , \mathbf { r }\) and \(\mathbf { s }\) respectively, where $$\mathbf { p } = - 3 \mathbf { i } + 4 \mathbf { j } - \mathbf { k } , \quad \mathbf { q } = 4 \mathbf { i } + 4 \mathbf { j } - 2 \mathbf { k } , \quad \mathbf { r } = \mathbf { i } - 2 \mathbf { j } + \mathbf { k } , \quad \mathbf { s } = 4 \mathbf { i } + \mathbf { k }$$ Forces of magnitude 20 N and \(2 \sqrt { } 13 \mathrm {~N}\) act along \(S Q\) and \(S R\) respectively. A third force acts at \(P\).
Given that the system of three forces reduces to a couple \(\mathbf { G }\), find
  1. the third force,
  2. the magnitude of \(\mathbf { G }\).
    (6)
    (Total 12 marks)
Edexcel M5 2006 January Q7
15 marks Challenging +1.8
7. At time \(t = 0\), a small body is projected vertically upwards. While ascending it picks up small drops of moisture from the atmosphere. The drops of moisture are at rest before they are picked up. At time \(t\), the combined body \(P\) has mass \(m\) and speed \(v\).
  1. Show that, while \(P\) is moving upwards, \(m \frac { \mathrm {~d} v } { \mathrm {~d} t } + v \frac { \mathrm {~d} m } { \mathrm {~d} t } = - m g\). The initial mass of \(P\) is \(M\), and \(m = M \mathrm { e } ^ { k t }\), where \(k\) is a positive constant.
  2. Show that, while \(P\) is moving upwards, \(\frac { \mathrm { d } } { \mathrm { d } t } \left( v \mathrm { e } ^ { k t } \right) = - g \mathrm { e } ^ { k t }\). Given that the initial projection speed of \(P\) is \(\frac { g } { 2 k }\),
  3. find, in terms of \(M\), the mass of \(P\) when it reaches its highest point.
    (Total 15 marks)
Edexcel M5 2006 January Q8
17 marks Challenging +1.2
8. Four uniform rods, each of mass \(m\) and length \(2 a\), are joined together at their ends to form a plane rigid square framework \(A B C D\) of side \(2 a\). The framework is free to rotate in a vertical plane about a fixed smooth horizontal axis through \(A\). The axis is perpendicular to the plane of the framework.
  1. Show that the moment of inertia of the framework about the axis is \(\frac { 40 m a ^ { 2 } } { 3 }\). The framework is slightly disturbed from rest when \(C\) is vertically above \(A\). Find
  2. the angular acceleration of the framework when \(A C\) is horizontal,
  3. the angular speed of the framework when \(A C\) is horizontal,
  4. the magnitude of the force acting on the framework at \(A\) when \(A C\) is horizontal.
Edexcel M5 2002 June Q1
5 marks Standard +0.8
  1. \hspace{0pt} [In this question \(\mathbf { i }\) and \(\mathbf { j }\) are horizontal unit vectors.]
A small smooth ring of mass 0.5 kg moves along a smooth horizontal wire. The only forces acting on the ring are its weight, the normal reaction from the wire, and a constant force ( \(5 \mathbf { i } + \mathbf { j } - 3 \mathbf { k }\) ) N. The ring is initially at rest at the point with position vector \(( \mathbf { i } + \mathbf { j } + \mathbf { k } ) \mathrm { m }\), relative to a fixed origin. Find the speed of the ring as it passes through the point with position vector \(( 3 \mathbf { i } + \mathbf { k } ) \mathrm { m }\).
Edexcel M5 2002 June Q2
6 marks Standard +0.8
2. Three forces, \(\mathbf { F } _ { 1 } , \mathbf { F } _ { 2 }\) and \(\mathbf { F } _ { 3 }\) act on a rigid body. \(\mathbf { F } _ { 1 } = ( 2 \mathbf { i } - \mathbf { j } + 3 \mathbf { k } ) \mathrm { N } , \mathbf { F } _ { 2 } = ( \mathbf { i } + \mathbf { j } - 4 \mathbf { k } )\) N and \(\mathbf { F } _ { 3 } = ( p \mathbf { i } + q \mathbf { j } + r \mathbf { k } ) \mathrm { N }\), where \(p , q\) and \(r\) are constants. All three forces act through the point with position vector ( \(3 \mathbf { i } - 2 \mathbf { j } + \mathbf { k }\) ) m, relative to a fixed origin. The three forces \(\mathbf { F } _ { 1 } , \mathbf { F } _ { 2 }\) and \(\mathbf { F } _ { 3 }\) are equivalent to a single force ( \(5 \mathbf { i } - 4 \mathbf { j } + 2 \mathbf { k }\) ) N, acting at the origin, together with a couple \(\mathbf { G }\).
  1. Find the values of \(p , q\) and \(r\).
  2. Find \(\mathbf { G }\).
Edexcel M5 2002 June Q3
7 marks Standard +0.3
3. At time \(t\) seconds, the position vector of a particle \(P\) is \(\mathbf { r }\) metres, relative to a fixed origin. The particle moves in such a way that $$\frac { \mathrm { d } ^ { 2 } \mathbf { r } } { \mathrm {~d} t ^ { 2 } } - 4 \frac { \mathrm {~d} \mathbf { r } } { \mathrm {~d} t } = \mathbf { 0 }$$ At \(t = 0 , P\) is moving with velocity ( \(8 \mathbf { i } - 6 \mathbf { j }\) ) \(\mathrm { m } \mathrm { s } ^ { - 1 }\).
Find the speed of \(P\) when \(t = \frac { 1 } { 2 } \ln 2\).
Edexcel M5 2002 June Q4
9 marks Challenging +1.2
4. A uniform plane lamina of mass \(m\) is in the shape of an equilateral triangle of side \(2 a\). Find, using integration, the moment of inertia of the lamina about one of its edges.
Edexcel M5 2002 June Q5
14 marks Challenging +1.8
5. A rocket is launched vertically upwards from rest. Initially, the total mass of the rocket and its fuel is 1000 kg . The rocket burns fuel at a rate of \(10 \mathrm {~kg} \mathrm {~s} ^ { - 1 }\). The burnt fuel is ejected vertically downwards with a speed of \(2000 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) relative to the rocket, and burning stops after one minute. At time \(t\) seconds, \(t \leq 60\), after the launch, the speed of the rocket is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Air resistance is assumed to be negligible.
  1. Show that $$- 9.8 ( 100 - t ) = ( 100 - t ) \frac { \mathrm { d } v } { \mathrm {~d} t } - 2000$$ (8)
  2. Find the speed of the rocket when burning stops.
    (6) \section*{6.} \begin{figure}[h]
    \captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{4c585ec7-7b3e-4ff8-b7c2-83f58ad82ae9-4_316_929_391_573}
    \end{figure} A rough uniform rod, of mass \(m\) and length \(4 a\), is rod is held on a rough horizontal table. The rod is perpendicular to the edge of the table and a length \(3 a\) projects horizontally over the edge, as shown in Fig. 1.