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Edexcel M4 Specimen Q4
11 marks Challenging +1.2
\includegraphics{figure_2} 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\(\mathbf{i}\) + \(\mathbf{j}\)) m s\(^{-1}\), and sphere \(B\) has mass 5 kg and velocity (\(-\mathbf{i}\) + \(\mathbf{j}\)) m 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, [5]
  2. the coefficient of restitution between the spheres. [6]
Edexcel M4 Specimen Q5
12 marks Challenging +1.3
An elastic string spring of modulus \(2mg\) 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 \(2m\omega v\), 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 \(\frac{\mathrm{d}^2 x}{\mathrm{d} t^2} + 2\omega \frac{\mathrm{d} x}{\mathrm{d} t} + 2\omega^2 x = 0\). [7]
  2. Find the general solution of this differential equation. [4]
  3. Hence find the period of the damped harmonic motion. [1]
Edexcel M4 Specimen Q6
14 marks Standard +0.3
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 m 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 m 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. [6]
  2. Find the shortest distance between the car and the tractor. [8]
Edexcel M4 Specimen Q7
16 marks Challenging +1.8
\includegraphics{figure_3} A uniform rod \(AB\) has mass \(m\) and length \(2a\). 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 \(mg\), to the end \(B\) of the rod. The end \(B\) is always vertically below \(R\) and angle \(\angle RAB = \theta\), as shown in Fig. 3.
  1. Show that the potential energy of the system is $$mga(2\sin^2\theta - 3\sin\theta) + \text{constant}.$$ [6]
  2. Hence determine the value of \(\theta\), \(0 < \frac{\pi}{2}\), for which the system is in equilibrium. [5]
  3. Determine whether this position of equilibrium is stable or unstable. [5]
OCR M4 2006 June Q1
5 marks Standard +0.3
A straight rod \(AB\) of length \(a\) has variable density. At a distance \(x\) from \(A\) its mass per unit length is \(k(a + 2x)\), where \(k\) is a positive constant. Find the distance from \(A\) of the centre of mass of the rod. [5]
OCR M4 2006 June Q2
8 marks Standard +0.3
A flywheel takes the form of a uniform disc of mass 8 kg and radius 0.15 m. It rotates freely about an axis passing through its centre and perpendicular to the disc. A couple of constant moment is applied to the flywheel. The flywheel turns through an angle of 75 radians while its angular speed increases from 10 rad s\(^{-1}\) to 25 rad s\(^{-1}\).
  1. Find the moment of the couple about the axis. [5]
When the flywheel is rotating with angular speed 25 rad s\(^{-1}\), it locks together with a second flywheel which is mounted on the same axis and is at rest. Immediately afterwards, both flywheels rotate together with the same angular speed 9 rad s\(^{-1}\).
  1. Find the moment of inertia of the second flywheel about the axis. [3]
OCR M4 2006 June Q3
8 marks Standard +0.8
The region bounded by the \(x\)-axis, the lines \(x = 1\) and \(x = 2\) and the curve \(y = \frac{1}{x^2}\) for \(1 \leq x \leq 2\), is occupied by a uniform lamina of mass 24 kg. The unit of length is the metre. Find the moment of inertia of this lamina about the \(x\)-axis. [8]
OCR M4 2006 June Q4
10 marks Challenging +1.8
\includegraphics{figure_4} A uniform rod \(AB\), of mass \(m\) and length \(2a\), is freely hinged to a fixed point at \(A\). A particle of mass \(2m\) is attached to the rod at \(B\). A light elastic string, with natural length \(a\) and modulus of elasticity \(5mg\), passes through a fixed smooth ring \(R\). One end of the string is fixed to \(A\) and the other end is fixed to the mid-point \(C\) of \(AB\). The ring \(R\) is at the same horizontal level as \(A\), and is at a distance \(a\) from \(A\). The rod \(AB\) and the ring \(R\) are in a vertical plane, and \(RC\) is at an angle \(\theta\) above the horizontal, where \(0 < \theta < \frac{1}{2}\pi\), so that the acute angle between \(AB\) and the horizontal is \(2\theta\) (see diagram).
  1. By considering the energy of the system, find the value of \(\theta\) for which the system is in equilibrium. [7]
  2. Determine whether this position of equilibrium is stable or unstable. [3]
OCR M4 2006 June Q5
11 marks Challenging +1.2
A uniform rectangular lamina \(ABCD\) has mass 20 kg and sides of lengths \(AB = 0.6\) m and \(BC = 1.8\) m. It rotates in its own vertical plane about a fixed horizontal axis which is perpendicular to the lamina and passes through the mid-point of \(AB\).
  1. Show that the moment of inertia of the lamina about the axis is 22.2 kg m\(^2\). [3]
\includegraphics{figure_5} The lamina is released from rest with \(BC\) horizontal and below the level of the axis. Air resistance may be neglected, but a frictional couple opposes the motion. The couple has constant moment 44.1 N m about the axis. The angle through which the lamina has turned is denoted by \(\theta\) (see diagram).
  1. Show that the angular acceleration is zero when \(\cos \theta = 0.25\). [3]
  2. Hence find the maximum angular speed of the lamina. [5]
OCR M4 2006 June Q6
13 marks Challenging +1.2
\includegraphics{figure_6} A ship \(P\) is moving with constant velocity 7 m s\(^{-1}\) in the direction with bearing 110°. A second ship \(Q\) is moving with constant speed 10 m s\(^{-1}\) in a straight line. At one instant \(Q\) is at the point \(X\), and \(P\) is 7400 m from \(Q\) on a bearing of 050° (see diagram). In the subsequent motion, the shortest distance between \(P\) and \(Q\) is 1790 m.
  1. Show that one possible direction for the velocity of \(Q\) relative to \(P\) has bearing 036°, to the nearest degree, and find the bearing of the other possible direction of this relative velocity. [3]
Given that the velocity of \(Q\) relative to \(P\) has bearing 036°, find
  1. the bearing of the direction in which \(Q\) is moving, [4]
  2. the magnitude of the velocity of \(Q\) relative to \(P\), [2]
  3. the time taken for \(Q\) to travel from \(X\) to the position where the two ships are closest together, [3]
  4. the bearing of \(P\) from \(Q\) when the two ships are closest together. [1]
OCR M4 2006 June Q7
17 marks Challenging +1.2
\includegraphics{figure_7} A uniform rod \(AB\) has mass \(m\) and length \(6a\). It is free to rotate in a vertical plane about a smooth fixed horizontal axis passing through the point \(C\) on the rod, where \(AC = a\). The angle between \(AB\) and the upward vertical is \(\theta\), and the force acting on the rod at \(C\) has components \(R\) parallel to \(AB\) and \(S\) perpendicular to \(AB\) (see diagram). The rod is released from rest in the position where \(\theta = \frac{1}{4}\pi\). Air resistance may be neglected.
  1. Find the angular acceleration of the rod in terms of \(a\), \(g\) and \(\theta\). [4]
  2. Show that the angular speed of the rod is \(\sqrt{\frac{2g(1 - 2\cos\theta)}{7a}}\). [3]
  3. Find \(R\) and \(S\) in terms of \(m\), \(g\) and \(\theta\). [6]
  4. When \(\cos\theta = \frac{1}{3}\), show that the force acting on the rod at \(C\) is vertical, and find its magnitude. [4]
OCR M4 2016 June Q1
4 marks Standard +0.3
A uniform square lamina, of mass 5 kg and side 0.2 m, is rotating about a fixed vertical axis that is perpendicular to the lamina and that passes through its centre. A couple of constant moment 0.06 N m is applied to the lamina. The lamina turns through an angle of 155 radians while its angular speed increases from 8 rad s\(^{-1}\) to \(\omega\) rad s\(^{-1}\). Find \(\omega\). [4]
OCR M4 2016 June Q2
9 marks Standard +0.3
\includegraphics{figure_2} Boat \(A\) is travelling with constant speed 7.9 m s\(^{-1}\) on a course with bearing 035°. Boat \(B\) is travelling with constant speed 10.5 m s\(^{-1}\) on a course with bearing 330°. At one instant, the boats are 1500 m apart with \(B\) on a bearing of 125° from \(A\) (see diagram).
  1. Find the magnitude and the bearing of the velocity of \(B\) relative to \(A\). [5]
  2. Find the shortest distance between \(A\) and \(B\) in the subsequent motion. [2]
  3. Find the time taken from the instant when \(A\) and \(B\) are 1500 m apart to the instant when \(A\) and \(B\) are at the point of closest approach. [2]
OCR M4 2016 June Q3
13 marks Challenging +1.8
\includegraphics{figure_3} Two uniform rods \(AB\) and \(BC\), each of length \(a\) and mass \(m\), are rigidly joined together so that \(AB\) is perpendicular to \(BC\). The rod \(AB\) is freely hinged to a fixed point at \(A\). The rods can rotate in a vertical plane about a smooth fixed horizontal axis through \(A\). One end of a light elastic string of natural length \(a\) and modulus of elasticity \(\lambda mg\) is attached to \(B\). The other end of the string is attached to a fixed point \(D\) vertically above \(A\), where \(AD = a\). The string \(BD\) makes an angle \(\theta\) radians with the downward vertical (see diagram).
  1. Taking \(D\) as the reference level for gravitational potential energy, show that the total potential energy \(V\) of the system is given by $$V = \frac{1}{2}mga(\sin 2\theta - 3\cos 2\theta) + \frac{1}{2}\lambda mga(2\cos \theta - 1)^2 - 2mga.$$ [5]
  2. Given that \(\theta = \frac{1}{3}\pi\) is a position of equilibrium, find the exact value of \(\lambda\). [4]
  3. Find \(\frac{d^2V}{d\theta^2}\) and hence determine whether the position of equilibrium at \(\theta = \frac{1}{3}\pi\) is stable or unstable. [4]
OCR M4 2016 June Q4
13 marks Standard +0.8
The region bounded by the curve \(y = 2e^{\frac{1}{2}x}\) for \(0 \leq x \leq 2\), the \(x\)-axis, the \(y\)-axis and the line \(x = 2\), is occupied by a uniform lamina.
  1. Find the exact value of the \(y\)-coordinate of the centre of mass of the lamina. [6]
As shown in the diagram below, a uniform lamina occupies the closed region bounded by the \(x\)-axis, the \(y\)-axis and the curve \(y = f(x)\) where $$f(x) = \begin{cases} 2e^{\frac{1}{2}x} & 0 \leq x \leq 2, \\ \frac{2}{3}(5-x)e & 2 \leq x \leq 5. \end{cases}$$ \includegraphics{figure_4}
  1. Find the exact value of the \(x\)-coordinate of the centre of mass of the lamina. [7]
OCR M4 2016 June Q5
18 marks Challenging +1.2
A uniform rod \(AB\) has mass \(2m\) and length \(4a\).
  1. Show by integration that the moment of inertia of the rod about an axis perpendicular to the rod through \(A\) is \(\frac{32}{3}ma^2\). [4]
The rod is initially at rest with \(B\) vertically below \(A\) and it is free to rotate in a vertical plane about a smooth fixed horizontal axis through \(A\). A particle of mass \(m\) is moving horizontally in the plane in which the rod is free to rotate. The particle has speed \(v\), and strikes the rod at \(B\). In the subsequent motion the particle adheres to the rod and the combined rigid body \(Q\), consisting of the rod and the particle, starts to rotate.
  1. Find, in terms of \(v\) and \(a\), the initial angular speed of \(Q\). [4]
At time \(t\) seconds the angle between \(Q\) and the downward vertical is \(\theta\) radians.
  1. Show that \(\dot{\theta}^2 = k\frac{g}{a}(\cos \theta - 1) + \frac{9v^2}{400a^2}\), stating the value of the constant \(k\). [4]
  2. Find, in terms of \(a\) and \(g\), the set of values of \(v^2\) for which \(Q\) makes complete revolutions. [2]
When \(Q\) is horizontal, the force exerted by the axis on \(Q\) has vertically upwards component \(R\).
  1. Find \(R\) in terms of \(m\) and \(g\). [4]
OCR M4 2016 June Q6
15 marks Challenging +1.2
\includegraphics{figure_6} A compound pendulum consists of a uniform rod \(AB\) of length 1 m and mass 3 kg, a particle of mass 1 kg attached to the rod at \(A\) and a circular disc of radius \(\frac{1}{5}\) m, mass 6 kg and centre \(C\). The end \(B\) of the rod is rigidly attached to a point on the circumference of the disc in such a way that \(ABC\) is a straight line. The pendulum is initially at rest with \(B\) vertically below \(A\) and it is free to rotate in a vertical plane about a smooth fixed horizontal axis passing through the point \(P\) on the rod where \(AP = x\) m and \(x < \frac{1}{3}\) (see diagram).
  1. Show that the moment of inertia of the pendulum about the axis of rotation is \((10x^2 - 19x + 12)\) kg m\(^2\). [6]
The pendulum is making small oscillations about the equilibrium position, such that at time \(t\) seconds the angular displacement that the pendulum makes with the downward vertical is \(\theta\) radians.
  1. Find the angular acceleration of the pendulum, in terms of \(x\), \(g\) and \(\theta\). [4]
  2. Show that the motion is approximately simple harmonic, and show that the approximate period of oscillations, in seconds, is given by \(2\pi\sqrt{\frac{20x^2 - 38x + 24}{(19-20x)g}}\). [2]
  3. Hence find the value of \(x\) for which the approximate period of oscillations is least. [3]
Edexcel M5 Q1
7 marks Standard +0.3
At time \(t = 0\), a particle \(P\) of mass \(3\) kg is at rest at the point \(A\) with position vector \((j - 3k)\) m. Two constant forces \(\mathbf{F}_1\) and \(\mathbf{F}_2\) then act on the particle \(P\) and it passes through the point \(B\) with position vector \((8i - 3j + 5k)\) m. Given that \(\mathbf{F}_1 = (4i - 2j + 5k)\) N and \(\mathbf{F}_2 = (8i - 4j + 7k)\) N and that \(\mathbf{F}_1\) and \(\mathbf{F}_2\) are the only two forces acting on \(P\), find the velocity of \(P\) as it passes through \(B\), giving your answer as a vector. [7]
Edexcel M5 Q2
11 marks Challenging +1.2
At time \(t\) seconds, the position vector of a particle \(P\) is \(\mathbf{r}\) metres, where \(\mathbf{r}\) satisfies the vector differential equation $$\frac{d^2\mathbf{r}}{dt^2} + 4\mathbf{r} = e^{2t} \mathbf{j}.$$ When \(t = 0\), \(P\) has position vector \((i + j)\) m and velocity \(2i\) m s\(^{-1}\). Find an expression for \(\mathbf{r}\) in terms of \(t\). [11]
Edexcel M5 Q3
9 marks Challenging +1.2
A spaceship is moving in a straight line in deep space and needs to increase its speed. This is done by ejecting fuel backwards from the spaceship at a constant speed \(c\) relative to the spaceship. When the speed of the spaceship is \(v\), its mass is \(m\).
  1. Show that, while the spaceship is ejecting fuel, $$\frac{dv}{dm} = -\frac{c}{m}.$$ [5]
The initial mass of the spaceship is \(m_0\) and at time \(t\) the mass of the spaceship is given by \(m = m_0(1 - kt)\), where \(k\) is a positive constant.
  1. Find the acceleration of the spaceship at time \(t\). [4]
Edexcel M5 Q4
13 marks Challenging +1.8
\includegraphics{figure_4} **Figure 1** A uniform lamina of mass \(M\) is in the shape of a right-angled triangle \(OAB\). The angle \(OAB\) is \(90°\), \(OA = a\) and \(AB = 2a\), as shown in Figure 1.
  1. Prove, using integration, that the moment of inertia of the lamina \(OAB\) about the edge \(OA\) is \(\frac{8}{3}Ma^2\). (You may assume without proof that the moment of inertia of a uniform rod of mass \(m\) and length \(2l\) about an axis through one end and perpendicular to the rod is \(\frac{4}{3}ml^2\).) [6]
The lamina \(OAB\) is free to rotate about a fixed smooth horizontal axis along the edge \(OA\) and hangs at rest with \(B\) vertically below \(A\). The lamina is then given a horizontal impulse of magnitude \(J\). The impulse is applied to the lamina at the point \(B\), in a direction which is perpendicular to the plane of the lamina. Given that the lamina first comes to instantaneous rest after rotating through an angle of \(120°\),
  1. find an expression for \(J\), in terms of \(M\), \(a\) and \(g\). [7]
Edexcel M5 Q5
16 marks Challenging +1.2
Two forces \(\mathbf{F}_1 = (2i + j)\) N and \(\mathbf{F}_2 = (-2j - k)\) N act on a rigid body. The force \(\mathbf{F}_1\) acts at the point with position vector \(\mathbf{r}_1 = (3i + j + k)\) m and the force \(\mathbf{F}_2\) acts at the point with position vector \(\mathbf{r}_2 = (i - 2j)\) m. A third force \(\mathbf{F}_3\) acts on the body such that \(\mathbf{F}_1\), \(\mathbf{F}_2\) and \(\mathbf{F}_3\) are in equilibrium.
  1. Find the magnitude of \(\mathbf{F}_3\). [4]
  1. Find a vector equation of the line of action of \(\mathbf{F}_3\). [8]
The force \(\mathbf{F}_3\) is replaced by a fourth force \(\mathbf{F}_4\), acting through the origin \(O\), such that \(\mathbf{F}_1\), \(\mathbf{F}_2\) and \(\mathbf{F}_4\) are equivalent to a couple.
  1. Find the magnitude of this couple. [4]
Edexcel M5 Q6
19 marks Challenging +1.8
A pendulum consists of a uniform rod \(AB\), of length \(4a\) and mass \(2m\), whose end \(A\) is rigidly attached to the centre \(O\) of a uniform square lamina \(PQRS\), of mass \(4m\) and side \(a\). The rod \(AB\) is perpendicular to the plane of the lamina. The pendulum is free to rotate about a fixed smooth horizontal axis \(L\) which passes through \(B\). The axis \(L\) is perpendicular to \(AB\) and parallel to the edge \(PQ\) of the square.
  1. Show that the moment of inertia of the pendulum about \(L\) is \(75ma^2\). [4]
The pendulum is released from rest when \(BA\) makes an angle \(\alpha\) with the downward vertical through \(B\), where \(\tan \alpha = \frac{3}{4}\). When \(BA\) makes an angle \(\theta\) with the downward vertical through \(B\), the magnitude of the component, in the direction \(AB\), of the force exerted by the axis \(L\) on the pendulum is \(X\).
  1. Find an expression for \(X\) in terms of \(m\), \(g\) and \(\theta\). [9]
Using the approximation \(\theta \approx \sin \theta\),
  1. find an estimate of the time for the pendulum to rotate through an angle \(\alpha\) from its initial rest position. [6]
Edexcel M5 Q1
7 marks Challenging +1.2
At time \(t = 0\), the position vector of a particle \(P\) is \(-3j\) m. At time \(t\) seconds, the position vector of \(P\) is \(\mathbf{r}\) metres and the velocity of \(P\) is \(\mathbf{v}\) m s\(^{-1}\). Given that $$\mathbf{v} - 2\mathbf{r} = 4e^t \mathbf{j},$$ find the time when \(P\) passes through the origin. [7]
Edexcel M5 Q2
13 marks Challenging +1.8
\includegraphics{figure_2} **Figure 1** A uniform circular disc has mass \(4m\), centre \(O\) and radius \(4a\). The line \(POQ\) is a diameter of the disc. A circular hole of radius \(2a\) is made in the disc with the centre of the hole at the point \(R\) on \(PQ\) where \(QR = 5a\), as shown in Figure 1. The resulting lamina is free to rotate about a fixed smooth horizontal axis \(L\) which passes through \(Q\) and is perpendicular to the plane of the lamina.
  1. Show that the moment of inertia of the lamina about \(L\) is \(69ma^2\). [7]
The lamina is hanging at rest with \(P\) vertically below \(Q\) when it is given an angular velocity \(\Omega\). Given that the lamina turns through an angle \(\frac{2\pi}{3}\) before it first comes to instantaneous rest,
  1. find \(\Omega\) in terms of \(g\) and \(a\). [6]