Questions M4 (327 questions)

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Edexcel M4 2014 June Q1
8 marks Standard +0.8
A particle \(A\) has constant velocity \((3\mathbf{i} + \mathbf{j})\) m s\(^{-1}\) and a particle \(B\) has constant velocity \((\mathbf{i} - \mathbf{k})\) m s\(^{-1}\). At time \(t = 0\) seconds, the position vectors of the particles \(A\) and \(B\) with respect to a fixed origin \(O\) are \((-6\mathbf{i} + 4\mathbf{j} - 3\mathbf{k})\) m and \((-2\mathbf{i} + 2\mathbf{j} + 3\mathbf{k})\) m respectively.
  1. Show that, in the subsequent motion, the minimum distance between \(A\) and \(B\) is \(4\sqrt{2}\) m. [6]
  2. Find the position vector of \(A\) at the instant when the distance between \(A\) and \(B\) is a minimum. [2]
Edexcel M4 2014 June Q2
11 marks Standard +0.8
A car of mass 1000 kg is moving along a straight horizontal road. The engine of the car is working at a constant rate of 25 kW. When the speed of the car is \(v\) m s\(^{-1}\), the resistance to motion has magnitude \(10v\) newtons.
  1. Show that, at the instant when \(v = 20\), the acceleration of the car is 1.05 m s\(^{-2}\). [3]
  2. Find the distance travelled by the car as it accelerates from a speed of 10 m s\(^{-1}\) to a speed of 20 m s\(^{-1}\). [8]
Edexcel M4 2014 June Q3
8 marks Challenging +1.2
A small ball is moving on a smooth horizontal plane when it collides obliquely with a smooth plane vertical wall. The coefficient of restitution between the ball and the wall is \(\frac{1}{3}\). The speed of the ball immediately after the collision is half the speed of the ball immediately before the collision. Find the angle through which the path of the ball is deflected by the collision. [8]
Edexcel M4 2014 June Q4
8 marks Challenging +1.2
At noon two ships \(A\) and \(B\) are 20 km apart with \(A\) on a bearing of 230° from \(B\). Ship \(B\) is moving at 6 km h\(^{-1}\) on a bearing of 015°. The maximum speed of \(A\) is 12 km h\(^{-1}\). Ship \(A\) sets a course to intercept \(B\) as soon as possible.
  1. Find the course set by \(A\), giving your answer as a bearing to the nearest degree. [4]
  2. Find the time at which \(A\) intercepts \(B\). [4]
Edexcel M4 2014 June Q5
12 marks Challenging +1.8
\includegraphics{figure_1} Two smooth uniform spheres \(A\) and \(B\) have equal radii. The mass of \(A\) is \(m\) and the mass of \(B\) is \(3m\). The spheres are moving on a smooth horizontal plane when they collide obliquely. Immediately before the collision, \(A\) is moving with speed \(3u\) at angle \(\alpha\) to the line of centres and \(B\) is moving with speed \(u\) at angle \(\beta\) to the line of centres, as shown in Figure 1. The coefficient of restitution between the two spheres is \(\frac{1}{5}\). It is given that \(\cos \alpha = \frac{1}{3}\) and \(\cos \beta = \frac{2}{3}\) and that \(\alpha\) and \(\beta\) are both acute angles.
  1. Find the magnitude of the impulse on \(A\) due to the collision in terms of \(m\) and \(u\). [8]
  2. Express the kinetic energy lost by \(A\) in the collision as a fraction of its initial kinetic energy. [4]
Edexcel M4 2014 June Q6
13 marks Challenging +1.8
A particle of mass \(m\) kg is attached to one end of a light elastic string of natural length \(a\) metres and modulus of elasticity \(5ma\) newtons. The other end of the string is attached to a fixed point \(O\) on a smooth horizontal plane. The particle is held at rest on the plane with the string stretched to a length \(2a\) metres and then released at time \(t = 0\). During the subsequent motion, when the particle is moving with speed \(v\) m s\(^{-1}\), the particle experiences a resistance of magnitude \(4mv\) newtons. At time \(t\) seconds after the particle is released, the length of the string is \((a + x)\) metres, where \(0 \leqslant x \leqslant a\).
  1. Show that, from \(t = 0\) until the string becomes slack, $$\frac{\mathrm{d}^2 x}{\mathrm{d}t^2} + 4\frac{\mathrm{d}x}{\mathrm{d}t} + 5x = 0$$ [3]
  2. Hence express \(x\) in terms of \(a\) and \(t\). [6]
  3. Find the speed of the particle at the instant when the string first becomes slack, giving your answer in the form \(ka\), where \(k\) is a constant to be found correct to 2 significant figures. [4]
Edexcel M4 2014 June Q7
15 marks Challenging +1.2
\includegraphics{figure_2} A bead \(B\) of mass \(m\) is threaded on a smooth circular wire of radius \(r\), which is fixed in a vertical plane. The centre of the circle is \(O\), and the highest point of the circle is \(A\). A light elastic string of natural length \(r\) and modulus of elasticity \(kmg\) has one end attached to the bead and the other end attached to \(A\). The angle between the string and the downward vertical is \(\theta\), and the extension in the string is \(x\), as shown in Figure 2. Given that the string is taut,
  1. show that the potential energy of the system is $$2mgr[(k-1)\cos^2 \theta - k\cos \theta] + \text{constant}$$ [6]
Given also that \(k = 3\),
  1. find the positions of equilibrium and determine their stability. [9]
Edexcel M4 Specimen Q1
6 marks Moderate -0.3
A particle \(P\) of mass 2 kg moves in a straight line along a smooth horizontal plane. The only horizontal force acting on \(P\) is a resistance of magnitude \(4v\) N, where \(v\) m s\(^{-1}\) is its speed. At time \(t = 0\) s, \(P\) has a speed of 5 m s\(^{-1}\). Find \(v\) in terms of \(t\). [6]
Edexcel M4 Specimen Q2
6 marks Moderate -0.3
\includegraphics{figure_1} A girl swims in still water at 1 m s\(^{-1}\). She swims across a river which is 336 m wide and is flowing at 0.6 m s\(^{-1}\). She sets off from a point \(A\) on one bank and lands at a point \(B\), which is directly opposite \(A\), on the other bank as shown in Fig. 1. Find
  1. the direction, relative to the earth, in which she swims, [3]
  2. the time that she takes to cross the river. [3]
Edexcel M4 Specimen Q3
10 marks Challenging +1.2
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 \(mkv^2\), 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. [9]
  2. State one physical factor which is ignored in this model. [1]
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]