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AQA M3 2016 June Q5
12 marks Challenging +1.8
A ball is projected from a point \(O\) above a smooth plane which is inclined at an angle of \(20°\) to the horizontal. The point \(O\) is at a perpendicular distance of \(1\) m from the inclined plane. The ball is projected with velocity \(22 \text{ m s}^{-1}\) at an angle of \(70°\) above the horizontal. The motion of the ball is in a vertical plane containing a line of greatest slope of the inclined plane. The ball strikes the inclined plane for the first time at a point \(A\). \includegraphics{figure_5}
    1. Find the time taken by the ball to travel from \(O\) to \(A\). [4 marks]
    2. Find the components of the velocity of the ball, parallel and perpendicular to the inclined plane, as it strikes the plane at \(A\). [4 marks]
  2. After striking \(A\), the ball rebounds and strikes the plane for a second time at a point further up than \(A\). The coefficient of restitution between the ball and the inclined plane is \(e\). Show that \(e < k\), where \(k\) is a constant to be determined. [4 marks]
AQA M3 2016 June Q6
14 marks Challenging +1.2
In this question use \(\cos 30° = \sin 60° = \frac{\sqrt{3}}{2}\). A smooth spherical ball, \(A\), is moving with speed \(u\) in a straight line on a smooth horizontal table when it hits an identical ball, \(B\), which is at rest on the table. Just before the collision, the direction of motion of \(A\) is parallel to a fixed smooth vertical wall. At the instant of collision, the line of centres of \(A\) and \(B\) makes an angle of \(60°\) with the wall, as shown in the diagram. \includegraphics{figure_6} The coefficient of restitution between \(A\) and \(B\) is \(e\).
  1. Show that the speed of \(B\) immediately after the collision is \(\frac{1}{4}u(1 + e)\) and find, in terms of \(u\) and \(e\), the components of the velocity of \(A\), parallel and perpendicular to the line of centres, immediately after the collision. [7 marks]
  2. Subsequently, \(B\) collides with the wall. After colliding with the wall, the direction of motion of \(B\) is parallel to the direction of motion of \(A\) after its collision with \(B\). Show that the coefficient of restitution between \(B\) and the wall is \(\frac{1 + e}{7 - e}\). [7 marks]
AQA M3 2016 June Q7
13 marks Challenging +1.8
A quad-bike, a truck and a car are moving on a large, open, horizontal surface in a desert plain. Relative to the quad-bike, which is travelling due west at its maximum speed of \(10 \text{ m s}^{-1}\), the truck is moving on a bearing of \(340°\). Relative to the car, which is travelling due east at a speed of \(15 \text{ m s}^{-1}\), the truck is moving on a bearing of \(300°\).
  1. Show that the speed of the truck is approximately \(24.7 \text{ m s}^{-1}\) and that it is moving on a bearing of \(318°\), correct to the nearest degree. [8 marks]
  2. At the instant when the truck is at a distance of \(400\) metres from the quad-bike, the bearing of the truck from the quad-bike is \(060°\). The truck continues to move with the same velocity as in part (a). The quad-bike continues to move at a speed of \(10 \text{ m s}^{-1}\). Find the bearing, to the nearest degree, on which the quad-bike should travel in order to approach the truck as closely as possible. [5 marks]
Edexcel M3 Q1
7 marks Standard +0.3
A bird of mass 0.5 kg, flying around a vertical feeding post at a constant speed of 6 ms\(^{-1}\), banks its wings to move in a horizontal circle of radius 2 m. The aerodynamic lift \(L\) newtons is perpendicular to the bird's wings, as shown. \includegraphics{figure_1} Modelling the bird as a particle, find, to the nearest degree, the angle that its wings make with the vertical. [7 marks]
Edexcel M3 Q2
7 marks Standard +0.8
A thin elastic string, of modulus \(\lambda\) N and natural length 20 cm, passes round two small, smooth pegs \(A\) and \(B\) on the same horizontal level to form a closed loop. \(AB = 10\) cm. The ends of the string are attached to a weight \(P\) of mass 0.7 kg. When \(P\) rests in equilibrium, \(APB\) forms an equilateral triangle. \includegraphics{figure_2}
  1. Find the value of \(\lambda\). [6 marks]
  2. State one assumption that you have made about the weight \(P\), explaining how you have used this assumption in your solution. [1 mark]
Edexcel M3 Q3
8 marks Standard +0.8
A particle \(P\) of mass 0.5 kg moves along a straight line. When \(P\) is at a distance \(x\) m from a fixed point \(O\) on the line, the force acting on it is directed towards \(O\) and has magnitude \(\frac{8}{x}\) N. When \(x = 2\), the speed of \(P\) is 4 ms\(^{-1}\). Find the speed of \(P\) when it is 0.5 m from \(O\). [8 marks]
Edexcel M3 Q4
9 marks Standard +0.3
A particle \(P\) of mass \(m\) kg is attached to one end of a light elastic string of natural length \(l\) m and modulus of elasticity \(\lambda\) N. The other end of the string is attached to a fixed point \(O\). \(P\) is released from rest at \(O\) and falls vertically downwards under gravity. The greatest distance below \(O\) reached by \(P\) is \(2l\) m.
  1. Show that \(\lambda = 4mg\). [3 marks]
  2. Find, in terms of \(l\) and \(g\), the speed with which \(P\) passes through the point \(A\), where \(OA = \frac{5l}{4}\) m. [6 marks]
Edexcel M3 Q5
12 marks Challenging +1.2
A uniform solid right circular cone has height \(h\) and base radius \(r\). The top part of the cone is removed by cutting through the cone parallel to the base at a height \(\frac{h}{2}\). \includegraphics{figure_5}
  1. Show that the centre of mass of the remaining solid is at a height \(\frac{11h}{56}\) above the base, along its axis of symmetry. [7 marks]
The remaining part of the solid is suspended from the point \(D\) on the circumference of its smaller circular face, and the axis of symmetry then makes an angle \(\alpha\) with the vertical, where \(\tan \alpha = \frac{1}{2}\).
  1. Find the value of the ratio \(h : r\). [5 marks]
Edexcel M3 Q6
15 marks Standard +0.8
A light elastic string, of natural length \(l\) m and modulus of elasticity \(\frac{mg}{2}\) newtons, has one end fastened to a fixed point \(O\). A particle \(P\), of mass \(m\) kg, is attached to the other end of the string. \(P\) hangs in equilibrium at the point \(E\), vertically below \(O\), where \(OE = (l + e)\) m
  1. Find the numerical value of the ratio \(e : l\). [2 marks]
\(P\) is now pulled down a further distance \(\frac{3l}{2}\) m from \(E\) and is released from rest. In the subsequent motion, the string remains taut. At time \(t\) s after being released, \(P\) is at a distance \(x\) m below \(E\).
  1. Write down a differential equation for the motion of \(P\) and show that the motion is simple harmonic. [4 marks]
  2. Write down the period of the motion. [2 marks]
  3. Find the speed with which \(P\) first passes through \(E\) again. [2 marks]
  4. Show that the time taken by \(P\) after it is released to reach the point \(A\) above \(E\), where \(AE = \frac{3l}{4}\) m, is \(\frac{2\pi}{3}\sqrt{\frac{2l}{g}}\) s. [5 marks]
Edexcel M3 Q7
17 marks Challenging +1.2
A particle \(P\) is attached to one end of a light inextensible string of length \(l\) m. The other end of the string is attached to a fixed point \(O\). When \(P\) is hanging at rest vertically below \(O\), it is given a horizontal speed \(u\) ms\(^{-1}\) and starts to move in a vertical circle. Given that the string becomes slack when it makes an angle of 120° with the downward vertical through \(O\),
  1. show that \(u^2 = \frac{7gl}{2}\). [10 marks]
  2. Find, in terms of \(l\), the greatest height above \(O\) reached by \(P\) in the subsequent motion. [7 marks]
Edexcel M3 Q1
7 marks Standard +0.3
A particle of mass \(m\) kg moves in a horizontal straight line. Its initial speed is \(u\) ms\(^{-1}\) and the only force acting on it is a variable resistance of magnitude \(mkv\) N, where \(v\) ms\(^{-1}\) is the speed of the particle after \(t\) seconds and \(k\) is a constant. Show that \(v = ue^{-kt}\). [7 marks]
Edexcel M3 Q2
7 marks Standard +0.3
A particle \(P\) of mass \(m\) kg moves in a horizontal circle at one end of a light inextensible string of length 40 cm, as shown. The other end of the string is attached to a fixed point \(O\). The angular velocity of \(P\) is \(\omega\) rad s\(^{-1}\). \includegraphics{figure_2} If the angle \(\theta\) which the string makes with the vertical must not exceed 60°, calculate the greatest possible value of \(\omega\). [7 marks]
Edexcel M3 Q3
8 marks Standard +0.3
A particle \(P\) of mass \(m\) kg is attached to one end of a light elastic string of natural length 0·5 m and modulus of elasticity \(\frac{mg}{2}\) N. The other end of the string is attached to a fixed point \(O\) and \(P\) hangs vertically below \(O\).
  1. Find the stretched length of the string when \(P\) rests in equilibrium. [3 marks]
  2. Find the elastic potential energy stored in the string in the equilibrium position. [2 marks]
\(P\), which is still attached to the string, is now held at rest at \(O\) and then lowered gently into its equilibrium position.
  1. Find the work done by the weight of the particle as it moves from \(O\) to the equilibrium position. [2 marks]
  2. Explain the discrepancy between your answers to parts (b) and (c). [1 mark]
Edexcel M3 Q4
8 marks Challenging +1.2
A particle \(P\), of mass \(m\) kg, is attached to two light elastic strings, each of natural length \(l\) m and modulus of elasticity \(3mg\) N. The other ends of the strings are attached to the fixed points \(A\) and \(B\), where \(AB\) is horizontal and \(AB = 2l\) m. \includegraphics{figure_4} If \(P\) rests in equilibrium vertically below the mid-point of \(AB\), with each string making an angle \(\theta\) with the vertical, show that $$\cot \theta - \cos \theta = \frac{1}{6}.$$ [8 marks]
Edexcel M3 Q5
14 marks Standard +0.3
A small bead \(P\), of mass \(m\) kg, can slide on a smooth circular ring, with centre \(O\) and radius \(r\) m, which is fixed in a vertical plane. \(P\) is projected from the lowest point \(L\) of the ring with speed \(\sqrt{(3gr)}\) ms\(^{-1}\). When \(P\) has reached a position such that \(OP\) makes an angle \(\theta\) with the downward vertical, as shown, its speed is \(v\) ms\(^{-1}\). \includegraphics{figure_5}
  1. Show that \(v^2 = gr(1 + 2 \cos \theta)\). [5 marks]
  2. Show that the magnitude of the reaction \(RN\) of the ring on the bead is given by $$R = mg(1 + 3 \cos \theta).$$ [4 marks]
  3. Find the values of \(\cos \theta\) when
    1. \(P\) is instantaneously at rest,
    2. the reaction \(R\) is instantaneously zero. [2 marks]
  4. Hence show that the ratio of the heights of \(P\) above \(L\) in cases (i) and (ii) is \(9:8\). [3 marks]
Edexcel M3 Q6
15 marks Standard +0.8
A light elastic string, of natural length 0·8 m, has one end fastened to a fixed point \(O\). The other end of the string is attached to a particle \(P\) of mass 0·5 kg. When \(P\) hangs in equilibrium, the length of the string is 1·5 m.
  1. Calculate the modulus of elasticity of the string. [3 marks]
\(P\) is displaced to a point 0·5 m vertically below its equilibrium position and released from rest. \begin{enumerate}[label=(\alph*)] \setcounter{enumi}{1} \item Show that the subsequent motion of \(P\) is simple harmonic, with period 1·68 s. [5 marks] \item Calculate the maximum speed of \(P\) during its motion. [3 marks] \item Show that the time taken for \(P\) to first reach a distance 0·25 m from the point of release is 0·28 s, to 2 significant figures. [4 marks] \end{enumerate]
Edexcel M3 Q7
16 marks Challenging +1.2
  1. Show that the centre of mass of a uniform solid hemisphere of radius \(r\) is at a distance \(\frac{3r}{8}\) from the centre \(O\) of the plane face. [7 marks]
The figure shows the vertical cross-section of a rough solid hemisphere at rest on a rough inclined plane inclined at an angle \(\theta\) to the horizontal, where \(\sin \theta = \frac{3}{10}\). \includegraphics{figure_7} \begin{enumerate}[label=(\alph*)] \setcounter{enumi}{1} \item Indicate on a copy of the figure the three forces acting on the hemisphere, clearly stating what they are, and paying special attention to their lines of action. [3 marks] \item Given that the plane face containing the diameter \(AB\) makes an angle \(\alpha\) with the vertical, show that \(\cos \alpha = \frac{4}{5}\). [6 marks] \end{enumerate]
Edexcel M3 Q1
7 marks Standard +0.3
A motorcyclist rides in a cylindrical well of radius 5 m. He maintains a horizontal circular path at a constant speed of 10 ms\(^{-1}\). The coefficient of friction between the wall and the wheels of the cycle is \(\mu\). \includegraphics{figure_1} Modelling the cyclist and his machine as a particle in contact with the wall, show that he will not slip downwards provided that \(\mu \geq 0.49\). [7 marks]
Edexcel M3 Q2
7 marks Standard +0.3
A particle \(P\) moves with simple harmonic motion in a straight line. The centre of oscillation is \(O\). When \(P\) is at a distance 1 m from \(O\), its speed is 8 ms\(^{-1}\). When it is at a distance 2 m from \(O\), its speed is 4 ms\(^{-1}\).
  1. Find the amplitude of the motion. [4 marks]
  2. Show that the period of motion is \(\frac{\pi}{2}\) s. [3 marks]
Edexcel M3 Q3
9 marks Standard +0.8
A particle of mass \(m\) kg is attached to the end \(B\) of a light elastic string \(AB\). The string has natural length \(l\) m and modulus of elasticity \(\lambda\) N. \includegraphics{figure_3} The end \(A\) is attached to a fixed point on a smooth plane inclined at an angle \(\alpha\) to the horizontal, as shown, and the particle rests in equilibrium with the length \(AB = \frac{5l}{4}\) m.
  1. Show that \(\lambda = 4 mg \sin \alpha\). [3 marks]
The particle is now moved and held at rest at \(A\) with the string slack. It is then gently released so that it moves down the plane along a line of greatest slope.
  1. Find the greatest distance from \(A\) that the particle reaches down the plane. [6 marks]
Edexcel M3 Q4
9 marks Standard +0.3
The acceleration \(a\) ms\(^{-2}\) of a particle \(P\) moving in a straight line away from a fixed point \(O\) is given by \(a = \frac{k}{1+t}\), where \(t\) is the time that has elapsed since \(P\) left \(O\), and \(k\) is a constant.
  1. By solving a suitable differential equation, find an expression for the velocity \(v\) ms\(^{-1}\) of \(P\) in terms of \(t\), \(k\) and another constant \(c\). [3 marks]
Given that \(v = 0\) when \(t = 0\) and that \(v = 4\) when \(t = 2\),
  1. show that \(v \ln 3 = 4 \ln (1 + t)\). [3 marks]
  2. Calculate the time when \(P\) has a speed of 8 ms\(^{-1}\). [3 marks]
Edexcel M3 Q5
13 marks Standard +0.3
A particle of mass \(m\) kg, at a distance \(x\) m from the centre of the Earth, experiences a force of magnitude \(\frac{km}{x^2}\) N towards the centre of the Earth, where \(k\) is a constant. Given that the radius of the Earth is \(6.37 \times 10^6\) m, and that a 3 kg mass experiences a force of 30 N at the surface of the Earth,
  1. calculate the value of \(k\), stating the units of your answer. [3 marks]
The 3 kg mass falls from rest at a distance \(x = 12.74 \times 10^6\) m from the centre of the Earth. Ignoring air resistance,
  1. show that it reaches the surface of the Earth with speed \(7.98 \times 10^3\) ms\(^{-1}\). [7 marks]
In a simplified model, the particle is assumed to fall with a constant acceleration 10 ms\(^{-2}\). According to this model it attains the same speed as in (b), \(7.98 \times 10^3\) ms\(^{-1}\), at a distance \((12.74 - d) \times 10^6\) m from the centre of the Earth.
  1. Find the value of \(d\). [3 marks]
Edexcel M3 Q6
15 marks Standard +0.8
A particle \(P\) of mass 0.4 kg hangs by a light, inextensible string of length 20 cm whose other end is attached to a fixed point \(O\). It is given a horizontal velocity of 1.4 ms\(^{-1}\) so that it begins to move in a vertical circle. If in the ensuing motion the string makes an angle of \(\theta\) with the downward vertical through \(O\), show that
  1. \(\theta\) cannot exceed 60°, [6 marks]
  2. the tension, \(T\) N, in the string is given by \(T = 3.92(3 \cos \theta - 1)\). [4 marks]
If the string breaks when \(\cos \theta = \frac{3}{5}\) and \(P\) is ascending,
  1. find the greatest height reached by \(P\) above the initial point of projection. [5 marks]
Edexcel M3 Q7
15 marks Challenging +1.8
A uniform solid sphere, of radius \(a\), is divided into two sections by a plane at a distance \(\frac{a}{2}\) from the centre and parallel to a diameter.
  1. Show that the centre of gravity of the smaller cap from its plane face is \(\frac{7a}{40}\). [9 marks]
This smaller cap is now placed on an inclined plane whose angle of inclination to the horizontal is \(\theta\). The plane is rough enough to prevent slipping and the cap rests with its curved surface in contact with the plane.
  1. If the maximum value of \(\theta\) for which this is possible without the cap turning over is 30°, find the corresponding maximum inclination of the axis of symmetry of the cap to the vertical. [6 marks]
Edexcel M3 Q1
7 marks Standard +0.8
A light spring, of natural length 30 cm, is fixed in a vertical position. When a small ball of mass 0.4 kg rests on top of it, the spring is compressed by 10 cm. The ball is then held at a height of 15 cm vertically above the top of the spring and released from rest. Calculate the maximum compression of the string in the resulting motion. [7 marks]