Questions — Edexcel (10514 questions)

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Edexcel M3 Q6
14 marks Challenging +1.2
The figure show a wine glass consisting of a hemispherical cup of radius \(r\), a cylindrical solid stem of height \(r\) and a circular base of radius \(r\). The cup has mass \(M\) and the stem has mass \(m\). Modelling the cup as a thin, uniform hemispherical shell, the base as a uniform lamina made of the same thin material as the cup, and the stem as a uniform solid cylinder,
  1. show that the mass of the circular base is \(\frac{1}{2}M\). [1 mark]
Given that the centre of mass of the glass is at a distance \(\frac{13r}{14}\) from the base along the vertical axis of symmetry,
  1. express \(M\) in terms of \(m\). [6 marks]
If the cup is now filled with liquid whose mass is \(2M\),
  1. show that the position of the centre of mass rises through a distance \(\frac{13r}{35}\). [6 marks]
  2. State an assumption that you have made about the liquid. [1 mark]
\includegraphics{figure_6}
Edexcel M3 Q7
15 marks Standard +0.8
A particle of mass \(m\) kg is attached to one end of an 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 \(O\). The particle hangs in equilibrium at a point \(C\).
    1. Prove that if the particle is slightly displaced in a vertical direction, it will perform simple harmonic motion about \(C\). [6 marks]
    2. Find the period, \(T\) s, of the motion in terms of \(l\), \(m\) and \(\lambda\). [1 mark]
    3. Explain the significance of the term 'slightly' as used in (i) above. [1 mark]
When an additional mass \(M\) is attached to the particle, it is found that the system oscillates about a point \(D\), at a distance \(d\) below \(C\), with period \(T_1\) s.
    1. Write down an expression for \(T_1\) in terms of \(l\), \(m\), \(M\) and \(\lambda\). [2 marks]
    2. Hence show that \(T_1^2 - T^2 = \frac{4\pi^2 d}{g}\). [5 marks]
Edexcel M3 Q1
7 marks Challenging +1.2
A particle of mass \(m\) kg is attached to one end of a light inextensible string of length \(l\) m whose other end is fixed to a point \(O\). The particle is made to move in a vertical circle with centre \(O\), with constant angular velocity \(\omega\) rad s\(^{-1}\). At a certain instant it is in the position shown, where the string makes an angle \(\theta\) radians with the downward vertical through \(O\). \includegraphics{figure_1}
  1. Find an expression, in terms of \(m\), \(l\) and \(\omega\), for the kinetic energy of the particle at this instant. [2 marks]
  2. Find an expression, in terms of \(m\), \(g\), \(l\) and \(\theta\), for the potential energy of the particle relative to the horizontal plane through the lowest point \(A\). [2 marks]
  3. Determine the position of the particle when the rate of increase of its total energy, with respect to time, is a maximum. [3 marks]
Edexcel M3 Q2
7 marks Standard +0.3
A particle moves along a straight line in such a way that its displacement \(x\) m from a fixed point \(O\) on the line, at time \(t\) seconds after it leaves \(O\), is given by \(x = p \sin \omega t + q \cos \omega t\) where \(p\), \(q\) and \(\omega\) are constants.
  1. Show that the motion of the particle is simple harmonic. [5 marks]
  2. If the particle leaves \(O\) with speed 15 ms\(^{-1}\), and \(\omega = 3\), find the amplitude of the motion. [2 marks]
Edexcel M3 Q3
9 marks Standard +0.3
A particle \(P\) of mass 0.2 kg moves in a horizontal circle on one end of an elastic string whose other end is attached to a fixed point \(O\). The angular velocity of \(P\) is \(\pi\) rad s\(^{-1}\). The natural length of the string is 1 m and, while \(P\) is in motion, the distance \(OP = 1.15\) m.
  1. Calculate, to 3 significant figures, the modulus of elasticity of the string. [6 marks]
The motion now ceases and \(P\) hangs at rest vertically below \(O\).
  1. Show that the extension in the string in this position is about 13 cm. [3 marks]
Edexcel M3 Q4
10 marks Challenging +1.2
A small stone \(P\) of mass \(m\) kg is attached to one end of a light elastic string of modulus \(3mg\) N and natural length \(2l\) m. The other end of the string is fixed to a point \(O\) at a height \(3l\) m above a horizontal surface. \(P\) is released from rest at \(O\); it hits the surface and rebounds to a height of \(2l\) m. The coefficient of restitution between \(P\) and the surface is \(e\). Calculate the value of \(e\). [9 marks] State one assumption (other than the string being light) that you have used in your solution. [1 mark]
Edexcel M3 Q5
13 marks Challenging +1.2
A small sphere \(S\), of mass \(m\) kg is released from rest at the surface of a liquid in a right circular cylinder whose axis is vertical. When \(S\) is moving downwards with speed \(v\) ms\(^{-1}\), the viscous resistive force acting upwards on it has magnitude \(v^2\) N.
  1. Write down a differential equation for the motion of \(S\), clearly defining any symbol(s) that you introduce. [4 marks]
  2. Find, in terms of \(m\), the distance \(S\) has fallen when its speed is \(\sqrt{\frac{mg}{2}}\) ms\(^{-1}\). [9 marks]
Edexcel M3 Q6
13 marks Challenging +1.8
The diagram shows two identical particles, each of mass \(m\) kg, connected by a thin, light inextensible string. \(P\) slides on the surface of a smooth right circular cylinder fixed with its axis, through \(O\), horizontal. \(Q\) moves vertically. \(OP\) makes an angle \(\theta\) radians with the horizontal. \includegraphics{figure_6} The system is released from rest in the position where \(\theta = 0\).
  1. Show that the vertical distance moved by \(Q\) is \(\frac{\theta}{\sin \theta}\) times the vertical distance moved by \(P\). [4 marks]
  2. In the position where \(\theta = \frac{\pi}{6}\), prove that the reaction of the cylinder on \(P\) has magnitude \(\left(1-\frac{\pi}{6}\right)mg\) N. [9 marks]
Edexcel M3 Q7
16 marks Challenging +1.8
A container consists of two sections made from the same material: a hollow portion formed by removing a cone (shaded in the figure) from a solid cylinder of radius \(r\) and height \(h\), and a solid hemisphere of radius \(r\). The vertex of the removed cone coincides with the centre \(O\) of the horizontal plane face of the hemisphere. \(CD\) is a diameter of this plane face. \includegraphics{figure_7}
  1. Show that the distance of the centre of mass of the container from the plane face of the hemisphere is \(\left|\frac{3}{8}(h-r)\right|\). Explain why the modulus sign is necessary. [9 marks]
  2. Find the ratio \(h : r\) in each of the following cases:
    1. When the container is suspended from the point \(C\), the angle made by \(CD\) with the vertical is equal to the angle which \(CD\) would make with the vertical if the hemisphere alone were suspended from \(C\). [4 marks]
    2. The container is able to stand without toppling in any position when it is placed with the surface of the hemispherical part in contact with a smooth horizontal table. [3 marks]
Edexcel M3 Q1
3 marks Easy -1.2
A particle of mass \(0.6\) kg moves in a horizontal circle with constant angular speed \(1.5\) radians per second. If the force directed towards the centre of the circle has magnitude \(0.27\) N, find the radius of the circular path. [3 marks]
Edexcel M3 Q2
8 marks Standard +0.3
The diagram shows a particle of mass \(0.7\) kg resting on a rough horizontal table. The coefficient of friction between the particle and the table is \(0.25\). A light elastic string, of natural length \(50\) cm and modulus of elasticity \(6.86\) N, is attached to the particle. The string is kept at an angle of \(60°\) to the horizontal and is gradually extended by pulling on it until the particle moves. Show that the particle starts to move when the extension in the string is \(17\) cm. \includegraphics{figure_2} [8 marks]
Edexcel M3 Q3
8 marks Standard +0.8
A smooth circular hoop of radius \(1\) m, with centre \(O\), is fixed in a vertical plane. A small ring \(Q\), of mass \(0.1\) kg, is threaded onto the hoop and held so that the angle \(QOH = 30°\), where \(H\) is the highest point of the hoop. \(Q\) is released from rest at this position. Find, in terms of \(g\), \begin{enumerate}[label=(\alph*)] \item the horizontal and vertical components of the acceleration of \(Q\) when it reaches the lowest point of the hoop; [5 marks] \item the magnitude of the reaction between \(Q\) and the hoop at this lowest point. [3 marks]
Edexcel M3 Q4
9 marks Standard +0.8
A particle \(P\) moves with simple harmonic motion in a straight line, with the centre of motion at the point \(O\) on the line. \(A\) and \(B\) are on opposite sides of \(O\), with \(OA = 4\) m, \(OB = 6\) m. When passing through \(A\) and \(B\), \(P\) has speed \(6\) ms\(^{-1}\) and \(4\) ms\(^{-1}\) respectively. \includegraphics{figure_4} \begin{enumerate}[label=(\alph*)] \item Find the amplitude of the motion. [6 marks] \item Show that the period of motion is \(2\pi\) s. [3 marks]
Edexcel M3 Q5
11 marks Challenging +1.2
  1. Prove that the centre of mass of a uniform solid hemisphere of radius \(r\) is at a distance \(\frac{3r}{8}\) from its plane face. [7 marks]
A solid cylinder of radius \(\frac{3r}{4}\) and height \(kr\), where \(k < 1\), is welded to a uniform hemisphere of radius \(r\) made of the same material, so that their axes of symmetry coincide. The figure shows the cross section of the resulting solid. If the centre of mass of this solid is at \(O\), the centre of the plane face of the hemisphere, \includegraphics{figure_5}
  1. find the value of \(k\). [4 marks]
Edexcel M3 Q6
16 marks Challenging +1.2
The gravitational attraction \(F\) N between two point masses \(m_1\) kg and \(m_2\) kg at a distance \(x\) m apart is given by \(F = \frac{km_1m_2}{x^2}\), where \(k\) is a constant. Given that a small body of mass \(1\) kg experiences a force of \(g\) N at the surface of the Earth, which has radius \(R\) m and mass \(M\) kg,
  1. show that \(k = \frac{gR^2}{M}\). [2 marks]
A small communications satellite of mass \(m\) kg is put into a circular orbit of radius \(r\) m around the Earth. Modelling the Earth as a particle of mass \(M\) kg, and using the value of \(k\) from (a),
  1. prove that the period of rotation, \(T\) s, of the satellite is given by \(T = \frac{2\pi}{R}\sqrt{\frac{r^3}{g}}\). [4 marks]
To cover transmission to any point on the Earth, three small satellites \(X\), \(Y\) and \(Z\), each of mass \(m\) kg, are placed in a common circular orbit of radius \(r\) and form an equilateral triangle as shown. \includegraphics{figure_6}
  1. Show on a copy of the diagram the direction of the three forces acting on \(X\). [1 mark]
  2. State, with a reason, the direction of the resultant force on \(X\). [2 marks]
  3. Show that the period of rotation of \(X\) is given by \(T\sqrt{\frac{3M}{2M + m\sqrt{3}}}\) s, where \(T\) s is the period found in (b). [7 marks]
Edexcel M3 Q7
17 marks Challenging +1.8
One end of a light elastic string, of natural length \(3l\) m, is attached to a fixed point \(O\). A particle of mass \(m\) kg is attached to the other end of the string. When the particle hangs freely in equilibrium, the string is extended by a length of \(l\) m. The particle is then pulled down through a further distance \(2l\) m and released from rest.
  1. Prove that as long as the string is taut, the particle performs simple harmonic motion about its equilibrium position. [5 marks]
  2. Show that the time between the release of the particle and the instant when the string becomes slack is \(\frac{2\pi}{3}\sqrt{\frac{l}{g}}\) s. [4 marks]
  3. Find the greatest height reached by the particle above its point of release. [4 marks]
  4. Show that the time \(T\) s taken to reach this greatest height from the moment of release is given by \(T = \left(\frac{2\pi}{3} + \sqrt{3}\right)\sqrt{\frac{l}{g}}\). [4 marks]
Edexcel M3 Q1
8 marks Standard +0.8
A particle \(P\) of mass \(m\) kg moves in a horizontal circle at one end of a light elastic string of natural length \(l\) m and modulus of elasticity \(mg\) N. The other end of the string is attached to a fixed point \(O\). Given that the string makes an angle of \(60°\) with the vertical,
  1. show that \(OP = 3l\) m. [4 marks]
  2. Find, in terms of \(l\) and \(g\), the angular speed of \(P\). [4 marks]
Edexcel M3 Q2
10 marks Challenging +1.2
A particle \(P\) of mass \(m\) kg moves vertically upwards under gravity, starting from ground level. It is acted on by a resistive force of magnitude \(m f(x)\) N, where \(f(x)\) is a function of the height \(x\) m of \(P\) above the ground. When \(P\) is at this height, its upward speed \(v\) ms\(^{-1}\) is given by $$v^2 = 2e^{-2gx} - 1.$$
  1. Write down a differential equation for the motion of \(P\) and hence determine \(f(x)\) in terms of \(g\) and \(x\). [5 marks]
  2. Show that the greatest height reached by \(P\) above the ground is \(\frac{1}{2g} \ln 2\) m. [2 marks]
Given that the work, in J, done by \(P\) against the resisting force as it moves from ground level to a point \(H\) m above the ground is equal to \(\int_0^H m f(x) dx\),
  1. show that the total work done by \(P\) against the resistance during its upward motion is \(\frac{1}{2}m(1 - \ln 2)\) J. [3 marks]
Edexcel M3 Q3
10 marks Challenging +1.2
A car of mass \(m\) kg moves round a curve of radius \(r\) m on a road which is banked at an angle \(\theta\) to the horizontal. When the speed of the car is \(u\) ms\(^{-1}\), the car experiences no sideways frictional force. Given that \(\tan \theta = \frac{u^2}{gr}\), show that the sideways frictional force on the car when its speed is \(\frac{u}{2}\) ms\(^{-1}\) has magnitude \(\frac{3}{4}mg \sin \theta\) N. [10 marks]
Edexcel M3 Q4
10 marks Challenging +1.2
Two light elastic strings, each of length \(l\) m and modulus of elasticity \(\lambda\) N, are attached to a particle \(P\) of mass \(m\) kg. The other ends of the strings are attached to fixed points \(A\) and \(B\) on the same horizontal level, where \(AB = 2l\) m. \(P\) is held vertically below the mid-point of \(AB\), with each string taut and inclined at \(30°\) to the horizontal, and released from rest. Given that \(P\) comes to instantaneous rest when each string makes an angle of \(60°\) with the horizontal, show that \(\lambda = \frac{3mg}{6 - 2\sqrt{3}}\). \includegraphics{figure_1} [10 marks]
Edexcel M3 Q5
11 marks Standard +0.8
A particle \(P\) is projected horizontally with speed \(u\) ms\(^{-1}\) from the highest point of a smooth sphere of radius \(r\) m and centre \(O\). It moves on the surface in a vertical plane, and at a particular instant the radius \(OP\) makes an angle \(\theta\) with the upward vertical, as shown. At this instant \(P\) has speed \(v\) ms\(^{-1}\) and the magnitude of the reaction between \(P\) and the sphere is \(X\) N. \includegraphics{figure_2}
  1. Assuming that \(u^2 < gr\), show that
    1. \(v^2 = u^2 + 2gr(1 - \cos \theta)\), [2 marks]
    2. \(X = mg\left(3\cos \theta - 2 - \frac{u^2}{gr}\right)\). [4 marks]
  2. Show that \(P\) leaves the surface of the sphere when \(\cos \theta = \frac{u^2 + 2gr}{3gr}\). [3 marks]
  3. Discuss what happens if \(u^2 \geq gr\). [2 marks]
Edexcel M3 Q6
12 marks Standard +0.3
A particle \(P\) of mass \(m\) kg hangs in equilibrium at one end of a light spring, of natural length \(l\) m and modulus of elasticity \(\lambda\) N, whose other end is fixed at a point vertically above \(P\). In this position the length of the spring is \((l + e)\) m. When \(P\) is displaced vertically through a small distance and released, it performs simple harmonic motion with 5 oscillations per second.
  1. Show that \(\frac{\lambda}{l} = 100n^2m\). [8 marks]
  2. Express \(e\) in terms of \(g\). [2 marks]
  3. Determine, in terms of \(m\) and \(l\), the magnitude of the tension in the spring when it is stretched to twice its natural length. [2 marks]
Edexcel M3 Q7
14 marks Challenging +1.2
  1. Prove that the centre of mass of a uniform solid right circular cone of height \(h\) and base radius \(r\) is at a distance \(\frac{3h}{4}\) from the vertex. [7 marks]
An item of confectionery consists of a thin wafer in the form of a hollow right circular cone of height \(h\) and mass \(m\), filled with solid chocolate, also of mass \(m\), to a depth of \(kh\) as shown. The centre of mass of the item is at \(O\), the centre of the horizontal plane face of the chocolate. \includegraphics{figure_3}
  1. Show that \(k = \frac{8h}{15}\). [3 marks]
In the packaging process, the cone has to move on a conveyor belt inclined at an angle \(\alpha\) to the horizontal as shown. If the belt is rough enough to prevent sliding, and the maximum value of \(\alpha\) for which the cone does not topple is \(45°\), \includegraphics{figure_4}
  1. find the radius of the base of the cone in terms of \(h\). [4 marks]
Edexcel M3 Q1
7 marks Standard +0.3
A small bead is threaded onto a smooth circular hoop, of radius \(r\) m, fixed in a vertical plane. It is projected with speed \(u\) ms\(^{-1}\) from the lowest point of the hoop. Find \(u\) in terms of \(g\) and \(r\) if
  1. the bead just reaches the highest point of the hoop, [3 marks]
  2. the reaction on the bead is zero when it is at the highest point of the hoop. [4 marks]
Edexcel M3 Q2
7 marks Standard +0.3
An ornamental tower is made from a solid right circular cylinder of mass \(M\) and height \(h\) by removing three identical cylindrical sections, each of height \(\frac{h}{8}\), equally spaced above a base of height \(\frac{h}{4}\), as shown. The tower is held in position by light, thin vertical strips \(AB\) and \(CD\). \includegraphics{figure_2} Find the distance of the centre of mass of the tower from its horizontal base. [7 marks]