Questions — Edexcel M3 (510 questions)

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Edexcel M3 2007 June Q5
11 marks Standard +0.3
A particle \(P\) moves on the \(x\)-axis with simple harmonic motion about the origin \(O\) as centre. When \(P\) is at a distance \(0.04\) m from \(O\), its speed is \(0.2\) m s\(^{-1}\) and the magnitude of its acceleration is \(1\) m s\(^{-2}\).
  1. Find the period of the motion. [3]
The amplitude of the motion is \(a\) metres. Find
  1. the value of \(a\), [3]
  2. the total time, within one complete oscillation, for which the distance \(OP\) is greater than \(\frac{3}{4}a\) metres. [5]
Edexcel M3 2007 June Q6
12 marks Standard +0.8
A particle \(P\) is free to move on the smooth inner surface of a fixed thin hollow sphere of internal radius \(a\) and centre \(O\). The particle passes through the lowest point of the spherical surface with speed \(U\). The particle loses contact with the surface when \(OP\) is inclined at an angle \(\alpha\) to the upward vertical.
  1. Show that \(U^2 = ag(2 + 3\cos \alpha)\). [7]
The particle has speed \(W\) as it passes through the level of \(O\). Given that \(\cos \alpha = \frac{1}{\sqrt{3}}\),
  1. show that \(W^2 = ag\sqrt{3}\). [5]
Edexcel M3 2007 June Q7
15 marks Challenging +1.2
\includegraphics{figure_1} A light elastic string, of natural length \(3l\) and modulus of elasticity \(\lambda\), has its ends attached to two points \(A\) and \(B\), where \(AB = 3l\) and \(AB\) is horizontal. A particle \(P\) of mass \(m\) is attached to the mid-point of the string. Given that \(P\) rests in equilibrium at a distance \(2l\) below \(AB\), as shown in Figure 1,
  1. show that \(\lambda = \frac{15mg}{16}\) [9]
The particle is pulled vertically downwards from its equilibrium position until the total length of the elastic string is \(7.8l\). The particle is released from rest.
  1. Show that \(P\) comes to instantaneous rest on the line \(AB\). [6]
Edexcel M3 2009 June Q1
9 marks Standard +0.3
A light elastic string has natural length \(8\) m and modulus of elasticity \(80\) N. The ends of the string are attached to fixed points \(P\) and \(Q\) which are on the same horizontal level and \(12\) m apart. A particle is attached to the mid-point of the string and hangs in equilibrium at a point \(4.5\) m below \(PQ\).
  1. Calculate the weight of the particle. [6]
  2. Calculate the elastic energy in the string when the particle is in this position. [3]
Edexcel M3 2009 June Q2
8 marks Standard +0.3
[The centre of mass of a uniform hollow cone of height \(h\) is \(\frac{1}{3}h\) above the base on the line from the centre of the base to the vertex.] \includegraphics{figure_1} A marker for the route of a charity walk consists of a uniform hollow cone fixed on to a uniform solid cylindrical ring, as shown in Figure 1. The hollow cone has base radius \(r\), height \(9h\) and mass \(m\). The solid cylindrical ring has outer radius \(r\), height \(2h\) and mass \(3m\). The marker stands with its base on a horizontal surface.
  1. Find, in terms of \(h\), the distance of the centre of mass of the marker from the horizontal surface. [5]
When the marker stands on a plane inclined at arctan \(\frac{1}{12}\) to the horizontal it is on the point of toppling over. The coefficient of friction between the marker and the plane is large enough to be certain that the marker will not slip.
  1. Find \(h\) in terms of \(r\). [3]
Edexcel M3 2009 June Q3
8 marks Standard +0.3
\includegraphics{figure_2} A particle \(P\) of mass \(m\) moves on the smooth inner surface of a hemispherical bowl of radius \(r\). The bowl is fixed with its rim horizontal as shown in Figure 2. The particle moves with constant angular speed \(\sqrt{\left(\frac{3g}{2r}\right)}\) in a horizontal circle at depth \(d\) below the centre of the bowl.
  1. Find, in terms of \(m\) and \(g\), the magnitude of the normal reaction of the bowl on \(P\). [4]
  2. Find \(d\) in terms of \(r\). [4]
Edexcel M3 2009 June Q4
9 marks Standard +0.3
The finite region bounded by the \(x\)-axis, the curve \(y = \frac{1}{x}\), the line \(x = \frac{1}{4}\) and the line \(x = 1\), is rotated through one complete revolution about the \(x\)-axis to form a uniform solid of revolution.
  1. Show that the volume of the solid is \(21\pi\). [4]
  2. Find the coordinates of the centre of mass of the solid. [5]
Edexcel M3 2009 June Q5
11 marks Challenging +1.2
One end of a light inextensible string of length \(l\) is attached to a fixed point \(A\). The other end is attached to a particle \(P\) of mass \(m\), which is held at a point \(B\) with the string taut and \(AP\) making an angle arccos \(\frac{1}{4}\) with the downward vertical. The particle is released from rest. When \(AP\) makes an angle \(\theta\) with the downward vertical, the string is taut and the tension in the string is \(T\).
  1. Show that $$T = 3mg \cos \theta - \frac{mg}{2}.$$ [6]
\includegraphics{figure_3} At an instant when \(AP\) makes an angle of \(60°\) to the downward vertical, \(P\) is moving upwards, as shown in Figure 3. At this instant the string breaks. At the highest point reached in the subsequent motion, \(P\) is at a distance \(d\) below the horizontal through \(A\).
  1. Find \(d\) in terms of \(l\). [5]
Edexcel M3 2009 June Q6
14 marks Challenging +1.2
A cyclist and her bicycle have a combined mass of \(100\) kg. She is working at a constant rate of \(80\) W and is moving in a straight line on a horizontal road. The resistance to motion is proportional to the square of her speed. Her initial speed is \(4\) m s\(^{-1}\) and her maximum possible speed under these conditions is \(20\) m s\(^{-1}\). When she is at a distance \(x\) m from a fixed point \(O\) on the road, she is moving with speed \(v\) m s\(^{-1}\) away from \(O\).
  1. Show that $$v \frac{dv}{dx} = \frac{8000 - v^3}{10000v}.$$ [5]
  2. Find the distance she travels as her speed increases from \(4\) m s\(^{-1}\) to \(8\) m s\(^{-1}\). [5]
  3. Use the trapezium rule, with 2 intervals, to estimate how long it takes for her speed to increase from \(4\) m s\(^{-1}\) to \(8\) m s\(^{-1}\). [4]
Edexcel M3 2009 June Q7
16 marks Challenging +1.2
\includegraphics{figure_4} \(A\) and \(B\) are two points on a smooth horizontal floor, where \(AB = 5\) m. A particle \(P\) has mass \(0.5\) kg. One end of a light elastic spring, of natural length \(2\) m and modulus of elasticity \(16\) N, is attached to \(P\) and the other end is attached to \(A\). The ends of another light elastic spring, of natural length \(1\) m and modulus of elasticity \(12\) N, are attached to \(P\) and \(B\), as shown in Figure 4.
  1. Find the extensions in the two springs when the particle is at rest in equilibrium. [5]
Initially \(P\) is at rest in equilibrium. It is then set in motion and starts to move towards \(B\). In the subsequent motion \(P\) does not reach \(A\) or \(B\).
  1. Show that \(P\) oscillates with simple harmonic motion about the equilibrium position. [4]
  2. Given that the initial speed of \(P\) is \(\sqrt{10}\) m s\(^{-1}\), find the proportion of time in each complete oscillation for which \(P\) stays within \(0.25\) m of the equilibrium position. [7]
Edexcel M3 2012 June Q1
9 marks Standard +0.3
A particle \(P\) is moving along the positive \(x\)-axis. At time \(t = 0\), \(P\) is at the origin \(O\). At time \(t\) seconds, \(P\) is \(x\) metres from \(O\) and has velocity \(v = 2e^{-t}\) m s\(^{-1}\) in the direction of \(x\) increasing.
  1. Find the acceleration of \(P\) in terms of \(x\). [3]
  2. Find \(x\) in terms of \(t\). [6]
Edexcel M3 2012 June Q2
8 marks Standard +0.3
A particle \(P\) moves in a straight line with simple harmonic motion about a fixed centre \(O\). The period of the motion is \(\frac{\pi}{2}\) seconds. At time \(t\) seconds the speed of \(P\) is \(v\) m s\(^{-1}\). When \(t = 0\), \(P\) is at \(O\) and \(v = 6\). Find
  1. the greatest distance of \(P\) from \(O\) during the motion, [3]
  2. the greatest magnitude of the acceleration of \(P\) during the motion, [2]
  3. the smallest positive value of \(t\) for which \(P\) is 1 m from \(O\). [3]
Edexcel M3 2012 June Q3
10 marks Standard +0.3
\includegraphics{figure_1} A particle \(Q\) of mass 5 kg is attached by two light inextensible strings to two fixed points \(A\) and \(B\) on a vertical pole. Each string has length 0.6 m and \(A\) is 0.4 m vertically above \(B\), as shown in Figure 1. Both strings are taut and \(Q\) is moving in a horizontal circle with constant angular speed 10 rad s\(^{-1}\). Find the tension in
  1. \(AQ\),
  2. \(BQ\). [10]
Edexcel M3 2012 June Q4
10 marks Standard +0.3
\includegraphics{figure_2} Figure 2 shows the cross-section \(AVBC\) of the solid \(S\) formed when a uniform right circular cone of base radius \(a\) and height \(a\), is removed from a uniform right circular cone of base radius \(a\) and height \(2a\). Both cones have the same axis \(VCO\), where \(O\) is the centre of the base of each cone.
  1. Show that the distance of the centre of mass of \(S\) from the vertex \(V\) is \(\frac{5}{4}a\). [5]
The mass of \(S\) is \(M\). A particle of mass \(kM\) is attached to \(S\) at \(B\). The system is suspended by a string attached to the vertex \(V\), and hangs freely in equilibrium. Given that \(VA\) is at an angle \(45°\) to the vertical through \(V\),
  1. find the value of \(k\). [5]
Edexcel M3 2012 June Q5
12 marks Standard +0.8
A fixed smooth sphere has centre \(O\) and radius \(a\). A particle \(P\) is placed on the surface of the sphere at the point \(A\), where \(OA\) makes an angle \(\alpha\) with the upward vertical through \(O\). The particle is released from rest at \(A\). When \(OP\) makes an angle \(\theta\) to the upward vertical through \(O\), \(P\) is on the surface of the sphere and the speed of \(P\) is \(v\). Given that \(\cos \alpha = \frac{3}{5}\)
  1. show that $$v^2 = \frac{2ga}{5}(3 - 5\cos \theta)$$ [4]
  2. find the speed of \(P\) at the instant when it loses contact with the sphere. [8]
Edexcel M3 2012 June Q6
12 marks Standard +0.8
\includegraphics{figure_3} Figure 3 shows a uniform equilateral triangular lamina \(PRT\) with sides of length \(2a\).
  1. Using calculus, prove that the centre of mass of \(PRT\) is at a distance \(\frac{2\sqrt{3}}{3}a\) from \(R\). [6]
\includegraphics{figure_4} The circular sector \(PQU\), of radius \(a\) and centre \(P\), and the circular sector \(TUS\), of radius \(a\) and centre \(T\), are removed from \(PRT\) to form the uniform lamina \(QRSU\) shown in Figure 4.
  1. Show that the distance of the centre of mass of \(QRSU\) from \(U\) is \(\frac{2a}{3\sqrt{3} - \pi}\). [6]
Edexcel M3 2012 June Q7
14 marks Standard +0.8
A particle \(B\) of mass 0.5 kg is attached to one end of a light elastic string of natural length 0.75 m and modulus of elasticity 24.5 N. The other end of the string is attached to a fixed point \(A\). The particle is hanging in equilibrium at the point \(E\), vertically below \(A\).
  1. Show that \(AE = 0.9\) m. [3]
The particle is held at \(A\) and released from rest. The particle first comes to instantaneous rest at the point \(C\).
  1. Find the distance \(AC\). [5]
  2. Show that while the string is taut, \(B\) is moving with simple harmonic motion with centre \(E\). [4]
  3. Calculate the maximum speed of \(B\). [2]
Edexcel M3 2014 June Q1
8 marks Standard +0.3
A particle \(P\) of mass \(0.25\) kg is moving along the positive \(x\)-axis under the action of a single force. At time \(t\) seconds \(P\) is \(x\) metres from the origin \(O\) and is moving away from \(O\) with speed \(v\) m s\(^{-1}\) where \(\frac{\mathrm{d}v}{\mathrm{d}x} = 3\). It is given that \(x = 2\) and \(v = 3\) when \(t = 0\)
  1. Find the magnitude of the force acting on \(P\) when \(x = 5\) [4]
  2. Find the value of \(t\) when \(x = 5\) [4]
Edexcel M3 2014 June Q2
13 marks Standard +0.8
\includegraphics{figure_1} A cone of semi-vertical angle \(60°\) is fixed with its axis vertical and vertex upwards. A particle of mass \(m\) is attached to one end of a light inextensible string of length \(l\). The other end of the string is attached to a fixed point vertically above the vertex of the cone. The particle moves in a horizontal circle on the smooth outer surface of the cone with constant angular speed \(\omega\), with the string making a constant angle \(60°\) with the horizontal, as shown in Figure 1.
  1. Find the tension in the string, in terms of \(m\), \(l\), \(\omega\) and \(g\). [7]
The particle remains on the surface of the cone.
  1. Show that the time for the particle to make one complete revolution is greater than $$2\pi\sqrt{\frac{l\sqrt{3}}{2g}}$$ [6]
Edexcel M3 2014 June Q3
11 marks Challenging +1.2
One end \(A\) of a light elastic string \(AB\), of modulus of elasticity \(mg\) and natural length \(a\), is fixed to a point on a rough plane inclined at an angle \(\theta\) to the horizontal. The other end \(B\) of the string is attached to a particle of mass \(m\) which is held at rest on the plane. The string \(AB\) lies along a line of greatest slope of the plane, with \(B\) lower than \(A\) and \(AB = a\). The coefficient of friction between the particle and the plane is \(\mu\), where \(\mu < \tan \theta\). The particle is released from rest.
  1. Show that when the particle comes to rest it has moved a distance \(2a(\sin \theta - \mu \cos \theta)\) down the plane. [6]
  2. Given that there is no further motion, show that \(\mu \geqslant \frac{1}{3} \tan \theta\). [5]
Edexcel M3 2014 June Q4
16 marks Challenging +1.2
\includegraphics{figure_2} A smooth sphere of radius \(a\) is fixed with a point \(A\) of its surface in contact with a fixed vertical wall. A particle is placed on the highest point of the sphere and is projected towards the wall and perpendicular to the wall with horizontal speed \(\sqrt{\frac{2ag}{5}}\), as shown in Figure 2. The particle leaves the surface of the sphere with speed \(V\).
  1. Show that \(V = \sqrt{\frac{4ag}{5}}\) [7]
The particle strikes the wall at the point \(X\).
  1. Find the distance \(AX\). [9]
Edexcel M3 2014 June Q5
13 marks Standard +0.8
\includegraphics{figure_3} A uniform solid right circular cylinder has height \(h\) and radius \(r\). The centre of one plane face is \(O\) and the centre of the other plane face is \(Y\). A cylindrical hole is made by removing a solid cylinder of radius \(\frac{1}{4}r\) and height \(\frac{1}{4}h\) from the end with centre \(O\). The axis of the cylinder removed is parallel to \(OY\) and meets the end with centre \(O\) at \(X\), where \(OX = \frac{1}{4}r\). One plane face of the cylinder removed coincides with the plane face through \(O\) of the original cylinder. The resulting solid \(S\) is shown in Figure 3.
  1. Show that the centre of mass of \(S\) is at a distance \(\frac{85h}{168}\) from the plane face containing \(O\). [7]
The solid \(S\) is freely suspended from \(O\). In equilibrium the line \(OY\) is inclined at an angle arctan(17) to the horizontal.
  1. Find \(r\) in terms of \(h\). [6]
Edexcel M3 2014 June Q6
14 marks Standard +0.8
A light elastic string, of natural length \(l\) and modulus of elasticity \(4mg\), has one end attached to a fixed point \(A\). The other end is attached to a particle \(P\) of mass \(m\). The particle hangs freely at rest in equilibrium at the point \(E\). The distance of \(E\) below \(A\) is \((l + e)\).
  1. Find \(e\) in terms of \(l\). [2]
At time \(t = 0\), the particle is projected vertically downwards from \(E\) with speed \(\sqrt{gl}\).
  1. Prove that, while the string is taut, \(P\) moves with simple harmonic motion. [5]
  2. Find the amplitude of the simple harmonic motion. [3]
  3. Find the time at which the string first goes slack. [4]
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]