Questions — Edexcel M3 (510 questions)

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Edexcel M3 2002 January Q4
10 marks Standard +0.3
\includegraphics{figure_2} Figure 2 shows the region \(R\) bounded by the curve with equation \(y^2 = rx\), where \(r\) is a positive constant, the \(x\)-axis and the line \(x = r\). A uniform solid of revolution \(S\) is formed by rotating \(R\) through one complete revolution about the \(x\)-axis.
  1. Show that the distance of the centre of mass of \(S\) from \(O\) is \(\frac{4}{5}r\). [6]
The solid is placed with its plane face on a plane which is inclined at an angle \(\alpha\) to the horizontal. The plane is sufficiently rough to prevent \(S\) from sliding. Given that \(S\) does not topple,
  1. find, to the nearest degree, the maximum value of \(\alpha\). [4]
Edexcel M3 2002 January Q5
10 marks Challenging +1.2
A cyclist is travelling around a circular track which is banked at 25° to the horizontal. The coefficient of friction between the cycle's tyres and the track is 0.6. The cyclist moves with constant speed in a horizontal circle of radius 40 m, without the tyres slipping. Find the maximum speed of the cyclist. [10]
Edexcel M3 2002 January Q6
13 marks Standard +0.3
The points \(O\), \(A\), \(B\) and \(C\) lie in a straight line, in that order, where \(OA = 0.6\) m, \(OB = 0.8\) m and \(OC = 1.2\) m. A particle \(P\), moving along this straight line, has a speed of \(\left(\frac{1}{10}\sqrt{5}\right)\) m s\(^{-1}\) at \(A\), \(\left(\frac{1}{5}\sqrt{5}\right)\) m s\(^{-1}\) at \(B\) and is instantaneously at rest at \(C\).
  1. Show that this information is consistent with \(P\) performing simple harmonic motion with centre \(O\). [5]
Given that \(P\) is performing simple harmonic motion with centre \(O\),
  1. show that the speed of \(P\) at \(O\) is 0.6 m s\(^{-1}\), [2]
  2. find the magnitude of the acceleration of \(P\) as it passes \(A\), [2]
  3. find, to 3 significant figures, the time taken for \(P\) to move directly from \(A\) to \(B\). [4]
Edexcel M3 2002 January Q7
15 marks Standard +0.8
\includegraphics{figure_3} Figure 3 shows a fixed hollow sphere of internal radius \(a\) and centre \(O\). A particle \(P\) of mass \(m\) is projected horizontally from the lowest point \(A\) of a sphere with speed \(\sqrt{\left(\frac{5}{4}ag\right)}\). It moves in a vertical circle, centre \(O\), on the smooth inner surface of the sphere. The particle passes through the point \(B\), which is in the same horizontal plane as \(O\). It leaves the surface of the sphere at the point \(C\), where \(OC\) makes an angle \(\theta\) with the upward vertical.
  1. Find, in terms of \(m\) and \(g\), the normal reaction between \(P\) and the surface of the sphere at \(B\). [4]
  2. Show that \(\theta = 60°\). [7]
After leaving the surface of the sphere, \(P\) meets it again at the point \(A\).
  1. Find, in terms of \(a\) and \(g\), the time \(P\) takes to travel from \(C\) to \(A\). [4]
Edexcel M3 2005 January Q1
7 marks Moderate -0.3
A particle \(P\) of mass 0.5 kg is attached to one end of a light inextensible string of length 1.5 m. The other end of the string is attached to a fixed point \(A\). The particle is moving, with the string taut, in a horizontal circle with centre \(O\) vertically below \(A\). The particle is moving with constant angular speed 2.7 rad s\(^{-1}\). Find
  1. the tension in the string, [4]
  2. the angle, to the nearest degree, that \(AP\) makes with the downward vertical. [3]
Edexcel M3 2005 January Q2
9 marks Standard +0.8
\includegraphics{figure_1} A child's toy consists of a uniform solid hemisphere, of mass \(M\) and base radius \(r\), joined to a uniform solid right circular cone of mass \(m\), where \(2m < M\). The cone has vertex \(O\), base radius \(r\) and height \(3r\). Its plane face, with diameter \(AB\), coincides with the plane face of the hemisphere, as shown in Figure 1.
  1. Show that the distance of the centre of mass of the toy from \(AB\) is $$\frac{3(M - 2m)}{8(M + m)}r.$$ [5]
The toy is placed with \(OA\) on a horizontal surface. The toy is released from rest and does not remain in equilibrium.
  1. Show that \(M > 26m\). [4]
Edexcel M3 2005 January Q3
9 marks Standard +0.8
\includegraphics{figure_2} A uniform lamina occupies the region \(R\) bounded by the \(x\)-axis and the curve $$y = \sin x, \quad 0 \leq x \leq \pi,$$ as shown in Figure 2.
  1. Show, by integration, that the \(y\)-coordinate of the centre of mass of the lamina is \(\frac{\pi}{8}\). [6]
\includegraphics{figure_3} A uniform prism \(S\) has cross-section \(R\). The prism is placed with its rectangular face on a table which is inclined at an angle \(\theta^{\circ}\) to the horizontal. The cross-section \(R\) lies in a vertical plane as shown in Figure 3. The table is sufficiently rough to prevent \(S\) sliding. Given that \(S\) does not topple,
  1. find the largest possible value of \(\theta\). [3]
Edexcel M3 2005 January Q4
10 marks Standard +0.8
[diagram]
In a game at a fair, a small target \(C\) moves horizontally with simple harmonic motion between the points \(A\) and \(B\), where \(AB = 4L\). The target moves inside a box and takes 3 s to travel from \(A\) to \(B\). A player has to shoot at \(C\), but \(C\) is only visible to the player when it passes a window \(PQ\), where \(PQ = b\). The window is initially placed with \(Q\) at the point as shown in Figure 4. The target \(C\) takes 0.75 s to pass from \(Q\) to \(P\).
  1. Show that \(b = (2 - \sqrt{2})L\). [5]
  2. Find the speed of \(C\) as it passes \(P\). [2]
\includegraphics{figure_5} For advanced players, the window \(PQ\) is moved to the centre of \(AB\) so that \(AP = QB\), as shown in Figure 5.
  1. Find the time, in seconds to 2 decimal places, taken for \(C\) to pass from \(Q\) to \(P\) in this new position. [3]
Edexcel M3 2005 January Q5
12 marks Standard +0.8
At time \(t = 0\), a particle \(P\) is at the origin \(O\), moving with speed 18 m s\(^{-1}\) along the \(x\)-axis, in the positive \(x\)-direction. At time \(t\) seconds (\(t > 0\)) the acceleration of \(P\) has magnitude \(\frac{3}{\sqrt{(t + 4)}}\) m s\(^{-2}\) and is directed towards \(O\).
  1. Show that, at time \(t\) seconds, the velocity of \(P\) is \([30 - 6\sqrt{(t + 4)}]\) m s\(^{-1}\). [5]
  2. Find the distance of \(P\) from \(O\) when \(P\) comes to instantaneous rest. [7]
Edexcel M3 2005 January Q6
14 marks Standard +0.3
A light spring of natural length \(L\) has one end attached to a fixed point \(A\). A particle \(P\) of mass \(m\) is attached to the other end of the spring. The particle is moving vertically. As it passes through the point \(B\) below \(A\), where \(AB = L\), its speed is \(\sqrt{(2gL)}\). The particle comes to instantaneous rest at a point \(C\), \(4L\) below \(A\).
  1. Show that the modulus of elasticity of the spring is \(\frac{8mg}{9}\). [4]
At the point \(D\) the tension in the spring is \(mg\).
  1. Show that \(P\) performs simple harmonic motion with centre \(D\). [5]
  2. Find, in terms of \(L\) and \(g\),
    1. the period of the simple harmonic motion,
    2. the maximum speed of \(P\).
    [5]
Edexcel M3 2005 January Q7
14 marks Challenging +1.8
\includegraphics{figure_6} A trapeze artiste of mass 60 kg is attached to the end \(A\) of a light inextensible rope \(OA\) of length 5 m. The artiste must swing in an arc of a vertical circle, centre \(O\), from a platform \(P\) to another platform \(Q\), where \(PQ\) is horizontal. The other end of the rope is attached to the fixed point \(O\) which lies in the vertical plane containing \(PQ\), with \(\angle POQ = 120^{\circ}\) and \(OP = OQ = 5\) m, as shown in Figure 6. As part of her act, the artiste projects herself from \(P\) with speed \(\sqrt{15}\) m s\(^{-1}\) in a direction perpendicular to the rope \(OA\) and in the plane \(POQ\). She moves in a circular arc towards \(Q\). At the lowest point of her path she catches a ball of mass \(m\) kg which is travelling towards her with speed 3 m s\(^{-1}\) and parallel to \(QP\). After catching the ball, she comes to rest at the point \(Q\). By modelling the artiste and the ball as particles and ignoring her air resistance, find
  1. the speed of the artiste immediately before she catches the ball, [4]
  2. the value of \(m\), [7]
  3. the tension in the rope immediately after she catches the ball. [3]
Edexcel M3 2011 January Q1
6 marks Standard +0.3
A particle \(P\) moves on the positive \(x\)-axis. When the distance of \(P\) from the origin \(O\) is \(x\) metres, the acceleration of \(P\) is \((7 - 2x)\) m s\(^{-2}\), measured in the positive \(x\)-direction. When \(t = 0\), \(P\) is at \(O\) and is moving in the positive \(x\)-direction with speed 6 m s\(^{-1}\). Find the distance of \(P\) from \(O\) when \(P\) first comes to instantaneous rest. [6]
Edexcel M3 2011 January Q2
8 marks Standard +0.3
\includegraphics{figure_1} A toy is formed by joining a uniform solid hemisphere, of radius \(r\) and mass \(4m\), to a uniform right circular solid cone of mass \(km\). The cone has vertex \(A\), base radius \(r\) and height \(2r\). The plane face of the cone coincides with the plane face of the hemisphere. The centre of the plane face of the hemisphere is \(O\) and \(OB\) is a radius of its plane face as shown in Figure 1. The centre of mass of the toy is at \(O\).
  1. Find the value of \(k\). [4]
A metal stud of mass \(2m\) is attached to the toy at \(A\). The toy is now suspended by a light string attached to \(B\) and hangs freely at rest. The angle between \(OB\) and the vertical is \(30°\).
  1. Find the value of \(\lambda\). [4]
Edexcel M3 2011 January Q3
10 marks Standard +0.8
\includegraphics{figure_2} The region \(R\) is bounded by the curve with equation \(y = e^x\), the line \(x = 1\), the line \(x = 2\) and the \(x\)-axis as shown in Figure 2. A uniform solid \(S\) is formed by rotating \(R\) through \(2\pi\) about the \(x\)-axis.
  1. Show that the volume of \(S\) is \(\frac{1}{2}\pi (e^4 - e^2)\). [4]
  2. Find, to 3 significant figures, the \(x\)-coordinate of the centre of mass of \(S\). [6]
Edexcel M3 2011 January Q4
11 marks Standard +0.3
A particle \(P\) moves along the \(x\)-axis. At time \(t\) seconds its displacement, \(x\) metres, from the origin \(O\) is given by \(x = 5 \sin (\frac{1}{4}\pi t)\).
  1. Prove that \(P\) is moving with simple harmonic motion. [3]
  2. Find the period and the amplitude of the motion. [2]
  3. Find the maximum speed of \(P\). [2]
The points \(A\) and \(B\) on the positive \(x\)-axis are such that \(OA = 2\) m and \(OB = 3\) m.
  1. Find the time taken by \(P\) to travel directly from \(A\) to \(B\). [4]
Edexcel M3 2011 January Q5
10 marks Standard +0.3
\includegraphics{figure_3} A small ball \(P\) of mass \(m\) is attached to the ends of two light inextensible strings of length \(l\). The other ends of the strings are attached to fixed points \(A\) and \(B\), where \(A\) is vertically above \(B\). Both strings are taut and \(AP\) is perpendicular to \(BP\) as shown in Figure 3. The system rotates about the line \(AB\) with constant angular speed \(\omega\). The ball moves in a horizontal circle.
  1. Find, in terms of \(m\), \(g\), \(l\) and \(\omega\), the tension in \(AP\) and the tension in \(BP\). [8]
  2. Show that \(\omega^2 \geq \frac{g\sqrt{2}}{l}\). [2]
Edexcel M3 2011 January Q6
13 marks Standard +0.8
\includegraphics{figure_4} A small ball of mass \(3m\) is attached to the ends of two light elastic strings \(AP\) and \(BP\), each of natural length \(l\) and modulus of elasticity \(kmg\). The ends \(A\) and \(B\) of the strings are attached to fixed points on the same horizontal level, with \(AB = 2l\). The mid-point of \(AB\) is \(C\). The ball hangs in equilibrium at a distance \(\frac{3}{4}l\) vertically below \(C\) as shown in Figure 4.
  1. Show that \(k = 10\) [7]
The ball is now pulled vertically downwards until it is at a distance \(\frac{15}{8}l\) below \(C\). The ball is released from rest.
  1. Find the speed of the ball as it reaches \(C\). [6]
Edexcel M3 2011 January Q7
17 marks Challenging +1.2
\includegraphics{figure_5} A particle \(P\) of mass \(m\) is attached to one end of a light rod of length \(l\). The other end of the rod is attached to a fixed point \(O\). The rod can turn freely in a vertical plane about \(O\). The particle is projected with speed \(u\) from a point \(A\), where \(OA\) makes an angle \(\alpha\) with the upward vertical through \(O\) and \(0 < \alpha < \frac{\pi}{2}\). When \(OP\) makes an angle \(\theta\) with the upward vertical through \(O\) the speed of \(P\) is \(v\) as shown in Figure 5.
  1. Show that \(v^2 = u^2 + 2gl (\cos \alpha - \cos \theta)\). [4]
It is given that \(\cos \alpha = \frac{3}{5}\) and that \(P\) moves in a complete vertical circle.
  1. Show that \(u > 2\sqrt{\frac{gl}{5}}\). [4]
As the rod rotates the least tension in the rod is \(T\) and the greatest tension is \(5T\).
  1. Show that \(u^2 = \frac{33}{10}gl\). [9]
Edexcel M3 2001 June Q1
7 marks Moderate -0.3
A particle \(P\) moves along the x-axis in the positive direction. At time \(t\) seconds, the velocity of \(P\) is \(v\) m s\(^{-1}\) and its acceleration is \(\frac{1}{5}e^{-2t}\) m s\(^{-2}\). When \(t = 0\) the speed of \(P\) is 10 m s\(^{-1}\).
  1. Express \(v\) in terms of \(t\). [4]
  2. Find, to 3 significant figures, the speed of \(P\) when \(t = 3\). [2]
  3. Find the limiting value of \(v\). [1]
Edexcel M3 2001 June Q2
7 marks Challenging +1.2
\includegraphics{figure_1} A smooth solid hemisphere, of radius 0.8 m and centre \(O\), is fixed with its plane face on a horizontal table. A particle of mass 0.5 kg is projected horizontally with speed \(u\) m s\(^{-1}\) from the highest point \(A\) of the hemisphere. The particle leaves the hemisphere at the point \(B\), which is a vertical distance of 0.2 m below the level of \(A\). The speed of the particle at \(B\) is \(v\) m s\(^{-1}\) and the angle between \(OA\) and \(OB\) is \(\theta\), as shown in Fig. 1.
  1. Find the value of \(\cos \theta\). [1]
  2. Show that \(v^2 = 5.88\). [3]
  3. Find the value of \(u\). [3]
Edexcel M3 2001 June Q3
10 marks Standard +0.3
\includegraphics{figure_2} A light horizontal spring, of natural length 0.25 m and modulus of elasticity 52 N, is fastened at one end to a point \(A\). The other end of the spring is fastened to a small wooden block \(B\) of mass 1.5 kg which is on a horizontal table, as shown in Fig. 2. The block is modelled as a particle. The table is initially assumed to be smooth. The block is released from rest when it is a distance 0.3 m from \(A\). By using the principle of the conservation of energy,
  1. find, to 3 significant figures, the speed of \(B\) when it is a distance 0.25 m from \(A\). [5]
It is now assumed that the table is rough and the coefficient of friction between \(B\) and the table is 0.6.
  1. Find, to 3 significant figures, the minimum distance from \(A\) at which \(B\) can rest in equilibrium. [5]
Edexcel M3 2001 June Q4
10 marks Standard +0.8
A projectile \(P\) is fired vertically upwards from a point on the earth's surface. When \(P\) is at a distance \(x\) from the centre of the earth its speed is \(v\). Its acceleration is directed towards the centre of the earth and has magnitude \(\frac{k}{x^2}\), where \(k\) is a constant. The earth may be assumed to be a sphere of radius \(R\).
  1. Show that the motion of \(P\) may be modelled by the differential equation $$v \frac{dv}{dx} = -\frac{gR^2}{x^2}.$$ [4]
The initial speed of \(P\) is \(U\), where \(U^2 < 2gR\). The greatest distance of \(P\) from the centre of the earth is \(X\).
  1. Find \(X\) in terms of \(U\), \(R\) and \(g\). [6]
Edexcel M3 2001 June Q5
11 marks Standard +0.3
\includegraphics{figure_3} An ornament \(S\) is formed by removing a solid right circular cone, of radius \(r\) and height \(\frac{1}{2}h\), from a solid uniform cylinder, of radius \(r\) and height \(h\), as shown in Fig. 3.
  1. Show that the distance of the centre of mass \(S\) from its plane face is \(\frac{19}{30}h\). [7]
The ornament is suspended from a point on the circular rim of its open end. It hangs in equilibrium with its axis of symmetry inclined at an angle \(\alpha\) to the horizontal. Given that \(h = 4r\),
  1. find, in degrees to one decimal place, the value of \(\alpha\). [4]
Edexcel M3 2001 June Q6
14 marks Standard +0.3
\includegraphics{figure_4} A particle \(P\) of mass \(m\) is attached to two light inextensible strings. The other ends of the string are attached to fixed points \(A\) and \(B\). The point \(A\) is a distance \(h\) vertically above \(B\). The system rotates about the line \(AB\) with constant angular speed \(\omega\). Both strings are taut and inclined at \(60°\) to \(AB\), as shown in Fig. 4. The particle moves in a circle of radius \(r\).
  1. Show that \(r = \frac{\sqrt{3}}{2}h\). [2]
  2. Find, in terms of \(m\), \(g\), \(h\) and \(\omega\), the tension in \(AP\) and the tension in \(BP\). [8]
The time taken for \(P\) to complete one circle is \(T\).
  1. Show that \(T < \pi\sqrt{\left(\frac{2h}{g}\right)}\). [4]
Edexcel M3 2001 June Q7
16 marks Challenging +1.2
\includegraphics{figure_5} A small ring \(R\) of mass \(m\) is free to slide on a smooth straight wire which is fixed at an angle of \(30°\) to the horizontal. The ring is attached to one end of a light elastic string of natural length \(a\) and modulus of elasticity \(\lambda\). The other end of the string is attached to a fixed point \(A\) of the wire, as shown in Fig. 5. The ring rests in equilibrium at the point \(B\), where \(AB = \frac{a}{2}\).
  1. Show that \(\lambda = 4mg\). [3]
The ring is pulled down to the point \(C\), where \(BC = \frac{1}{4}a\), and released from rest. At time \(t\) after \(R\) is released the extension of the string is \((\frac{1}{4}a + x)\).
  1. Obtain a differential equation for the motion of \(R\) while the string remains taut, and show that it represents simple harmonic motion with period \(\pi\sqrt{\left(\frac{a}{g}\right)}\). [6]
  2. Find, in terms of \(g\), the greatest magnitude of the acceleration of \(R\) while the string remains taut. [2]
  3. Find, in terms of \(a\) and \(g\), the time taken for \(R\) to move from the point at which it first reaches maximum speed to the point where the string becomes slack for the first time. [5]