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Edexcel M2 Q5
12 marks Standard +0.3
5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{0ea2267e-6c46-4a4f-9a38-c242de57901d-3_405_718_1169_555} \captionsetup{labelformat=empty} \caption{Fig. 2}
\end{figure} During a cricket match, a batsman hits the ball giving it an initial velocity of \(22 \mathrm {~ms} ^ { - 1 }\) at an angle \(\alpha\) to the horizontal where \(\sin \alpha = \frac { 7 } { 8 }\). When the batsman strikes the ball it is 1.6 metres above the ground, as shown in Figure 2, and it subsequently moves freely under gravity.
  1. Find, correct to 3 significant figures, the maximum height above the ground reached by the ball. The ball is caught by a fielder when it is 0.2 metres above the ground.
  2. Find the length of time for which the ball is in the air. Assuming that the fielder who caught the ball ran at a constant speed of \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\),
  3. find, correct to 3 significant figures, the maximum distance that the fielder could have been from the ball when it was struck.
Edexcel M2 Q6
16 marks Standard +0.3
6. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{0ea2267e-6c46-4a4f-9a38-c242de57901d-4_433_282_196_726} \captionsetup{labelformat=empty} \caption{Fig. 3}
\end{figure} Figure 3 shows a uniform rectangular lamina \(A B C D\) of mass \(8 m\) in which the sides \(A B\) and \(B C\) are of length \(a\) and \(2 a\) respectively. Particles of mass \(2 m , 6 m\) and \(4 m\) are fixed to the lamina at the points \(A , B\) and \(D\) respectively.
  1. Write down the distance of the centre of mass from \(A D\).
  2. Show that the distance of the centre of mass from \(A B\) is \(\frac { 4 } { 5 } a\). Another particle of mass \(k m\) is attached to the lamina at the point \(B\).
  3. Show that the distance of the centre of mass from \(A D\) is now given by \(\frac { ( 10 + k ) a } { 20 + k }\).
    (4 marks)
    Given that when the lamina is suspended freely from the point \(A\) the side \(A B\) makes an angle of \(45 ^ { \circ }\) with the vertical,
  4. find the value of \(k\).
Edexcel M2 Q7
17 marks Standard +0.3
7. Particle \(A\) of mass 7 kg is moving with speed \(u _ { 1 }\) on a smooth horizontal surface when it collides directly with particle \(B\) of mass 4 kg moving in the same direction as \(A\) with speed \(u _ { 2 }\). After the impact, \(A\) continues to move in the same direction but its speed has been halved. Given that the coefficient of restitution between the particles is \(e\),
  1. show that \(8 u _ { 2 } ( e + 1 ) = u _ { 1 } ( 8 e - 3 )\). Given also that \(u _ { 1 } = 14 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(u _ { 2 } = 3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\),
  2. find \(e\),
  3. show that the percentage of the kinetic energy of the particles lost as a result of the impact is \(9.6 \%\), correct to 2 significant figures.
Edexcel M3 Q1
7 marks Moderate -0.3
  1. A bird of mass 0.5 kg , flying around a vertical feeding post at a constant speed of \(6 \mathrm {~ms} ^ { - 1 }\), 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.
    Modelling the bird as a particle, find, to the nearest degree, the \includegraphics[max width=\textwidth, alt={}, center]{430c3b75-57aa-42ff-867e-304b85e7d521-1_303_472_349_1505}
    angle that its wings make with the vertical.
  2. A thin elastic string, of modulus \(\lambda \mathrm { N }\) and natural length 20 cm , passes round two small, smooth pegs \(A\) and \(B\) on the same horizontal level to form a closed loop. \(A B = 10 \mathrm {~cm}\). The ends of the string are attached to a weight \(P\) of mass 0.7 kg .
    When \(P\) rests in equilibrium, \(A P B\) forms an equilateral triangle. \includegraphics[max width=\textwidth, alt={}, center]{430c3b75-57aa-42ff-867e-304b85e7d521-1_346_371_836_1560}
    1. Find the value of \(\lambda\).
    2. State one assumption that you have made about the weight \(P\), explaining how you have used this assumption in your solution.
    3. A particle \(P\) of mass 0.5 kg moves along a straight line. When \(P\) is at a distance \(x \mathrm {~m}\) from a fixed point \(O\) on the line, the force acting on it is directed towards \(O\) and has magnitude \(\frac { 8 } { x ^ { 2 } } \mathrm {~N}\). When \(x = 2\), the speed of \(P\) is \(4 \mathrm {~ms} ^ { - 1 }\).
      Find the speed of \(P\) when it is 0.5 m from \(O\).
    4. A particle \(P\) of mass \(m \mathrm {~kg}\) is attached to one end of a light elastic string of natural length \(l \mathrm {~m}\) and modulus of elasticity \(\lambda \mathrm { N }\). 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 \(2 l \mathrm {~m}\).
    5. Show that \(\lambda = 4 m g\).
    6. Find, in terms of \(l\) and \(g\), the speed with which \(P\) passes through the point \(A\), where
    $$O A = \frac { 5 l } { 4 } \mathrm {~m} .$$ (6 marks) \section*{MECHANICS 3 (A) TEST PAPER 1 Page 2}
Edexcel M3 Q5
12 marks Challenging +1.2
5.
\includegraphics[max width=\textwidth, alt={}]{430c3b75-57aa-42ff-867e-304b85e7d521-2_389_412_265_386}
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 \(\frac { h } { 2 }\).
  1. Show that the centre of mass of the remaining solid is at a height \(\frac { 11 h } { 56 }\) above the base, along its axis of symmetry. 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 \(\tan \alpha = \frac { 1 } { 2 }\).
  2. Find the value of the ratio \(h : r\).
Edexcel M3 Q6
15 marks Challenging +1.2
6. A light elastic string, of natural length \(l \mathrm {~m}\) and modulus of elasticity \(\frac { m g } { 2 }\) newtons, has one end fastened to a fixed point \(O\). A particle \(P\), of mass \(m \mathrm {~kg}\), is attached to the other end of the string. \(P\) hangs in equilibrium at the point \(E\), vertically below \(O\), where \(O E = ( l + e ) \mathrm { m }\)
  1. Find the numerical value of the ratio \(e : l\). \(P\) is now pulled down a further distance \(\frac { 3 l } { 2 } \mathrm {~m}\) from \(E\) and is released from rest.
    In the subsequent motion, the string remains taut. At time \(t \mathrm {~s}\) after being released, \(P\) is at a distance \(x \mathrm {~m}\) below \(E\).
  2. Write down a differential equation for the motion of \(P\) and show that the motion is simple harmonic.
  3. Write down the period of the motion.
  4. Find the speed with which \(P\) first passes through \(E\) again.
  5. Show that the time taken by \(P\) after it is released to reach the point \(A\) above \(E\), where $$A E = \frac { 3 l } { 4 } \mathrm {~m} , \text { is } \frac { 2 \pi } { 3 } \sqrt { \frac { 2 l } { g } } \mathrm {~s} .$$
Edexcel M3 Q7
17 marks Challenging +1.2
  1. A particle \(P\) is attached to one end of a light inextensible string of length \(l \mathrm {~m}\). 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 \(u \mathrm {~ms} ^ { - 1 }\) and starts to move in a vertical circle.
Given that the string becomes slack when it makes an angle of \(120 ^ { \circ }\) with the downward vertical through \(O\),
  1. show that \(u ^ { 2 } = \frac { 7 g l } { 2 }\).
  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
  1. A particle of mass \(m \mathrm {~kg}\) moves in a horizontal straight line. Its initial speed is \(u \mathrm {~ms} ^ { - 1 }\) and the only force acting on it is a variable resistance of magnitude \(m k v \mathrm {~N}\), where \(v \mathrm {~ms} ^ { - 1 }\) is the speed of the particle after \(t\) seconds and \(k\) is a constant.
    Show that \(v = u e ^ { - k t }\).
  2. A particle \(P\) of mass \(m \mathrm {~kg}\) 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 \(\omega \mathrm { rad } \mathrm { s } ^ { - 1 }\).
    If the angle \(\theta\) which the string makes with the vertical must not \includegraphics[max width=\textwidth, alt={}, center]{e756c147-2711-4ee4-9ec6-7680fe84a0c5-1_314_401_722_1576}
    (7 marks)
    exceed \(60 ^ { \circ }\), calculate the greatest possible value of \(\omega\).
  3. A particle \(P\) of mass \(m \mathrm {~kg}\) is attached to one end of a light elastic string of natural length 0.5 m and modulus of elasticity \(\frac { m g } { 2 } \mathrm {~N}\). 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.
    2. Find the elastic potential energy stored in the string in the equilibrium position. \(P\), which is still attached to the string, is now held at rest at \(O\) and then lowered gently into its equilibrium position.
    3. Find the work done by the weight of the particle as it moves from \(O\) to the equilibrium position.
    4. Explain the discrepancy between your answers to parts (b) and (c).
    5. A particle \(P\), of mass \(m \mathrm {~kg}\), is attached to two light elastic strings, each of natural length \(l \mathrm {~m}\) and modulus of elasticity 3 mg N . The other ends of the strings are attached to the fixed points \(A\) and \(B\), where \(A B\) is horizontal and \(A B = 2 l \mathrm {~m}\). If \(P\) rests in equilibrium vertically below the mid-point of \(A B\), with each string making an angle \(\theta\) with the vertical, show that \includegraphics[max width=\textwidth, alt={}, center]{e756c147-2711-4ee4-9ec6-7680fe84a0c5-1_410_474_2052_1510}
    $$\cot \theta - \cos \theta = \frac { 1 } { 6 } .$$ \section*{MECHANICS 3 (A) TEST PAPER 2 Page 2}
Edexcel M3 Q5
18 marks Standard +0.3
  1. A small bead \(P\), of mass \(m \mathrm {~kg}\), can slide on a smooth circular ring, with centre \(O\) and radius \(r \mathrm {~m}\), which is fixed in a vertical plane. \(P\) is projected from the lowest point \(L\) of the ring with speed \(\sqrt { } ( 3 g r ) \mathrm { ms } ^ { - 1 }\). When \(P\) has reached a position such that \(O P\) makes an angle \(\theta\) with the downward vertical, as shown, its speed is \(v \mathrm {~ms} ^ { - 1 }\). \includegraphics[max width=\textwidth, alt={}, center]{e756c147-2711-4ee4-9ec6-7680fe84a0c5-2_355_337_262_1590}
    1. Show that \(v ^ { 2 } = g r ( 1 + 2 \cos \theta )\).
    2. Show that the magnitude of the reaction \(R N\) of the ring on the bead is given by
    $$R = m g ( 1 + 3 \cos \theta ) .$$
  2. Find the values of \(\cos \theta\) when
    1. \(P\) is instantaneously at rest, (ii) the reaction \(R\) is instantaneously zero.
  3. Hence show that the ratio of the heights of \(P\) above \(L\) in cases (i) and (ii) is \(9 : 8\).
Edexcel M3 Q6
18 marks Standard +0.8
6. 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. \(P\) is displaced to a point 0.5 m vertically below its equilibrium position and released from rest.
  2. Show that the subsequent motion of \(P\) is simple harmonic, with period 1.68 s .
  3. Calculate the maximum speed of \(P\) during its motion.
  4. 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.
Edexcel M3 Q7
16 marks Standard +0.8
7. (a) Show that the centre of mass of a uniform solid hemisphere of radius \(r\) is at a distance \(\frac { 3 r } { 8 }\) from the centre \(O\) of the plane face. 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 \(\sin \theta = \frac { 3 } { 10 }\).
(b) Indicate on a copy of the figure the three forces acting on the hemisphere, clearly stating what they are, and paying \includegraphics[max width=\textwidth, alt={}, center]{e756c147-2711-4ee4-9ec6-7680fe84a0c5-2_356_520_2042_1457}
(c) Given that the plane face containing the diameter \(A B\) makes an angle \(\alpha\) with the vertical, show that \(\cos \alpha = \frac { 4 } { 5 }\).
Edexcel M3 Q1
7 marks Standard +0.8
  1. One end of a light inextensible string of length \(2 r \mathrm {~m}\) is attached to a fixed point \(O\). A particle of mass \(m \mathrm {~kg}\) is attached to the other end \(Q\) of the string, so that it can move in a vertical plane. The string is held taut and horizontal and the particle is projected vertically downwards with a speed \(\sqrt { } ( g r ) \mathrm { ms } ^ { - 1 }\). When the string is vertical it begins to wrap round a small, smooth peg \(X\) at a distance \(r \mathrm {~m}\) vertically below \(O\). The particle continues to move.
    1. Find the speed of the particle when it reaches \(O\), in terms of \(g\) and \(r\).
    2. Show that, when \(Q X\) is horizontal, the tension in the string is 3 mgN .
    3. A particle moving along the \(x\)-axis describes simple harmonic motion about the origin \(O\). The period of its motion is \(\frac { \pi } { 2 }\) seconds. When it is at a distance 1 m from \(O\), its speed is \(3 \mathrm {~ms} ^ { - 1 }\). Calculate
    4. the amplitude of its motion,
    5. the maximum acceleration of the particle,
    6. the least time that it takes to move from \(O\) to a point 0.25 m from \(O\).
    7. A particle \(P\) of mass \(m \mathrm {~kg}\) is attached to the mid-point of a light elastic string of natural length \(8 l \mathrm {~m}\) and modulus of elasticity \(\lambda \mathrm { N }\). The two ends of the string are attached to fixed points \(A\) and \(B\) on the same horizontal level, where \(A B = 81 \mathrm {~m} . P\) is released from rest at the mid-point of \(A B\).
    8. If \(P\) comes to instantaneous rest at a depth \(3 / \mathrm { m }\) below \(A B\), find an expression for \(\lambda\) in terms of \(m\) and \(g\).
    9. Using this value of \(\lambda\), show that the speed \(v \mathrm {~ms} ^ { - 1 }\) of \(P\) when it passes through the point \(2 l \mathrm {~m}\) below \(A B\) is given by \(v ^ { 2 } = 4 ( 24 \sqrt { 5 } - 53 ) g l\).
    10. A particle \(P\) of mass 0.8 kg moves along a straight line \(O L\) and is acted on by a resistive force of magnitude \(R \mathrm {~N}\) directed towards the fixed point \(O\). When the displacement of \(P\) from \(O\) is \(x \mathrm {~m} , R = \frac { 0 \cdot 8 x v ^ { 2 } } { 1 + x ^ { 2 } }\), where \(v \mathrm {~ms} ^ { - 1 }\) is the speed of \(P\) at that instant.
    11. Write down a differential equation for the motion of \(P\).
    Given that \(v = 2\) when \(x = 0\),
  2. find the speed with which \(P\) passes through the point \(A\), where \(O A = 1 \mathrm {~m}\). \section*{MECHANICS 3 (A) TEST PAPER 3 Page 2}
Edexcel M3 Q5
10 marks Standard +0.3
  1. The diagram shows a uniform solid right circular cone of mass \(m \mathrm {~kg}\), height \(h \mathrm {~m}\) and base radius \(r \mathrm {~m}\) suspended by two vertical strings attached to the points \(P\) and \(Q\) on the circumference of the base. The vertex \(O\) of the cone is vertically below \(P\).
    1. Show that the tension in the string attached at \(Q\) is \(\frac { 3 m g } { 8 } \mathrm {~N}\). \includegraphics[max width=\textwidth, alt={}, center]{309da227-759c-475e-b12e-dcd9e338a417-2_296_277_269_1668}
    2. Find, in terms of \(m\) and \(g\), the tension in the other string.
    3. Two identical particles \(P\) and \(Q\) are connected by a light inextensible string passing through a small smooth-edged hole in a smooth table, as shown. \(P\) moves on the table in a horizontal circle of radius 0.2 m and \(Q\) hangs at rest. \includegraphics[max width=\textwidth, alt={}, center]{309da227-759c-475e-b12e-dcd9e338a417-2_309_430_859_1476}
    4. Calculate the number of revolutions made per minute by \(P\).
      (5 marks) \(Q\) is now also made to move in a horizontal circle of radius 0.2 m below the table. The part of the string between \(Q\) and the table makes an angle of \(45 ^ { \circ }\) with the vertical.
    5. Show that the numbers of revolutions per minute made by \(P\) and \(Q\) respectively are in the ratio \(2 ^ { 1 / 4 } : 1\). \includegraphics[max width=\textwidth, alt={}, center]{309da227-759c-475e-b12e-dcd9e338a417-2_293_428_1213_1499}
    6. A particle \(P\) of mass \(m \mathrm {~kg}\) is fixed to one end of a light elastic string of natural length \(l \mathrm {~m}\) and modulus of elasticity \(k m g \mathrm {~N}\). The other end of the string is fixed to a point \(X\) on a horizontal plane. \(P\) rests at \(O\), where \(O X = l \mathrm {~m}\), with the string just taut. It is then pulled away from \(X\) through a distance \(\frac { 3 l } { 4 } \mathrm {~m}\) and released from rest. On this side of \(O\), the plane is smooth.
    7. Show that, as long as the string is taut, \(P\) performs simple harmonic motion.
    8. Given that \(P\) first returns to \(O\) with speed \(\sqrt { } ( g l ) \mathrm { ms } ^ { - 1 }\), find the value of \(k\).
    9. On the other side of \(O\) the plane is rough, the coefficient of friction between \(P\) and the plane being \(\mu\). If \(P\) does not reach \(X\) in the subsequent motion, show that \(\mu > \frac { 1 } { 2 }\). ( 4 marks)
    10. If, further, \(\mu = \frac { 3 } { 4 }\), show that the time which elapses after \(P\) is released and before it comes to rest is \(\frac { 1 } { 24 } ( 9 \pi + 32 ) \sqrt { \frac { l } { g } }\) s.
      (6 marks)
Edexcel M3 Q1
7 marks Standard +0.3
  1. A motorcyclist rides in a cylindrical well of radius 5 m . He maintains a horizontal circular path at a constant speed of \(10 \mathrm {~ms} ^ { - 1 }\). The coefficient of friction between the wall and the wheels of the cycle is \(\mu\).
    Modelling the cyclist and his machine as a particle in contact \includegraphics[max width=\textwidth, alt={}, center]{d1acee30-26a7-44d0-b4f5-ab721451b8b8-1_359_263_370_1595}
    with the wall, show that he will not slip downwards provided that \(\mu \geq 0.49\).
  2. 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 \(8 \mathrm {~ms} ^ { - 1 }\). When it is at a distance 2 m from \(O\), its speed is \(4 \mathrm {~ms} ^ { - 1 }\).
    1. Find the amplitude of the motion.
    2. Show that the period of motion is \(\frac { \pi } { 2 } \mathrm {~s}\).
    3. A particle of mass \(m \mathrm {~kg}\) is attached to the end \(B\) of a light elastic string \(A B\). The string has natural length \(l \mathrm {~m}\) and modulus of elasticity \(\lambda . \mathrm { N }\).
    The end \(A\) is attached to a fixed point on a smooth plane \includegraphics[max width=\textwidth, alt={}, center]{d1acee30-26a7-44d0-b4f5-ab721451b8b8-1_289_543_1329_1425}
    inclined at an angle \(\alpha\) to the horizontal, as shown, and the particle rests in equilibrium with the length \(A B = \frac { 5 l } { 4 } \mathrm {~m}\).
Edexcel M3 Q4
9 marks Standard +0.3
4. The acceleration \(a \mathrm {~ms} ^ { - 2 }\) of a particle \(P\) moving in a straight line away from a fixed point \(O\) is given by \(a = \frac { k } { 1 + t }\), where \(t \mathrm {~s}\) 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 \(v \mathrm {~ms} ^ { - 1 }\) of \(P\) in terms of \(t , k\) and another constant \(c\). Given that \(v = 0\) when \(t = 0\) and that \(v = 4\) when \(t = 2\),
  2. show that \(v \ln 3 = 4 \ln ( 1 + t )\).
  3. Calculate the time when \(P\) has a speed of \(8 \mathrm {~ms} ^ { - 1 }\). \section*{MECHANICS 3 (A)TEST PAPER 4 Page 2}
Edexcel M3 Q5
13 marks Standard +0.3
  1. A particle of mass \(m \mathrm {~kg}\), at a distance \(x \mathrm {~m}\) from the centre of the Earth, experiences a force of magnitude \(\frac { k m } { x ^ { 2 } } \mathrm {~N}\) towards the centre of the Earth, where \(k\) is a constant. Given that the radius of the Earth is \(6.37 \times 10 ^ { 6 } \mathrm {~m}\), 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.
    The 3 kg mass falls from rest at a distance \(x = 12.74 \times 10 ^ { 6 } \mathrm {~m}\) from the centre of the Earth. Ignoring air resistance,
  2. show that it reaches the surface of the Earth with speed \(7.98 \times 10 ^ { 3 } \mathrm {~ms} ^ { - 1 }\). In a simplified model, the particle is assumed to fall with a constant acceleration \(10 \mathrm {~ms} ^ { - 2 }\). According to this model it attains the same speed as in (b), \(7.98 \times 10 ^ { 3 } \mathrm {~ms} ^ { - 1 }\), at a distance \(( 12 \cdot 74 - d ) \times 10 ^ { 6 } \mathrm {~m}\) from the centre of the Earth.
  3. Find the value of \(d\).
Edexcel M3 Q6
15 marks Challenging +1.2
6. 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 \(1.4 \mathrm {~ms} ^ { - 1 }\) 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 \(60 ^ { \circ }\),
  2. the tension, \(T \mathrm {~N}\), in the string is given by \(T = 3.92 ( 3 \cos \theta - 1 )\). If the string breaks when \(\cos \theta = \frac { 3 } { 5 }\) and \(P\) is ascending,
  3. find the greatest height reached by \(P\) above the initial point of projection.
Edexcel M3 Q7
15 marks Challenging +1.8
7. 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 \(\frac { 7 a } { 40 }\). 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.
  2. If the maximum value of \(\theta\) for which this is possible without the cap turning over is \(30 ^ { \circ }\), find the corresponding maximum inclination of the axis of symmetry of the cap to the vertical.
    (6 marks)
Edexcel M3 Q1
7 marks Standard +0.3
  1. 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.
  2. Aliya, whose mass is \(m \mathrm {~kg}\), is playing rounders. She rounds the first base at a speed of \(v \mathrm {~ms} ^ { - 1 }\), making the turn on a horizontal circular path of radius \(r \mathrm {~m}\).
    1. Write down, in terms of \(m , v\) and \(r\), the magnitude of the horizontal force acting on her.
    2. Show that if she continues on the same circular path, the reaction force exerted on her by the ground must act at an angle \(\theta\) to the vertical, where \(\tan \theta = \frac { v ^ { 2 } } { g r }\).
    3. A particle \(P\) of mass 0.2 kg is suspended by two identical light inelastic strings, with one end of each string attached to \(P\) and the other ends fixed to points \(O\) and \(X\) on the same horizontal level. Both strings are inclined at \(30 ^ { \circ }\) to the horizontal.
    4. Find the tension in the strings when \(P\) is at rest.
    The string \(X P\) is suddenly cut, so that \(P\) begins to move in a vertical circle with centre \(O\).
  3. Find the tension in the string \(O P\) when it makes an angle of \(60 ^ { \circ }\) with the horizontal.
Edexcel M3 Q4
11 marks Challenging +1.2
4. The radius of the Earth is \(R \mathrm {~m}\). The force of attraction towards the centre of the Earth experienced by a body of mass \(m \mathrm {~kg}\) at a distance \(x \mathrm {~m}\) from the centre is of magnitude \(\frac { k m } { x ^ { 2 } } \mathrm {~N}\), where \(k\) is a constant.
  1. Show that \(k = g R ^ { 2 }\). Two satellites \(A\) and \(B\), each of mass \(m \mathrm {~kg}\), are moving in circular orbits around the Earth at distances \(3 R \mathrm {~m}\) and \(4 R \mathrm {~m}\) respectively from the centre of the Earth. Given that the satellites move in the same plane and that they lie along the same radial line from the centre at any time,
  2. show that the ratio of the speed of \(B\) to that of \(A\) is \(4 : 3\). If, in addition, the satellites are linked with a taut, straight wire in the same plane and along the same radial line,
  3. find, in terms of \(m\) and \(g\), the magnitude of the force in the wire.
    (8 marks) \section*{MECHANICS 3 (A) TEST PAPER 5 Page 2}
Edexcel M3 Q5
13 marks Standard +0.8
  1. A light inelastic string of length \(l \mathrm {~m}\) passes through a small smooth ring which is fixed at a point \(O\) and is free to rotate about a vertical axis through \(O\). Particles \(P\) and \(Q\), of masses 0.06 kg and 0.04 kg respectively, are attached to the ends of the string.
    1. \(Q\) describes a horizontal circle with centre \(P\), while \(P\) hangs at rest at a depth \(d \mathrm {~m}\) below \(O\). Show that \(d = \frac { 2 l } { 5 }\).
      (6 marks) \includegraphics[max width=\textwidth, alt={}, center]{d8a0a9aa-a0af-48e6-a94c-12caaf0870db-2_273_365_406_1448}
    2. \(P\) and \(Q\) now both move in horizontal circles with the same angular velocity \(\omega \mathrm { rad } \mathrm { s } ^ { - 1 }\) about a vertical axis through \(O\). Show that \(O Q = \frac { 3 l } { 5 } \mathrm {~m}\). (7 marks) \includegraphics[max width=\textwidth, alt={}, center]{d8a0a9aa-a0af-48e6-a94c-12caaf0870db-2_250_433_694_1489}
    3. 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 \includegraphics[max width=\textwidth, alt={}, center]{d8a0a9aa-a0af-48e6-a94c-12caaf0870db-2_268_241_1005_1704}
      material as the cup, and the stem as a uniform solid cylinder,
    4. show that the mass of the circular base is \(\frac { 1 } { 2 } M\).
    Given that the centre of mass of the glass is at a distance \(\frac { 13 r } { 14 }\) from the base along the vertical axis of symmetry,
  2. express \(M\) in terms of \(m\). If the cup is now filled with liquid whose mass is \(2 M\),
  3. show that the position of the centre of mass rises through a distance \(\frac { 13 r } { 35 }\).
  4. State an assumption that you have made about the liquid.
Edexcel M3 Q7
15 marks Challenging +1.2
7. A particle of mass \(m \mathrm {~kg}\) is attached to one end of an elastic string of natural length \(l \mathrm {~m}\) and modulus of elasticity \(\lambda \mathrm { N }\). 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 \(C\).
    2. Find the period, \(T \mathrm {~s}\), of the motion in terms of \(l , m\) and \(\lambda\).
    3. Explain the significance of the term 'slightly' as used in (i) above. 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 \(T _ { 1 } \mathrm {~s}\).
    1. Write down an expression for \(T _ { 1 }\) in terms of \(l , m , M\) and \(\lambda\).
    2. Hence show that \(T _ { 1 } ^ { 2 } - T ^ { 2 } = \frac { 4 \pi ^ { 2 } d } { g }\).
Edexcel M3 Q1
7 marks Standard +0.3
  1. A particle of mass \(m \mathrm {~kg}\) is attached to one end of a light inextensible string of length \(l \mathrm {~m}\) 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 \(\omega \mathrm { rad } \mathrm { s } ^ { - 1 }\). 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[max width=\textwidth, alt={}, center]{3321c06a-29c3-430a-99a8-ec3a245abf10-1_341_328_347_1631}
    1. Find an expression, in terms of \(m , l\) and \(\omega\), for the kinetic energy of the particle at this instant.
    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 \(A\).
    3. Determine the position of the particle when the rate of increase of its total energy, with respect to time, is a maximum.
    4. A particle moves along a straight line in such a way that its displacement \(x \mathrm {~m}\) 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.
    5. Show that the motion of the particle is simple harmonic.
    6. If the particle leaves \(O\) with speed \(15 \mathrm {~ms} ^ { - 1 }\), and \(\omega = 3\), find the amplitude of the motion.
    7. 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 \(\pi \mathrm { rad } \mathrm { s } ^ { - 1 }\). The natural length of the string is 1 m and, while \(P\) is in motion, the distance \(O P = 1.15 \mathrm {~m}\).
    8. Calculate, to 3 significant figures, the modulus of elasticity of the string.
    The motion now ceases and \(P\) hangs at rest vertically below \(O\).
  2. Show that the extension in the string in this position is about 13 cm .
Edexcel M3 Q4
10 marks Challenging +1.2
4. A small stone \(P\) of mass \(m \mathrm {~kg}\) is attached to one end of a light elastic string of modulus 3 mg N and natural length \(2 l \mathrm {~m}\). The other end of the string is fixed to a point \(O\) at a height \(3 l \mathrm {~m}\) above a horizontal surface. \(P\) is released from rest at \(O\); it hits the surface and rebounds to a height of \(2 l \mathrm {~m}\). The coefficient of restitution between \(P\) and the surface is \(e\).
Calculate the value of \(e\). State one assumption (other than the string being light) that you have used in your solution.
(1 mark) \section*{MECHANICS 3 (A) TEST PAPER 6 Page 2}
Edexcel M3 Q5
13 marks Challenging +1.2
  1. A small sphere \(S\), of mass \(m \mathrm {~kg}\) 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 \(v \mathrm {~ms} ^ { - 1 }\), the viscous resistive force acting upwards on it has magnitude \(v ^ { 2 } \mathrm {~N}\).
    1. Write down a differential equation for the motion of \(S\), clearly defining any symbol(s) that you introduce.
    2. Find, in terms of \(m\), the distance \(S\) has fallen when its speed is \(\sqrt { \frac { m g } { 2 } } \mathrm {~ms} ^ { - 1 }\).
    3. The diagram shows two identical particles, each of mass \(m \mathrm {~kg}\), 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. \(O P\) makes an angle \(\theta\) radians with the horizontal.
    The system is released from rest in the position where \(\theta = 0\). \includegraphics[max width=\textwidth, alt={}, center]{3321c06a-29c3-430a-99a8-ec3a245abf10-2_355_307_818_1629}
  2. Show that the vertical distance moved by \(Q\) is \(\frac { \theta } { \sin \theta }\) times the vertical distance moved by \(P\).
  3. In the position where \(\theta = \frac { \pi } { 6 }\), prove that the reaction of the cylinder on \(P\) has magnitude \(\left( 1 - \frac { \pi } { 6 } \right) m g \mathrm {~N}\).