Questions — Edexcel (10514 questions)

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Edexcel M3 2024 June Q6
13 marks Standard +0.8
6. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{176ae8f8-7de9-4993-825a-6067614526ae-16_739_921_299_699} \captionsetup{labelformat=empty} \caption{Figure 4}
\end{figure} A fixed solid sphere has centre \(O\) and radius \(r\).
A particle \(P\) of mass \(m\) is held at rest on the smooth surface of the sphere at \(A\), the highest point of the sphere.
The particle \(P\) is then projected horizontally from \(A\) with speed \(u\) and moves on the surface of the sphere.
At the instant when \(P\) reaches the point \(B\) on the sphere, where angle \(A O B = \theta , P\) is moving with speed \(v\), as shown in Figure 4. At this instant, \(P\) loses contact with the surface of the sphere.
  1. Show that $$\cos \theta = \frac { 2 g r + u ^ { 2 } } { 3 g r }$$ In the subsequent motion, the particle \(P\) crosses the horizontal through \(O\) at the point \(C\), also shown in Figure 4. At the instant \(P\) passes through \(C , P\) is moving at an angle \(\alpha\) to the horizontal.
    Given that \(u ^ { 2 } = \frac { 2 g r } { 5 }\)
  2. find the exact value of \(\tan \alpha\).
Edexcel M3 2024 June Q7
15 marks Standard +0.3
  1. A particle \(P\) of mass \(m\) is attached to one end of a light elastic string of natural length \(l\) and modulus of elasticity 2 mg . The other end of the string is attached to a fixed point \(A\) on a smooth horizontal table. The particle \(P\) is at rest at the point \(B\) on the table, where \(A B = l\).
At time \(t = 0 , P\) is projected along the table with speed \(U\) in the direction \(A B\).
At time \(t\)
  • the elastic string has not gone slack
  • \(B P = x\)
  • the speed of \(P\) is \(v\)
    1. Show that
$$v ^ { 2 } = U ^ { 2 } - \frac { 2 g x ^ { 2 } } { l }$$
  • By differentiating this equation with respect to \(x\), prove that, before the elastic string goes slack, \(P\) moves with simple harmonic motion with period \(\pi \sqrt { \frac { 2 l } { g } }\) Given that \(U = \sqrt { \frac { g l } { 2 } }\)
  • find, in terms of \(l\) and \(g\), the exact total time, from the instant it is projected from \(B\), that it takes \(P\) to travel a total distance of \(\frac { 3 } { 4 } l\) along the table.
    •
    Edexcel M3 2021 October Q1
    6 marks Standard +0.3
    1. A particle \(P\) is moving in a straight line with simple harmonic motion of period 4 s . The centre of the motion is the point \(O\)
    At time \(t = 0 , P\) passes through \(O\) At time \(t = 0.5 \mathrm {~s} , P\) is moving with speed \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\)
    1. Show that the amplitude of the motion is \(\frac { 4 \sqrt { 2 } } { \pi } \mathrm {~m}\)
    2. Find the maximum speed of \(P\)
    Edexcel M3 2021 October Q2
    11 marks Standard +0.8
    2. In this question solutions relying on calculator technology are not acceptable. A particle \(P\) of mass 2 kg is moving along the positive \(x\)-axis.
    At time \(t\) seconds, where \(t \geqslant 0 , P\) is \(x\) metres from the origin \(O\) and is moving away from \(O\) with speed \(v \mathrm {~ms} ^ { - 1 }\) where \(v = \frac { 1 } { \sqrt { ( 2 x + 1 ) } }\)
    1. Find the magnitude of the resultant force acting on \(P\) when its speed is \(\frac { 1 } { 3 } \mathrm {~ms} ^ { - 1 }\) When \(t = 0 , P\) is at \(O\)
    2. Find the value of \(t\) when \(P\) is 7.5 m from \(O\)
    Edexcel M3 2021 October Q3
    12 marks Standard +0.3
    3. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{9777abb8-a564-40d5-8d96-d5649913737b-08_307_437_244_756} \captionsetup{labelformat=empty} \caption{Figure 1}
    \end{figure} A particle \(P\) of mass \(m\) is attached to one end of a light elastic spring of natural length \(l\) and modulus of elasticity \(k m g\), where \(k\) is a constant. The other end of the spring is fixed to horizontal ground. The particle \(P\) rests in equilibrium, with the spring vertical, at the point \(E\).
    The point \(E\) is at a height \(\frac { 3 } { 5 } l\) above the ground, as shown in Figure 1.
    1. Show that \(k = \frac { 5 } { 2 }\) The particle \(P\) is now moved a distance \(\frac { 1 } { 4 } l\) vertically downwards from \(E\) and released from rest. Air resistance is modelled as being negligible.
    2. Show that \(P\) moves with simple harmonic motion.
    3. Find the speed of \(P\) as it passes through \(E\).
    4. Find the time from the instant \(P\) is released to the first instant it passes through \(E\).
    Edexcel M3 2021 October Q4
    11 marks Standard +0.8
    1. A light elastic string has natural length \(2 a\) and modulus of elasticity \(2 m g\).
    One end of the elastic string is attached to a fixed point \(O\). A particle \(P\) of mass \(\frac { 1 } { 2 } m\) is attached to the other end of the elastic string. The point \(A\) is vertically below \(O\) with \(O A = 4 a\). Particle \(P\) is held at \(A\) and released from rest. The speed of \(P\) at the instant when it has moved a distance \(a\) upwards is \(\sqrt { 3 a g }\) Air resistance to the motion of \(P\) is modelled as having magnitude \(k m g\), where \(k\) is a constant. Using the model and the work-energy principle,
    1. show that \(k = \frac { 1 } { 4 }\) Particle \(P\) is now held at \(O\) and released from rest. As \(P\) moves downwards, it reaches its maximum speed as it passes through the point \(B\).
    2. Find the distance \(O B\).
    Edexcel M3 2021 October Q5
    11 marks Standard +0.8
    5. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{9777abb8-a564-40d5-8d96-d5649913737b-16_730_634_246_657} \captionsetup{labelformat=empty} \caption{Figure 2}
    \end{figure} A small smooth ring \(R\) of mass \(m\) is threaded on to a thin smooth fixed vertical pole. One end of a light inextensible string of length \(2 l\) is attached to a point \(A\) on the pole. The other end of the string is attached to \(R\). A particle \(P\) of mass \(m\) is attached to the midpoint of the string. The particle \(P\) moves with constant angular speed in a horizontal circle, with both halves of the string taut, and \(A R = \frac { 6 l } { 5 }\), as shown in Figure 2. It may be assumed that in this motion the string does not wrap itself around the pole and that at any instant, the triangle \(A P R\) lies in a vertical plane.
    1. Show that the tension in the lower half of the string is \(\frac { 5 m g } { 3 }\)
    2. Find, in terms of \(l\) and \(g\), the time for \(P\) to complete one revolution.
    Edexcel M3 2021 October Q6
    10 marks Standard +0.8
    6. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{9777abb8-a564-40d5-8d96-d5649913737b-20_534_551_248_699} \captionsetup{labelformat=empty} \caption{Figure 3}
    \end{figure} A light rod of length \(a\) is free to rotate in a vertical plane about a horizontal axis through one end \(O\). A particle \(P\) of mass \(m\) is attached to the other end of the rod. The particle \(P\) is held at rest with the rod making an angle \(\alpha\) with the upward vertical through \(O\), where \(\tan \alpha = \frac { 3 } { 4 }\) The particle \(P\) is then projected with speed \(u\) in a direction which is perpendicular to the rod. At the instant when the rod makes an angle \(\theta\) with the upward vertical through \(O\), the speed of \(P\) is \(v\), as shown in Figure 3. Air resistance is assumed to be negligible.
    1. Show that \(v ^ { 2 } = u ^ { 2 } + \frac { 2 a g } { 5 } ( 4 - 5 \cos \theta )\) It is given that \(u ^ { 2 } = \frac { 6 a g } { 5 }\) and \(P\) moves in complete vertical circles. When \(\theta = \beta\), the force exerted on \(P\) by the rod is zero.
    2. Find the value of \(\cos \beta\)
    Edexcel M3 2021 October Q7
    14 marks Challenging +1.2
    \hspace{0pt} [You may assume that the volume of a cone of height \(h\) and base radius \(r\) is \(\frac { 1 } { 3 } \pi r ^ { 2 } h\).] A uniform solid right circular cone \(C\), with vertex \(V\), has base radius \(r\) and height \(h\).
    1. Show that the centre of mass of \(C\) is \(\frac { 3 } { 4 } h\) from \(V\) A solid \(F\), shown below in Figure 4, is formed by removing the solid right circular cone \(C ^ { \prime }\) from \(C\), where cone \(C ^ { \prime }\) has height \(\frac { 1 } { 3 } h\) and vertex \(V\) \begin{figure}[h]
      \includegraphics[alt={},max width=\textwidth]{9777abb8-a564-40d5-8d96-d5649913737b-24_666_670_854_639} \captionsetup{labelformat=empty} \caption{Figure 4}
      \end{figure}
    2. Show that the distance of the centre of mass of \(F\) from its larger plane face is \(\frac { 3 } { 13 } h\) The solid \(F\) rests in equilibrium with its curved surface in contact with a horizontal plane.
    3. Show that \(13 r ^ { 2 } \leqslant 17 h ^ { 2 }\)
      \includegraphics[max width=\textwidth, alt={}]{9777abb8-a564-40d5-8d96-d5649913737b-28_2642_1844_116_114}
    Edexcel M3 2018 Specimen Q1
    8 marks Standard +0.3
    1. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{bb73b211-7629-4ed7-9b71-91841c29bb85-02_397_526_561_715} \captionsetup{labelformat=empty} \caption{Figure 1}
    \end{figure} A hemispherical bowl, of internal radius \(r\), is fixed with its circular rim upwards and horizontal. A particle \(P\) of mass \(m\) moves on the smooth inner surface of the bowl. The particle moves with constant angular speed in a horizontal circle. The centre of the circle is at a distance \(\frac { 1 } { 2 } r\) vertically below the centre of the bowl, as shown in Figure 1.
    The time taken by \(P\) to complete one revolution of its circular path is \(T\).
    Show that \(T = \pi \sqrt { \frac { 2 r } { g } }\).
    Edexcel M3 2018 Specimen Q2
    9 marks Standard +0.3
    2. A spacecraft \(S\) of mass \(m\) moves in a straight line towards the centre of the Earth. The Earth is modelled as a sphere of radius \(R\) and \(S\) is modelled as a particle. When \(S\) is at a distance \(x , x \geqslant R\), from the centre of the Earth, the force exerted by the Earth on \(S\) is directed towards the centre of the Earth. The force has magnitude \(\frac { K } { x ^ { 2 } }\), where \(K\) is a constant.
    1. Show that \(K = m g R ^ { 2 }\) When \(S\) is at a distance \(3 R\) from the centre of the Earth, the speed of \(S\) is \(V\). Assuming that air resistance can be ignored,
    2. find, in terms of \(g , R\) and \(V\), the speed of \(S\) as it hits the surface of the Earth.
      VIIIV SIHI NI JIIIM ION OCVIIV SIHI NI JAHM ION OOVI4V SIHI NI JIIIM I ON OO
    Edexcel M3 2018 Specimen Q3
    12 marks Standard +0.3
    3. At time \(t = 0\), a particle \(P\) is at the origin \(O\), moving with speed \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in the positive \(x\) direction. At time \(t\) seconds, \(t \geqslant 0\), the acceleration of \(P\) has magnitude \(2 ( t + 4 ) ^ { - \frac { 1 } { 2 } } \mathrm {~ms} ^ { - 2 }\) and is directed towards \(O\).
    1. Show that, at time \(t\) seconds, the velocity of \(P\) is \(16 - 4 ( t + 4 ) ^ { \frac { 1 } { 2 } } \mathrm {~ms} ^ { - 1 }\)
    2. Find the distance of \(P\) from \(O\) when \(P\) comes to instantaneous rest.
      VIIIV SIHI NI JIIIM ION OCVIIV SIHI NI JAHM ION OOVI4V SIHI NI JIIIM I ON OO
    Edexcel M3 2018 Specimen Q4
    12 marks Standard +0.3
    4. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{bb73b211-7629-4ed7-9b71-91841c29bb85-12_403_497_251_712} \captionsetup{labelformat=empty} \caption{Figure 2}
    \end{figure} A particle of mass \(3 m\) is attached to one end of a light inextensible string of length \(a\). The other end of the string is attached to a fixed point \(O\). The particle is held at the point \(A\), where \(O A\) is horizontal and \(O A = a\). The particle is projected vertically downwards from \(A\) with speed \(u\), as shown in Figure 2. The particle moves in complete vertical circles.
    1. Show that \(u ^ { 2 } \geqslant 3 a g\). Given that the greatest tension in the string is three times the least tension in the string, (b) show that \(u ^ { 2 } = 6 a g\).
      VIIIV SIHI NI JIIYM ION OCVIIVV SIHI NI JIIIAM ION OOVEYV SIHIL NI JIIIM ION OO
    Edexcel M3 2018 Specimen Q5
    17 marks Challenging +1.8
    5. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{bb73b211-7629-4ed7-9b71-91841c29bb85-16_193_931_269_520} \captionsetup{labelformat=empty} \caption{Figure 3}
    \end{figure} Two fixed points \(A\) and \(B\) are 5 m apart on a smooth horizontal floor. A particle \(P\) of mass 0.5 kg is attached to one end of a light elastic string, of natural length 2 m and modulus of elasticity 20 N . The other end of the string is attached to \(A\). A second light elastic string, of natural length 1.2 m and modulus of elasticity 15 N , has one end attached to \(P\) and the other end attached to \(B\). Initially \(P\) rests in equilibrium at the point \(O\), as shown in Figure 3.
    1. Show that \(A O = 3 \mathrm {~m}\). The particle is now pulled towards \(A\) and released from rest at the point \(C\), where \(A C B\) is a straight line and \(O C = 1 \mathrm {~m}\).
    2. Show that, while both strings are taut, \(P\) moves with simple harmonic motion.
    3. Find the speed of \(P\) at the instant when the string \(P B\) becomes slack. The particle first comes to instantaneous rest at the point \(D\).
    4. Find the distance \(D B\).
    Edexcel M3 2018 Specimen Q6
    17 marks Standard +0.8
    6. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{bb73b211-7629-4ed7-9b71-91841c29bb85-20_442_723_237_605} \captionsetup{labelformat=empty} \caption{Figure 4}
    \end{figure} The shaded region \(R\) is bounded by part of the curve with equation \(y = x ^ { 2 } + 3\), the \(x\)-axis, the \(y\)-axis and the line with equation \(x = 2\), as shown in Figure 4. The unit of length on each axis is one centimetre. The region \(R\) is rotated through \(2 \pi\) radians about the \(x\)-axis to form a uniform solid \(S\).
    Using algebraic integration,
    1. show that the volume of \(S\) is \(\frac { 202 } { 5 } \pi \mathrm {~cm} ^ { 3 }\),
    2. show that, to 2 decimal places, the centre of mass of \(S\) is 1.30 cm from \(O\). \begin{figure}[h]
      \includegraphics[alt={},max width=\textwidth]{bb73b211-7629-4ed7-9b71-91841c29bb85-20_483_469_1402_767} \captionsetup{labelformat=empty} \caption{Figure 5}
      \end{figure} A uniform right circular solid cone, of base radius 7 cm and height 6 cm , is joined to \(S\) to form a solid \(T\). The base of the cone coincides with the larger plane face of \(S\), as shown in Figure 5. The vertex of the cone is \(V\).
      The mass per unit volume of \(S\) is twice the mass per unit volume of the cone.
    3. Find the distance from \(V\) to the centre of mass of \(T\). The point \(A\) lies on the circumference of the base of the cone. The solid \(T\) is suspended from \(A\) and hangs freely in equilibrium.
    4. Find the size of the angle between \(V A\) and the vertical.
      Leave
      blank
      Q6
      VIIIV SIHI NI JAIIM ION OCVIIIV SIHI NI JIHMM ION OOVI4V SIHI NI JIIYM IONOO
    Edexcel M3 Q2
    Challenging +1.2
    2. \begin{figure}[h]
    \captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{ab85ec29-b1fc-45a9-9343-09feb33ab6c5-004_513_399_303_785}
    \end{figure} A uniform solid right circular cone has base radius \(a\) and semi-vertical angle \(\alpha\), where \(\tan \alpha = \frac { 1 } { 3 }\). The cone is freely suspended by a string attached at a point \(A\) on the rim of its base, and hangs in equilibrium with its axis of symmetry making an angle of \(\theta ^ { \circ }\) with the upward vertical, as shown in Figure 1. Find, to one decimal place, the value of \(\theta\).
    Edexcel M3 Q4
    Challenging +1.2
    4. \begin{figure}[h]
    \captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{ab85ec29-b1fc-45a9-9343-09feb33ab6c5-006_574_510_324_726}
    \end{figure} A particle \(P\) of mass \(m\) is attached to one end of a light inextensible string of length \(a\). The other end of the string is attached to a point \(O\). The point \(A\) is vertically below \(O\), and \(O A = a\). The particle is projected horizontally from \(A\) with speed \(\sqrt { } ( 3 a g )\). When \(O P\) makes an angle \(\theta\) with the upward vertical through \(O\) and the string is still taut, the tension in the string is \(T\) and the speed of \(P\) is \(v\), as shown in Figure 2.
    1. Find, in terms of \(a , g\) and \(\theta\), an expression for \(v ^ { 2 }\).
    2. Show that \(T = ( 1 - 3 \cos \theta ) m g\). The string becomes slack when \(P\) is at the point \(B\).
    3. Find, in terms of \(a\), the vertical height of \(B\) above \(A\). After the string becomes slack, the highest point reached by \(P\) is \(C\).
    4. Find, in terms of \(a\), the vertical height of \(C\) above \(B\).
    Edexcel M3 Q5
    Standard +0.8
    5. \begin{figure}[h]
    \captionsetup{labelformat=empty} \caption{Figure 3} \includegraphics[alt={},max width=\textwidth]{ab85ec29-b1fc-45a9-9343-09feb33ab6c5-008_531_691_299_657}
    \end{figure} One end of a light inextensible string is attached to a fixed point \(A\). The other end of the string is attached to a fixed point \(B\), vertically below \(A\), where \(A B = h\). A small smooth ring \(R\) of mass \(m\) is threaded on the string. The ring \(R\) moves in a horizontal circle with centre \(B\), as shown in Figure 3. The upper section of the string makes a constant angle \(\theta\) with the downward vertical and \(R\) moves with constant angular speed \(\omega\). The ring is modelled as a particle.
    1. Show that \(\omega ^ { 2 } = \frac { g } { h } \left( \frac { 1 + \sin \theta } { \sin \theta } \right)\).
    2. Deduce that \(\omega > \sqrt { \frac { 2 g } { h } }\). Given that \(\omega = \sqrt { \frac { 3 g } { h } }\),
    3. find, in terms of \(m\) and \(g\), the tension in the string.
    Edexcel M3 Q6
    Standard +0.3
    6. \begin{figure}[h]
    \captionsetup{labelformat=empty} \caption{Figure 4} \includegraphics[alt={},max width=\textwidth]{ab85ec29-b1fc-45a9-9343-09feb33ab6c5-010_515_1015_319_477}
    \end{figure} The shaded region \(R\) is bounded by the curve with equation \(y = \frac { 1 } { 2 x ^ { 2 } }\), the \(x\)-axis and the lines \(x = 1\) and \(x = 2\), as shown in Figure 4. The unit of length on each axis is 1 m . A uniform solid \(S\) has the shape made by rotating \(R\) through \(360 ^ { \circ }\) about the \(x\)-axis.
    1. Show that the centre of mass of \(S\) is \(\frac { 2 } { 7 } \mathrm {~m}\) from its larger plane face. \begin{figure}[h]
      \captionsetup{labelformat=empty} \caption{Figure 5} \includegraphics[alt={},max width=\textwidth]{ab85ec29-b1fc-45a9-9343-09feb33ab6c5-010_616_431_1420_778}
      \end{figure} A sporting trophy \(T\) is a uniform solid hemisphere \(H\) joined to the solid \(S\). The hemisphere has radius \(\frac { 1 } { 2 } \mathrm {~m}\) and its plane face coincides with the larger plane face of \(S\), as shown in Figure 5. Both \(H\) and \(S\) are made of the same material.
    2. Find the distance of the centre of mass of \(T\) from its plane face.
    Edexcel M3 2003 January Q1
    5 marks Standard +0.3
    1. \begin{figure}[h]
    \captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{044c5866-0a12-4309-8ced-b463e1615fb0-2_383_789_335_681}
    \end{figure} A particle of mass 5 kg is attached to one end of two light elastic strings. The other ends of the strings are attached to a hook on a beam. The particle hangs in equilibrium at a distance 120 cm below the hook with both strings vertical, as shown in Fig. 1. One string has natural length 100 cm and modulus of elasticity 175 N . The other string has natural length 90 cm and modulus of elasticity \(\lambda\) newtons. Find the value of \(\lambda\).
    (5)
    Edexcel M3 2003 January Q2
    9 marks Standard +0.3
    2. \begin{figure}[h]
    \captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{044c5866-0a12-4309-8ced-b463e1615fb0-2_389_601_1362_693}
    \end{figure} A light inextensible string of length \(8 l\) has its ends fixed to two points \(A\) and \(B\), where \(A\) is vertically above \(B\). A small smooth ring of mass \(m\) is threaded on the string. The ring is moving with constant speed in a horizontal circle with centre \(B\) and radius 3l, as shown in Fig. 2. Find
    1. the tension in the string,
    2. the speed of the ring.
    3. State briefly in what way your solution might no longer be valid if the ring were firmly attached to the string.
      (1) \section*{3.} \section*{Figure 3}
      \includegraphics[max width=\textwidth, alt={}]{044c5866-0a12-4309-8ced-b463e1615fb0-3_564_1051_438_541}
      A child's toy consists of a uniform solid hemisphere attached to a uniform solid cylinder. The plane face of the hemisphere coincides with the plane face of the cylinder, as shown in Fig. 3. The cylinder and the hemisphere each have radius \(r\), and the height of the cylinder is \(h\). The material of the hemisphere is 6 times as dense as the material of the cylinder. The toy rests in equilibrium on a horizontal plane with the cylinder above the hemisphere and the axis of the cylinder vertical.
    Edexcel M3 2003 January Q4
    11 marks Standard +0.3
    4. A piston \(P\) in a machine moves in a straight line with simple harmonic motion about a point \(O\), which is the centre of the oscillations. The period of the oscillations is \(\pi \mathrm { s }\). When \(P\) is 0.5 m from \(O\), its speed is \(2.4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find
    1. the amplitude of the motion,
    2. the maximum speed of \(P\) during the motion,
    3. the maximum magnitude of the acceleration of \(P\) during the motion,
    4. the total time, in s to 2 decimal places, in each complete oscillation for which the speed of \(P\) is greater than \(2.4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
    Edexcel M3 2003 January Q5
    12 marks Standard +0.8
    5. A car of mass 800 kg moves along a horizontal straight road. At time \(t\) seconds, the resultant force acting on the car has magnitude \(\frac { 48000 } { ( t + 2 ) ^ { 2 } }\) newtons, acting in the direction of the motion of the car. When \(t = 0\), the car is at rest.
    1. Show that the speed of the car approaches a limiting value as \(t\) increases and find this value.
    2. Find the distance moved by the car in the first 6 seconds of its motion.
    Edexcel M3 2003 January Q6
    12 marks Standard +0.8
    6. A light elastic string has natural length 4 m and modulus of elasticity 58.8 N . A particle \(P\) of mass 0.5 kg is attached to one end of the string. The other end of the string is attached to a vertical point \(A\). The particle is released from rest at \(A\) and falls vertically.
    1. Find the distance travelled by \(P\) before it immediately comes to instantaneous rest for the first time. The particle is now held at a point 7 m vertically below \(A\) and released from rest.
    2. Find the speed of the particle when the string first becomes slack.
    Edexcel M3 2003 January Q7
    16 marks Challenging +1.2
    7. \begin{figure}[h]
    \captionsetup{labelformat=empty} \caption{Figure 4} \includegraphics[alt={},max width=\textwidth]{044c5866-0a12-4309-8ced-b463e1615fb0-5_604_596_391_760}
    \end{figure} Part of a hollow spherical shell, centre \(O\) and radius \(a\), is removed to form a bowl with a plane circular rim. The bowl is fixed with the circular rim uppermost and horizontal. The point \(A\) is the lowest point of the bowl. The point \(B\) is on the rim of the bowl and \(\angle A O B = 120 ^ { \circ }\), as shown in Fig. 4. A smooth small marble of mass \(m\) is placed inside the bowl at \(A\) and given an initial horizontal speed \(u\). The direction of motion of the marble lies in the vertical plane \(A O B\). The marble stays in contact with the bowl until it reaches \(B\). When the marble reaches \(B\), its speed is \(v\).
    1. Find an expression for \(v ^ { 2 }\).
    2. For the case when \(u ^ { 2 } = 6 g a\), find the normal reaction of the bowl on the marble as the marble reaches \(B\).
    3. Find the least possible value of \(u\) for the marble to reach \(B\). The point \(C\) is the other point on the rim of the bowl lying in the vertical plane \(O A B\).
    4. Find the value of \(u\) which will enable the marble to leave the bowl at \(B\) and meet it again at the point \(C\).