Questions M3 (745 questions)

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Edexcel M3 2021 October Q7
  1. \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
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
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
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
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
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
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.
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    Q6

    \hline &
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    VIIIV SIHI NI JAIIM ION OCVIIIV SIHI NI JIHMM ION OOVI4V SIHI NI JIIYM IONOO
Edexcel M3 Q2
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
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
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
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 2002 January Q1
  1. A particle \(P\) of mass 0.2 kg moves away from the origin along the positive \(x\)-axis. It moves under the action of a force directed away from the origin \(O\), of magnitude \(\frac { 5 } { x + 1 } \mathrm {~N}\), where \(O P = x\) metres. Given that the speed of \(P\) is \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when \(x = 0\), find the value of \(x\), to 3 significant figures, when the speed of \(P\) is \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
    (8)
  2. One end of a light elastic string, of natural length 2 m and modulus of elasticity 19.6 N , is attached to a fixed point \(A\). A small ball \(B\) of mass 0.5 kg is attached to the other end of the string. The ball is released from rest at \(A\) and first comes to instantaneous rest at the point \(C\), vertically below \(A\).
    1. Find the distance \(A C\).
    2. Find the instantaneous acceleration of \(B\) at \(C\).
      (3)
    \section*{3.} Figure 1
    \includegraphics[max width=\textwidth, alt={}, center]{a46d3d34-2381-4e73-837f-a60663fb1419-3_520_967_264_531} A rod \(A B\), of mass \(2 m\) and length \(2 a\), is suspended from a fixed point \(C\) by two light strings \(A C\) and \(B C\). The rod rests horizontally in equilibrium with \(A C\) making an angle \(\alpha\) with the rod, where tan \(\alpha = \frac { 3 } { 4 }\), and with \(A C\) perpendicular to \(B C\), as shown in Fig. 1.
  3. Give a reason why the rod cannot be uniform.
  4. Show that the tension in \(B C\) is \(\frac { 8 } { 5 } m g\) and find the tension in \(A C\). The string \(B C\) is elastic, with natural length \(a\) and modulus of elasticity \(k m g\), where \(k\) is constant.
  5. Find the value of \(k\).
    (4)
Edexcel M3 2002 January Q4
4. Figure 2
\includegraphics[max width=\textwidth, alt={}, center]{a46d3d34-2381-4e73-837f-a60663fb1419-4_532_907_229_691} Figure 2 shows the region \(R\) bounded by the curve with equation \(y ^ { 2 } = r x\), 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 \(\frac { 2 } { 3 } r\).
    (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,
  2. find, to the nearest degree, the maximum value of \(\alpha\).
    (4)
Edexcel M3 2002 January Q5
5. A cyclist is travelling around a circular track which is banked at \(25 ^ { \circ }\) 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
6. The points \(O , A , B\) and \(C\) lie in a straight line, in that order, where \(O A = 0.6 \mathrm {~m}\), \(O B = 0.8 \mathrm {~m}\) and \(O C = 1.2 \mathrm {~m}\). A particle \(P\), moving along this straight line, has a speed of \(\left( \frac { 3 } { 10 } \sqrt { 3 } \right) \mathrm { m } \mathrm { s } ^ { - 1 }\) at \(A , \left( \frac { 1 } { 5 } \sqrt { 5 } \right) \mathrm { m } \mathrm { s } ^ { - 1 }\) at \(B\) and is instantaneously at rest at \(C\).
  1. Show that this information is consistent with \(P\) performing simple harmonic motion with centre \(O\). Given that \(P\) is performing simple harmonic motion with centre \(O\),
  2. show that the speed of \(P\) at \(O\) is \(0.6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\),
  3. find the magnitude of the acceleration of \(P\) as it passes \(A\),
  4. find, to 3 significant figures, the time taken for \(P\) to move directly from \(A\) to \(B\). \section*{7.} \begin{figure}[h]
    \captionsetup{labelformat=empty} \caption{Figure 3} \includegraphics[alt={},max width=\textwidth]{a46d3d34-2381-4e73-837f-a60663fb1419-6_694_690_270_758}
    \end{figure} 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 \(\sqrt { } \left( \frac { 7 } { 2 } a g \right)\). 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 \(O C\) makes an angle \(\theta\) with the upward vertical.
  5. Find, in terms of \(m\) and \(g\), the normal reaction between \(P\) and the surface of the sphere at \(B\).
  6. Show that \(\theta = 60 ^ { \circ }\). After leaving the surface of the sphere, \(P\) meets it again at the point \(A\).
  7. Find, in terms of \(a\) and \(g\), the time \(P\) takes to travel from \(C\) to \(A\).
Edexcel M3 2003 January Q1
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
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
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
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
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
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 \(C\).
Edexcel M3 2004 January Q2
2. A particle \(P\) moves along the \(x\)-axis. At time \(t\) seconds its acceleration is \(\left( - 4 \mathrm { e } ^ { - 2 t } \right) \mathrm { m } \mathrm { s } ^ { - 2 }\) in the direction of \(x\) increasing. When \(t = 0 , P\) is at the origin \(O\) and is moving with speed \(1 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in the direction of \(x\) increasing.
  1. Find an expression for the velocity of \(P\) at time \(t\).
  2. Find the distance of \(P\) from \(O\) when \(P\) comes to instantaneous rest.
    (6)
Edexcel M3 2004 January Q3
3. Above the earth's surface, the magnitude of the force on a particle due to the earth's gravity is inversely proportional to the square of the distance of the particle from the centre of the earth. Assuming that the earth is a sphere of radius \(R\), and taking \(g\) as the acceleration due to gravity at the surface of the earth,
  1. prove that the magnitude of the gravitational force on a particle of mass \(m\) when it is a distance \(x ( x \geq R )\) from the centre of the earth is \(\frac { m g R ^ { 2 } } { x ^ { 2 } }\). A particle is fired vertically upwards from the surface of the earth with initial speed \(u\), where \(u ^ { 2 } = \frac { 3 } { 2 } g R\). Ignoring air resistance,
  2. find, in terms of \(g\) and \(R\), the speed of the particle when it is at a height \(2 R\) above the surface of the earth.
Edexcel M3 2004 January Q4
4. A particle \(P\) of mass \(m\) is attached to one end of a light elastic string of length \(a\) and modulus of elasticity \(\frac { 1 } { 2 } m g\). The other end of the string is fixed at the point \(A\) which is at a height \(2 a\) above a smooth horizontal table. The particle is held on the table with the string making an angle \(\beta\) with the horizontal, where \(\tan \beta = \frac { 3 } { 4 }\).
  1. Find the elastic energy stored in the string in this position. The particle is now released. Assuming that \(P\) remains on the table,
  2. find the speed of \(P\) when the string is vertical. By finding the vertical component of the tension in the string when \(P\) is on the table and \(A P\) makes an angle \(\theta\) with the horizontal,
  3. show that the assumption that \(P\) remains in contact with the table is justified.
Edexcel M3 2004 January Q5
5. A piston in a machine is modelled as a particle of mass 0.2 kg attached to one end \(A\) of a light elastic spring, of natural length 0.6 m and modulus of elasticity 48 N . The other end \(B\) of the spring is fixed and the piston is free to move in a horizontal tube which is assumed to be smooth. The piston is released from rest when \(A B = 0.9 \mathrm {~m}\).
  1. Prove that the motion of the piston is simple harmonic with period \(\frac { \pi } { 10 } \mathrm {~s}\).
    (5)
  2. Find the maximum speed of the piston.
    (2)
  3. Find, in terms of \(\pi\), the length of time during each oscillation for which the length of the spring is less than 0.75 m .
    (5)