4.10f Simple harmonic motion: x'' = -omega^2 x

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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 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 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 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 2004 January Q5
    12 marks Standard +0.3
    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)
    Edexcel M3 2009 January Q4
    11 marks Standard +0.3
    4. A small shellfish is attached to a wall in a harbour. The rise and fall of the water level is modelled as simple harmonic motion and the shellfish as a particle. On a particular day the minimum depth of water occurs at 1000 hours and the next time that this minimum depth occurs is at 2230 hours. The shellfish is fixed in a position 5 m above the level of the minimum depth of the water and 11 m below the level of the maximum depth of the water. Find
    1. the speed, in metres per hour, at which the water level is rising when it reaches the shellfish,
    2. the earliest time after 1000 hours on this day at which the water reaches the shellfish.
    Edexcel M3 2010 January Q2
    9 marks Standard +0.3
    2. A particle \(P\) moves in a straight line with simple harmonic motion of period 2.4 s about a fixed origin \(O\). At time \(t\) seconds the speed of \(P\) is \(v \mathrm {~ms} ^ { - 1 }\). When \(t = 0 , P\) is at \(O\). When \(t = 0.4 , v = 4\). Find
    1. the greatest speed of \(P\),
    2. the magnitude of the greatest acceleration of \(P\).
    Edexcel M3 2012 January Q2
    8 marks Standard +0.3
    2. A particle \(P\) is moving in a straight line with simple harmonic motion. The centre of the oscillation is the fixed point \(C\), the amplitude of the oscillation is 0.5 m and the time to complete one oscillation is \(\frac { 2 \pi } { 3 }\) seconds. The point \(A\) is on the path of \(P\) and 0.2 m from \(C\). Find
    1. the magnitude and direction of the acceleration of \(P\) when it passes through \(A\),
    2. the speed of \(P\) when it passes through \(A\),
    3. the time \(P\) takes to move directly from \(C\) to \(A\).
    Edexcel M3 2013 January Q5
    12 marks Challenging +1.2
    5. A particle \(P\) is moving in a straight line with simple harmonic motion on a smooth horizontal floor. The particle comes to instantaneous rest at points \(A\) and \(B\) where \(A B\) is 0.5 m . The mid-point of \(A B\) is \(O\). The mid-point of \(O A\) is \(C\). The mid-point of \(O B\) is \(D\). The particle takes 0.2 s to travel directly from \(C\) to \(D\). At time \(t = 0 , P\) is moving through \(O\) towards \(A\).
    1. Show that the period of the motion is \(\frac { 6 } { 5 } \mathrm {~s}\).
    2. Find the distance of \(P\) from \(B\) when \(t = 2 \mathrm {~s}\).
    3. Find the maximum magnitude of the acceleration of \(P\).
    4. Find the maximum speed of \(P\).
    Edexcel M3 2004 June Q7
    16 marks Challenging +1.3
    7. A particle \(P\) of mass 0.3 kg is attached to one end of a light elastic spring. The other end of the spring is attached to a fixed point \(O\) on a smooth horizontal table. The spring has natural length 2 m and modulus of elasticity 21.6 N . The particle \(P\) is placed on the table at the point \(A\), where \(O A = 2 \mathrm {~m}\). The particle \(P\) is now pulled away from \(O\) to the point \(B\), where \(O A B\) is a straight line with \(O B = 3.5 \mathrm {~m}\). It is then released from rest.
    1. Prove that \(P\) moves with simple harmonic motion of period \(\frac { \pi } { 3 } \mathrm {~s}\).
    2. Find the speed of \(P\) when it reaches \(A\). The point \(C\) is the mid-point of \(A B\).
    3. Find, in terms of \(\pi\), the time taken for \(P\) to reach \(C\) for the first time. Later in the motion, \(P\) collides with a particle \(Q\) of mass 0.2 kg which is at rest at \(A\).
      After the impact, \(P\) and \(Q\) coalesce to form a single particle \(R\).
    4. Show that \(R\) also moves with simple harmonic motion and find the amplitude of this motion. END
    Edexcel M3 2005 June Q6
    14 marks Standard +0.3
    6. The rise and fall of the water level in a harbour is modelled as simple harmonic motion. On a particular day the maximum and minimum depths of water in the harbour are 10 m and 4 m and these occur at 1100 hours and 1700 hours respectively.
    1. Find the speed, in \(\mathrm { m } \mathrm { h } ^ { - 1 }\), at which the water level in the harbour is falling at 1600 hours on this particular day.
    2. Find the total time, between 1100 hours and 2300 hours on this particular day, for which the depth of water in the harbour is less than 5.5 m .
      (Total 14 marks)
    Edexcel M3 2008 June Q2
    11 marks Standard +0.3
    2. A particle \(P\) moves with simple harmonic motion and comes to rest at two points \(A\) and \(B\) which are 0.24 m apart on a horizontal line. The time for \(P\) to travel from \(A\) to \(B\) is 1.5 s . The midpoint of \(A B\) is \(O\). At time \(t = 0 , P\) is moving through \(O\), towards \(A\), with speed \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
    1. Find the value of \(u\).
    2. Find the distance of \(P\) from \(B\) when \(t = 2 \mathrm {~s}\).
    3. Find the speed of \(P\) when \(t = 2 \mathrm {~s}\).
    Edexcel M3 2010 June Q7
    15 marks Challenging +1.2
    1. A light elastic string, of natural length \(3 a\) and modulus of elasticity \(6 m g\), has one end attached to a fixed point \(A\). A particle \(P\) of mass \(2 m\) is attached to the other end of the string and hangs in equilibrium at the point \(O\), vertically below \(A\).
      1. Find the distance \(A O\).
      The particle is now raised to point \(C\) vertically below \(A\), where \(A C > 3 a\), and is released from rest.
    2. Show that \(P\) moves with simple harmonic motion of period \(2 \pi \sqrt { } \left( \frac { a } { g } \right)\). It is given that \(O C = \frac { 1 } { 4 } a\).
    3. Find the greatest speed of \(P\) during the motion. The point \(D\) is vertically above \(O\) and \(O D = \frac { 1 } { 8 } a\). The string is cut as \(P\) passes through \(D\), moving upwards.
    4. Find the greatest height of \(P\) above \(O\) in the subsequent motion.
    Edexcel M3 2011 June Q7
    15 marks Challenging +1.2
    1. A particle \(P\) of mass 0.5 kg is attached to the mid-point of a light elastic string of natural length 1.4 m and modulus of elasticity 2 N . The ends of the string are attached to the points \(A\) and \(B\) on a smooth horizontal table, where \(A B = 2 \mathrm {~m}\). The mid-point of \(A B\) is \(O\) and the point \(C\) is on the table between \(O\) and \(B\) where \(O C = 0.2 \mathrm {~m}\). At time \(t = 0\) the particle is released from rest at \(C\). At time \(t\) seconds the length of the string \(A P\) is \(( 1 + x ) \mathrm { m }\).
      1. Show that the tension in \(B P\) is \(\frac { 2 } { 7 } ( 3 - 10 x ) \mathrm { N }\).
      2. Find, in terms of \(x\), the tension in \(A P\).
      3. Show that \(P\) performs simple harmonic motion with period \(2 \pi \sqrt { } \left( \frac { 7 } { 80 } \right)\) s.
      4. Find the greatest speed of \(P\) during the motion.
      The point \(D\) lies between \(O\) and \(A\), where \(O D = 0.1 \mathrm {~m}\).
    2. Find the time taken by \(P\) to move directly from \(C\) to \(D\).
    Edexcel M3 2013 June Q7
    14 marks Challenging +1.2
    1. Two points \(A\) and \(B\) are 4 m apart on a smooth horizontal surface. A light elastic string, of natural length 0.8 m and modulus of elasticity 15 N , has one end attached to the point A. A light elastic string, of natural length 0.8 m and modulus of elasticity 10 N , has one end attached to the point \(B\). A particle \(P\) of mass 0.2 kg is attached to the free end of each string. The particle rests in equilibrium on the surface at the point \(C\) on the straight line between \(A\) and \(B\).
      1. Show that the length of \(A C\) is 1.76 m .
      The particle \(P\) is now held at the point \(D\) on the line \(A B\) such that \(A D = 2.16 \mathrm {~m}\). The particle is then released from rest and in the subsequent motion both strings remain taut.
    2. Show that \(P\) moves with simple harmonic motion.
    3. Find the speed of \(P\) as it passes through the point \(C\).
    4. Find the time from the instant when \(P\) is released from \(D\) until the instant when \(P\) is first moving with speed \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
    Edexcel M3 2013 June Q6
    14 marks Standard +0.8
    6. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{f6ab162c-8473-4464-ad62-87a359d85ab3-10_191_972_276_484} \captionsetup{labelformat=empty} \caption{Figure 5}
    \end{figure} The points \(A\) and \(B\) are 3.75 m apart on a smooth horizontal floor. A particle \(P\) has mass 0.8 kg . One end of a light elastic spring, of natural length 1.5 m and modulus of elasticity 24 N , is attached to \(P\) and the other end is attached to \(A\). The ends of another light elastic spring, of natural length 0.75 m and modulus of elasticity 18 N , are attached to \(P\) and \(B\). The particle \(P\) rests in equilibrium at the point \(O\), where \(A O B\) is a straight line, as shown in Figure 5.
    1. Show that \(A O = 2.4 \mathrm {~m}\). The point \(C\) lies on the straight line \(A O B\) between \(O\) and \(B\). The particle \(P\) is held at \(C\) and released from rest.
    2. Show that \(P\) moves with simple harmonic motion. The maximum speed of \(P\) is \(\sqrt { } 2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
    3. Find the time taken by \(P\) to travel 0.3 m from \(C\).
    Edexcel M3 2014 June Q7
    15 marks Standard +0.3
    7. A particle \(P\) of mass \(m\) is attached to one end of a light elastic spring of natural length \(l\). The other end of the spring is attached to a fixed point \(A\). The particle is hanging freely in equilibrium at the point \(B\), where \(A B = 1.5 l\)
    1. Show that the modulus of elasticity of the spring is \(2 m g\). The particle is pulled vertically downwards from \(B\) to the point \(C\), where \(A C = 1.8 \mathrm { l }\), and released from rest.
    2. Show that \(P\) moves in simple harmonic motion with centre \(B\).
    3. Find the greatest magnitude of the acceleration of \(P\). The midpoint of \(B C\) is \(D\). The point \(E\) lies vertically below \(A\) and \(A E = 1.2 l\)
    4. Find the time taken by \(P\) to move directly from \(D\) to \(E\).
    Edexcel M3 2015 June Q6
    15 marks Challenging +1.2
    6. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{00388805-5d60-4327-a10e-c0df74a0cb75-11_186_1042_223_452} \captionsetup{labelformat=empty} \caption{Figure 3}
    \end{figure} Two points \(A\) and \(B\) are 6 m apart on a smooth horizontal floor. A particle \(P\) of mass 0.5 kg is attached to one end of a light elastic spring, of natural length 2.5 m and modulus of elasticity 20 N . The other end of the spring is attached to \(A\). A second light elastic spring, of natural length 1.5 m and modulus of elasticity 18 N , has one end attached to \(P\) and the other end attached to \(B\), as shown in Figure 3. Initially \(P\) rests in equilibrium at the point \(O\), where \(A O B\) is a straight line.
    1. Find the length of \(A O\). The particle \(P\) now receives an impulse of magnitude 6 N s acting in the direction \(O B\) and \(P\) starts to move towards \(B\).
    2. Show that \(P\) moves with simple harmonic motion about \(O\).
    3. Find the amplitude of the motion.
    4. Find the time taken by \(P\) to travel 1.2 m from \(O\).
    Edexcel M3 2016 June Q7
    17 marks Challenging +1.2
    7. A particle \(P\) of mass 0.5 kg is attached to one end of a light elastic spring, of natural length 1.2 m and modulus of elasticity 15 N . The other end of the spring is attached to a fixed point \(A\) on a smooth horizontal table. The particle is placed on the table at the point \(B\) where \(A B = 1.2 \mathrm {~m}\). The particle is pulled away from \(B\) to the point \(C\), where \(A B C\) is a straight line and \(B C = 0.8 \mathrm {~m}\), and is then released from rest.
      1. Show that \(P\) moves with simple harmonic motion with centre \(B\).
      2. Find the period of this motion.
    2. Find the speed of \(P\) when it reaches \(B\). The point \(D\) is the midpoint of \(A B\).
    3. Find the time taken for \(P\) to move directly from \(C\) to \(D\). When \(P\) first comes to instantaneous rest a particle \(Q\) of mass 0.3 kg is placed at \(B\). When \(P\) reaches \(B\) again, \(P\) strikes and adheres to \(Q\) to form a single particle \(R\).
    4. Show that \(R\) also moves with simple harmonic motion.
    5. Find the amplitude of this motion.
    Edexcel M3 2017 June Q7
    17 marks Challenging +1.2
    7. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{698b44b5-801c-45ec-b9de-021e44487edb-24_173_968_223_488} \captionsetup{labelformat=empty} \caption{Figure 6}
    \end{figure} The fixed points \(A\) and \(B\) are 4 m apart on a smooth horizontal floor. One end of a light elastic string, of natural length 1.8 m and modulus of elasticity 45 N , is attached to a particle \(P\) and the other end is attached to \(A\). One end of another light elastic string, of natural length 1.2 m and modulus of elasticity 20 N , is attached to \(P\) and the other end is attached to \(B\). The particle \(P\) rests in equilibrium at the point \(O\), where \(A O B\) is a straight line, as shown in Figure 6.
    1. Show that \(A O = 2.2 \mathrm {~m}\). The point \(C\) lies on the straight line \(A O B\) with \(A C = 2.7 \mathrm {~m}\). The mass of \(P\) is 0.6 kg . The particle \(P\) is held at \(C\) and then released from rest.
    2. Show that, while both strings are taut, \(P\) moves with simple harmonic motion with centre \(O\). The point \(D\) lies on the straight line \(A O B\) with \(A D = 1.8 \mathrm {~m}\). When \(P\) reaches \(D\) the string \(P B\) breaks.
    3. Find the time taken by \(P\) to move directly from \(C\) to \(A\).
    Edexcel M3 2018 June Q7
    17 marks Challenging +1.2
    7. A particle \(P\) of mass 0.5 kg is attached to one end of a light elastic string. The string has natural length \(l\) metres and modulus of elasticity 29.4 N . The other end of the string is attached to a fixed point \(A\). The particle hangs freely in equilibrium at the point \(B\), where \(B\) is vertically below \(A\) and \(A B = 1.4 \mathrm {~m}\).
    1. Show that \(l = 1.2\) The point \(C\) is vertically below \(A\) and \(A C = 1.8 \mathrm {~m}\). The particle is pulled down to \(C\) and released from rest.
    2. Show that, while the string is taut, \(P\) moves with simple harmonic motion.
    3. Calculate the speed of \(P\) at the instant when the string first becomes slack. The particle first comes to instantaneous rest at the point \(D\).
    4. Find the time taken by \(P\) to return directly from \(D\) to \(C\).
    Edexcel M3 Q4
    11 marks Challenging +1.2
    4. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{45c51316-7d58-4c16-9b5f-1d7421060a88-4_332_1056_251_459} \captionsetup{labelformat=empty} \caption{Fig. 2}
    \end{figure} A small smooth bead \(B\) of mass 0.2 kg is threaded on a smooth horizontal wire. The point \(A\) is on the same horizontal level as the wire and at a perpendicular distance \(d\) from the wire. The point \(O\) is the point on the wire nearest to \(A\), as shown in Fig. 2. The bead experiences a force of magnitude \(5 ( A B )\) newtons in the direction \(B A\) towards \(A\). Initially \(B\) is at rest with \(O B = 2 \mathrm {~m}\).
    1. Prove that \(B\) moves with simple harmonic motion about \(O\), with period \(\frac { 2 \pi } { 5 } \mathrm {~s}\).
    2. Find the greatest speed of \(B\) in the motion.
    3. Find the time when \(B\) has first moved a distance 3 m from its initial position.
    Edexcel M3 Specimen Q7
    15 marks Challenging +1.2
    7. A particle \(P\) of mass \(m\) is attached to one end of a light elastic string of natural length \(a\) and modulus of elasticity 6 mg . The other end of the string is attached to a fixed point \(O\). When the particle hangs in equilibrium with the string vertical, the extension of the string is \(e\).
    1. Find \(e\).
      (2) The particle is now pulled down a vertical distance \(\frac { 1 } { 3 } a\) below its equilibrium position and released from rest. At time \(t\) after being released, during the time when the string remains taut, the extension of the string is \(e + x\). By forming a differential equation for the motion of \(P\) while the string remains taut,
    2. show that during this time \(P\) moves with simple harmonic motion of period \(2 \pi \sqrt { \frac { a } { 6 g } }\).
      (6)
    3. Show that, while the string remains taut, the greatest speed of \(P\) is \(\frac { 1 } { 3 } \sqrt { } ( 6 g a )\).
    4. Find \(t\) when the string becomes slack for the first time. \section*{END}
    CAIE FP1 2013 November Q16 OR
    Challenging +1.8
    Given that $$y ^ { 2 } \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } - 6 y ^ { 2 } \frac { \mathrm {~d} y } { \mathrm {~d} x } + 2 y \left( \frac { \mathrm {~d} y } { \mathrm {~d} x } \right) ^ { 2 } + 3 y ^ { 3 } = 25 \mathrm { e } ^ { - 2 x }$$ and that \(v = y ^ { 3 }\), show that $$\frac { \mathrm { d } ^ { 2 } v } { \mathrm {~d} x ^ { 2 } } - 6 \frac { \mathrm {~d} v } { \mathrm {~d} x } + 9 v = 75 \mathrm { e } ^ { - 2 x }$$ Find the particular solution for \(y\) in terms of \(x\), given that when \(x = 0 , y = 2\) and \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 1\).