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

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Edexcel M3 2002 June Q6
14 marks Standard +0.3
A light elastic string, of natural length \(4a\) and modulus of elasticity \(8mg\), has one end attached to a fixed point \(A\). A particle \(P\) of mass \(m\) is attached to the other end of the string and hangs in equilibrium at the point \(O\).
  1. Find the distance \(AO\). [2]
The particle is now pulled down to a point \(C\) vertically below \(O\), where \(OC = d\). It is released from rest. In the subsequent motion the string does not become slack.
  1. Show that \(P\) moves with simple harmonic motion of period \(\pi\sqrt{\frac{2a}{g}}\). [7]
The greatest speed of \(P\) during this motion is \(\frac{1}{2}\sqrt{(ga)}\).
  1. Find \(d\) in terms of \(a\). [3]
Instead of being pulled down a distance \(d\), the particle is pulled down a distance \(a\). Without further calculation,
  1. describe briefly the subsequent motion of \(P\). [2]
Edexcel M3 2003 June Q5
13 marks Standard +0.3
A particle \(P\) of mass \(0.8\) kg is attached to one end \(A\) of a light elastic spring \(OA\), of natural length \(60\) cm and modulus of elasticity \(12\) N. The spring is placed on a smooth horizontal table and the end \(O\) is fixed. The particle \(P\) is pulled away from \(O\) to a point \(B\), where \(OB = 85\) cm, and is released from rest.
  1. Prove that the motion of \(P\) is simple harmonic with period \(\frac{2\pi}{5}\) s. [5]
  2. Find the greatest magnitude of the acceleration of \(P\) during the motion. [2]
Two seconds after being released from rest, \(P\) passes through the point \(C\).
  1. Find, to 2 significant figures, the speed of \(P\) as it passes through \(C\). [5]
  2. State the direction in which \(P\) is moving 2 s after being released. [1]
Edexcel M3 2006 June Q3
11 marks Standard +0.3
A particle \(P\) of mass \(0.2\) kg oscillates with simple harmonic motion between the points \(A\) and \(B\), coming to rest at both points. The distance \(AB\) is \(0.2\) m, and \(P\) completes \(5\) oscillations every second.
  1. Find, to \(3\) significant figures, the maximum resultant force exerted on \(P\). [6]
When the particle is at \(A\), it is struck a blow in the direction \(BA\). The particle now oscillates with simple harmonic motion with the same frequency as previously but twice the amplitude.
  1. Find, to \(3\) significant figures, the speed of the particle immediately after it has been struck. [5]
Edexcel M3 2007 June Q5
11 marks Standard +0.3
A particle \(P\) moves on the \(x\)-axis with simple harmonic motion about the origin \(O\) as centre. When \(P\) is at a distance \(0.04\) m from \(O\), its speed is \(0.2\) m s\(^{-1}\) and the magnitude of its acceleration is \(1\) m s\(^{-2}\).
  1. Find the period of the motion. [3]
The amplitude of the motion is \(a\) metres. Find
  1. the value of \(a\), [3]
  2. the total time, within one complete oscillation, for which the distance \(OP\) is greater than \(\frac{3}{4}a\) metres. [5]
Edexcel M3 2009 June Q7
16 marks Challenging +1.2
\includegraphics{figure_4} \(A\) and \(B\) are two points on a smooth horizontal floor, where \(AB = 5\) m. A particle \(P\) has mass \(0.5\) kg. One end of a light elastic spring, of natural length \(2\) m and modulus of elasticity \(16\) N, is attached to \(P\) and the other end is attached to \(A\). The ends of another light elastic spring, of natural length \(1\) m and modulus of elasticity \(12\) N, are attached to \(P\) and \(B\), as shown in Figure 4.
  1. Find the extensions in the two springs when the particle is at rest in equilibrium. [5]
Initially \(P\) is at rest in equilibrium. It is then set in motion and starts to move towards \(B\). In the subsequent motion \(P\) does not reach \(A\) or \(B\).
  1. Show that \(P\) oscillates with simple harmonic motion about the equilibrium position. [4]
  2. Given that the initial speed of \(P\) is \(\sqrt{10}\) m s\(^{-1}\), find the proportion of time in each complete oscillation for which \(P\) stays within \(0.25\) m of the equilibrium position. [7]
Edexcel M3 2012 June Q2
8 marks Standard +0.3
A particle \(P\) moves in a straight line with simple harmonic motion about a fixed centre \(O\). The period of the motion is \(\frac{\pi}{2}\) seconds. At time \(t\) seconds the speed of \(P\) is \(v\) m s\(^{-1}\). When \(t = 0\), \(P\) is at \(O\) and \(v = 6\). Find
  1. the greatest distance of \(P\) from \(O\) during the motion, [3]
  2. the greatest magnitude of the acceleration of \(P\) during the motion, [2]
  3. the smallest positive value of \(t\) for which \(P\) is 1 m from \(O\). [3]
Edexcel M3 2012 June Q7
14 marks Standard +0.8
A particle \(B\) of mass 0.5 kg is attached to one end of a light elastic string of natural length 0.75 m and modulus of elasticity 24.5 N. The other end of the string is attached to a fixed point \(A\). The particle is hanging in equilibrium at the point \(E\), vertically below \(A\).
  1. Show that \(AE = 0.9\) m. [3]
The particle is held at \(A\) and released from rest. The particle first comes to instantaneous rest at the point \(C\).
  1. Find the distance \(AC\). [5]
  2. Show that while the string is taut, \(B\) is moving with simple harmonic motion with centre \(E\). [4]
  3. Calculate the maximum speed of \(B\). [2]
Edexcel M3 2014 June Q6
14 marks Standard +0.8
A light elastic string, of natural length \(l\) and modulus of elasticity \(4mg\), has one end attached to a fixed point \(A\). The other end is attached to a particle \(P\) of mass \(m\). The particle hangs freely at rest in equilibrium at the point \(E\). The distance of \(E\) below \(A\) is \((l + e)\).
  1. Find \(e\) in terms of \(l\). [2]
At time \(t = 0\), the particle is projected vertically downwards from \(E\) with speed \(\sqrt{gl}\).
  1. Prove that, while the string is taut, \(P\) moves with simple harmonic motion. [5]
  2. Find the amplitude of the simple harmonic motion. [3]
  3. Find the time at which the string first goes slack. [4]
OCR MEI C4 2013 June Q6
18 marks Standard +0.3
The motion of a particle is modelled by the differential equation $$v \frac{dv}{dt} + 4x = 0,$$ where \(x\) is its displacement from a fixed point, and \(v\) is its velocity. Initially \(x = 1\) and \(v = 4\).
  1. Solve the differential equation to show that \(v^2 = 20 - 4x^2\). [4]
Now consider motion for which \(x = \cos 2t + 2 \sin 2t\), where \(x\) is the displacement from a fixed point at time \(t\).
  1. Verify that, when \(t = 0\), \(x = 1\). Use the fact that \(v = \frac{dx}{dt}\) to verify that when \(t = 0\), \(v = 4\). [4]
  2. Express \(x\) in the form \(R \cos(2t - \alpha)\), where \(R\) and \(\alpha\) are constants to be determined, and obtain the corresponding expression for \(v\). Hence or otherwise verify that, for this motion too, \(v^2 = 20 - 4x^2\). [7]
  3. Use your answers to part (iii) to find the maximum value of \(x\), and the earliest time at which \(x\) reaches this maximum value. [3]
Edexcel M3 Q6
15 marks Standard +0.8
A light elastic string, of natural length \(l\) m and modulus of elasticity \(\frac{mg}{2}\) newtons, has one end fastened to a fixed point \(O\). A particle \(P\), of mass \(m\) kg, is attached to the other end of the string. \(P\) hangs in equilibrium at the point \(E\), vertically below \(O\), where \(OE = (l + e)\) m
  1. Find the numerical value of the ratio \(e : l\). [2 marks]
\(P\) is now pulled down a further distance \(\frac{3l}{2}\) m from \(E\) and is released from rest. In the subsequent motion, the string remains taut. At time \(t\) s after being released, \(P\) is at a distance \(x\) m below \(E\).
  1. Write down a differential equation for the motion of \(P\) and show that the motion is simple harmonic. [4 marks]
  2. Write down the period of the motion. [2 marks]
  3. Find the speed with which \(P\) first passes through \(E\) again. [2 marks]
  4. Show that the time taken by \(P\) after it is released to reach the point \(A\) above \(E\), where \(AE = \frac{3l}{4}\) m, is \(\frac{2\pi}{3}\sqrt{\frac{2l}{g}}\) s. [5 marks]
Edexcel M3 Q6
15 marks Standard +0.8
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. [3 marks]
\(P\) is displaced to a point 0·5 m vertically below its equilibrium position and released from rest. \begin{enumerate}[label=(\alph*)] \setcounter{enumi}{1} \item Show that the subsequent motion of \(P\) is simple harmonic, with period 1·68 s. [5 marks] \item Calculate the maximum speed of \(P\) during its motion. [3 marks] \item 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. [4 marks] \end{enumerate]
Edexcel M3 Q2
7 marks Standard +0.3
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 ms\(^{-1}\). When it is at a distance 2 m from \(O\), its speed is 4 ms\(^{-1}\).
  1. Find the amplitude of the motion. [4 marks]
  2. Show that the period of motion is \(\frac{\pi}{2}\) s. [3 marks]
Edexcel M3 Q7
15 marks Standard +0.8
A particle of mass \(m\) kg is attached to one end of an elastic string of natural length \(l\) m and modulus of elasticity \(\lambda\) 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 \(C\). [6 marks]
    2. Find the period, \(T\) s, of the motion in terms of \(l\), \(m\) and \(\lambda\). [1 mark]
    3. Explain the significance of the term 'slightly' as used in (i) above. [1 mark]
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 \(T_1\) s.
    1. Write down an expression for \(T_1\) in terms of \(l\), \(m\), \(M\) and \(\lambda\). [2 marks]
    2. Hence show that \(T_1^2 - T^2 = \frac{4\pi^2 d}{g}\). [5 marks]
Edexcel M3 Q2
7 marks Standard +0.3
A particle moves along a straight line in such a way that its displacement \(x\) 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.
  1. Show that the motion of the particle is simple harmonic. [5 marks]
  2. If the particle leaves \(O\) with speed 15 ms\(^{-1}\), and \(\omega = 3\), find the amplitude of the motion. [2 marks]
Edexcel M3 Q4
9 marks Standard +0.8
A particle \(P\) moves with simple harmonic motion in a straight line, with the centre of motion at the point \(O\) on the line. \(A\) and \(B\) are on opposite sides of \(O\), with \(OA = 4\) m, \(OB = 6\) m. When passing through \(A\) and \(B\), \(P\) has speed \(6\) ms\(^{-1}\) and \(4\) ms\(^{-1}\) respectively. \includegraphics{figure_4} \begin{enumerate}[label=(\alph*)] \item Find the amplitude of the motion. [6 marks] \item Show that the period of motion is \(2\pi\) s. [3 marks]
Edexcel M3 Q7
17 marks Challenging +1.8
One end of a light elastic string, of natural length \(3l\) m, is attached to a fixed point \(O\). A particle of mass \(m\) kg is attached to the other end of the string. When the particle hangs freely in equilibrium, the string is extended by a length of \(l\) m. The particle is then pulled down through a further distance \(2l\) m and released from rest.
  1. Prove that as long as the string is taut, the particle performs simple harmonic motion about its equilibrium position. [5 marks]
  2. Show that the time between the release of the particle and the instant when the string becomes slack is \(\frac{2\pi}{3}\sqrt{\frac{l}{g}}\) s. [4 marks]
  3. Find the greatest height reached by the particle above its point of release. [4 marks]
  4. Show that the time \(T\) s taken to reach this greatest height from the moment of release is given by \(T = \left(\frac{2\pi}{3} + \sqrt{3}\right)\sqrt{\frac{l}{g}}\). [4 marks]
Edexcel M3 Q6
12 marks Standard +0.3
A particle \(P\) of mass \(m\) kg hangs in equilibrium at one end of a light spring, of natural length \(l\) m and modulus of elasticity \(\lambda\) N, whose other end is fixed at a point vertically above \(P\). In this position the length of the spring is \((l + e)\) m. When \(P\) is displaced vertically through a small distance and released, it performs simple harmonic motion with 5 oscillations per second.
  1. Show that \(\frac{\lambda}{l} = 100n^2m\). [8 marks]
  2. Express \(e\) in terms of \(g\). [2 marks]
  3. Determine, in terms of \(m\) and \(l\), the magnitude of the tension in the spring when it is stretched to twice its natural length. [2 marks]
Edexcel M3 Q7
15 marks Challenging +1.8
A particle \(P\) of mass \(m\) kg is fixed to one end of a light elastic string of modulus \(mg\) N and natural length \(l\) m. The other end of the string is attached to a fixed point \(O\) on a rough horizontal table. Initially \(P\) is at rest in limiting equilibrium on the table at the point \(X\) where \(OX = \frac{5l}{4}\) m.
  1. Find the coefficient of friction between \(P\) and the table. [2 marks]
\(P\) is now given a small displacement \(x\) m horizontally along \(OX\), away from \(O\). While \(P\) is in motion, the frictional resistance remains constant at its limiting value.
  1. Show that as long as the string remains taut, \(P\) performs simple harmonic motion with \(X\) as the centre. [4 marks]
If \(P\) is held at the point where the extension in the string is \(l\) m and then released,
  1. show that the string becomes slack after a time \(\left(\frac{\pi}{2} + \arcsin\left(\frac{1}{3}\right)\right)\sqrt{\frac{l}{g}}\) s. [5 marks]
  2. Determine the speed of \(P\) when it reaches \(O\). [4 marks]
Edexcel M3 Q6
12 marks Standard +0.3
The figure shows a swing consisting of two identical vertical light springs attached symmetrically to a light horizontal cross-bar and supported from a strong fixed horizontal beam. When a mass of 24 kg is placed at the mid-point of the cross-bar, both springs extend by 30 cm to the position \(A\), as shown. \includegraphics{figure_6} Each spring has natural length \(l\) m and modulus of elasticity \(\lambda\) N.
  1. Show that \(\lambda = 392l\). [2 marks]
The 24 kg mass is left on the bar and the bar is then displaced downwards by a further 20 cm.
  1. Prove that the system comprising the bar and the mass now performs simple harmonic motion with the centre of oscillation at the level \(A\). [5 marks]
  2. Calculate the number of oscillations made per second in this motion. [3 marks]
  3. Find the maximum acceleration which the mass experiences during the motion. [2 marks]
OCR M3 2009 June Q6
13 marks Standard +0.3
\includegraphics{figure_6} A particle \(P\) of mass \(m\) kg is attached to one end of a light inextensible string of length \(L\) m. The other end of the string is attached to a fixed point \(O\). The particle is held at rest with the string taut and then released. \(P\) starts to move and in the subsequent motion the angular displacement of \(OP\), at time \(t\) s, is \(\theta\) radians from the downward vertical (see diagram). The initial value of \(\theta\) is \(0.05\).
  1. Show that \(\frac{d^2\theta}{dt^2} = -\frac{g}{L} \sin \theta\). [2]
  2. Hence show that the motion of \(P\) is approximately simple harmonic. [2]
  3. Given that the period of the approximate simple harmonic motion is \(\frac{4}{3}\pi\) s, find the value of \(L\). [2]
  4. Find the value of \(\theta\) when \(t = 0.7\) s, and the value of \(t\) when \(\theta\) next takes this value. [4]
  5. Find the speed of \(P\) when \(t = 0.7\) s. [3]
OCR M3 2010 June Q5
11 marks Standard +0.8
\includegraphics{figure_5} Particles \(P_1\) and \(P_2\) are each moving with simple harmonic motion along the same straight line. \(P_1\)'s motion has centre \(C_1\), period \(2\pi\) s and amplitude \(3\) m; \(P_2\)'s motion has centre \(C_2\), period \(\frac{4}{3}\pi\) s and amplitude \(4\) m. The points \(C_1\) and \(C_2\) are \(6.5\) m apart. The displacements of \(P_1\) and \(P_2\) from their centres of oscillation at time \(t\) s are denoted by \(x_1\) m and \(x_2\) m respectively. The diagram shows the positions of the particles at time \(t = 0\), when \(x_1 = 3\) and \(x_2 = 4\).
  1. State expressions for \(x_1\) and \(x_2\) in terms of \(t\), which are valid until the particles collide. [3]
The particles collide when \(t = 5.99\), correct to \(3\) significant figures.
  1. Find the distance travelled by \(P_2\) before the collision takes place. [4]
  2. Find the velocities of \(P_1\) and \(P_2\) immediately before the collision, and state whether the particles are travelling in the same direction or in opposite directions. [4]
OCR M3 2011 June Q4
11 marks Standard +0.3
One end of a light inextensible string of length \(2\) m is attached to a fixed point \(O\). A particle \(P\) of mass \(0.2\) kg is attached to the other end of the string. \(P\) is held at rest with the string taut so that \(OP\) makes an angle of \(0.15\) radians with the downward vertical. \(P\) is released and \(t\) seconds afterwards \(OP\) makes an angle of \(\theta\) radians with the downward vertical.
  1. Show that \(\frac{d^2\theta}{dt^2} = -4.9 \sin \theta\) and give a reason why the motion is approximately simple harmonic. [3]
Using the simple harmonic approximation,
  1. obtain an expression for \(\theta\) in terms of \(t\) and hence find the values of \(t\) at the first and second occasions when \(\theta = -0.1\), [5]
  2. find the angular speed of \(OP\) and the linear speed of \(P\) when \(t = 0.5\). [3]
OCR M3 2015 June Q3
13 marks Challenging +1.2
\includegraphics{figure_3} A small object \(P\) is attached to one end of each of two vertical light elastic strings. One string is of natural length \(0.4\) m and has modulus of elasticity \(10\) N; the other string is of natural length \(0.5\) m and has modulus of elasticity \(12\) N. The upper ends of both strings are attached to a fixed horizontal beam and \(P\) hangs in equilibrium \(0.6\) m below the beam (see diagram).
  1. Show that the weight of \(P\) is \(7.4\) N and find the total elastic potential energy stored in the two strings when \(P\) is hanging in equilibrium. [6]
\(P\) is then held at a point \(0.7\) m below the beam with the strings vertical. \(P\) is released from rest.
  1. Show that, throughout the subsequent motion, \(P\) performs simple harmonic motion, and find the period. [7]
OCR M3 2015 June Q6
11 marks Standard +0.3
A particle \(P\) starts from rest from a point \(A\) and moves in a straight line with simple harmonic motion about a point \(O\). At time \(t\) seconds after the motion starts the displacement of \(P\) from \(O\) is \(x\) m towards \(A\). The particle \(P\) is next at rest when \(t = 0.25\pi\) having travelled a distance of \(1.2\) m.
  1. Find the maximum velocity of \(P\). [3]
  2. Find the value of \(x\) and the velocity of \(P\) when \(t = 0.7\). [4]
  3. Find the other values of \(t\), for \(0 < t < 1\), at which \(P\)'s speed is the same as when \(t = 0.7\). Find also the corresponding values of \(x\). [4]
OCR M3 2016 June Q7
17 marks Challenging +1.2
A particle \(P\) of mass \(m\) kg is attached to one end of a light elastic string of modulus of elasticity \(24mg\) N and natural length \(0.6\) m. The other end of the string is attached to a fixed point \(O\); the particle \(P\) rests in equilibrium at a point \(A\) with the string vertical.
  1. Show that the distance \(OA\) is \(0.625\) m. [2]
Another particle \(Q\), of mass \(3m\) kg, is released from rest from a point \(0.4\) m above \(P\) and falls onto \(P\). The two particles coalesce.
  1. Show that the combined particle initially moves with speed \(2.1\) m s\(^{-1}\). [3]
  2. Show that the combined particle initially performs simple harmonic motion, and find the centre of this motion and its amplitude. [5]
  3. Find the time that elapses between \(Q\) being released from rest and the combined particle first reaching the highest point of its subsequent motion. [7]