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

Two points \(A\) and \(B\) lie on a smooth horizontal table with \(A B = 4 a\). One end of a light elastic spring, of natural length \(a\) and modulus of elasticity \(2 m g\), is attached to \(A\). The other end of the spring is attached to a particle \(P\) of mass \(m\). Another light elastic spring, of natural length \(a\) and modulus of elasticity \(m g\), has one end attached to \(B\) and the other end attached to \(P\). The particle \(P\) is on the table at rest and in equilibrium.

1. Show that \(A P = \frac { 5 a } { 3 }\). The particle \(P\) is now moved along the table from its equilibrium position through a distance \(0.5 a\) towards \(B\) and released from rest at time \(t = 0\). At time \(t , P\) is moving with speed \(v\) and has displacement \(x\) from its equilibrium position. There is a resistance to motion of magnitude \(4 m \omega v\) where \(\omega = \sqrt { } \left( \frac { g } { a } \right)\).
2. Show that \(\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } + 4 \omega \frac { \mathrm {~d} x } { \mathrm {~d} t } + 3 \omega ^ { 2 } x = 0\).
3. Find the velocity, \(\frac { \mathrm { d } x } { \mathrm {~d} t }\), of \(P\) in terms of \(a , \omega\) and \(t\).

6. Two points \(A\) and \(B\) are in a vertical line, with \(A\) above \(B\) and \(A B = 4 a\). One end of a light elastic spring, of natural length \(a\) and modulus of elasticity \(3 m g\), is attached to \(A\). The other end of the spring is attached to a particle \(P\) of mass \(m\). Another light elastic spring, of natural length \(a\) and modulus of elasticity \(m g\), has one end attached to \(B\) and the other end attached to \(P\). The particle \(P\) hangs at rest in equilibrium.

1. Show that \(A P = \frac { 7 a } { 4 }\) The particle \(P\) is now pulled down vertically from its equilibrium position towards \(B\) and at time \(t = 0\) it is released from rest. At time \(t\), the particle \(P\) is moving with speed \(v\) and has displacement \(x\) from its equilibrium position. The particle \(P\) is subject to air resistance of magnitude \(m k v\), where \(k\) is a positive constant.
2. Show that $$\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } + k \frac { \mathrm {~d} x } { \mathrm {~d} t } + \frac { 4 g } { a } x = 0$$
3. Find the range of values of \(k\) which would result in the motion of \(P\) being a damped oscillation.

5. A toy car of mass 0.5 kg is attached to one end \(A\) of a light elastic string \(A B\), of natural length 1.5 m and modulus of elasticity 27 N . Initially the car is at rest on a smooth horizontal floor and the string lies in a straight line with \(A B = 1.5 \mathrm {~m}\). The end \(B\) is moved in a straight horizontal line directly away from the car, with constant speed \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\). At time \(t\) seconds after \(B\) starts to move, the extension of the string is \(x\) metres and the car has moved a distance \(y\) metres. The effect of air resistance on the car can be ignored. By modelling the car as a particle, show that, while the string remains taut,

1. 1. \(x + y = u t\)
   2. \(\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } + 36 x = 0\)
2. Hence show that the string becomes slack when \(t = \frac { \pi } { 6 }\)
3. Find, in terms of \(u\), the speed of the car when \(t = \frac { \pi } { 12 }\)
4. Find, in terms of \(u\), the distance the car has travelled when it first reaches end \(B\) of the string.
   

6. \begin{figure}[h] \end{figure} Figure 3 shows a uniform rod \(A B\), of length \(2 l\) and mass \(4 m\). A particle of mass \(2 m\) is attached to the rod at \(B\). The rod can turn freely in a vertical plane about a fixed smooth horizontal axis through \(A\). One end of a light elastic spring, of natural length \(2 l\) and modulus of elasticity \(k m g\), where \(k > 4\), is attached to the rod at \(B\). The other end of the spring is attached to a fixed point \(C\) which is vertically above \(A\), where \(A C = 2 l\). The angle \(B A C\) is \(2 \theta\), where \(\frac { \pi } { 6 } < \theta \leqslant \frac { \pi } { 2 }\)

1. Show that the potential energy of the system is $$4 m g l \left\{ ( k - 4 ) \sin ^ { 2 } \theta - k \sin \theta \right\} + \text { constant }$$ Given that there is a position of equilibrium with \(\theta \neq \frac { \pi } { 2 }\)
2. show that \(k > 8\) Given that \(k = 10\)
3. determine the stability of this position of equilibrium.

7 \includegraphics[max width=\textwidth, alt={}, center]{a9e010ce-c3a8-4f95-a154-fd16ef3e5e5b-4_622_767_269_689} Particles \(P\) and \(Q\), with masses \(3 m\) and \(2 m\) respectively, are connected by a light inextensible string passing over a smooth light pulley. The particle \(P\) is connected to the floor by a light spring \(S _ { 1 }\) with natural length \(a\) and modulus of elasticity mg . The particle \(Q\) is connected to the floor by a light spring \(S _ { 2 }\) with natural length \(a\) and modulus of elasticity \(2 m g\). The sections of the string not in contact with the pulley, and the two springs, are vertical. Air resistance may be neglected. The particles \(P\) and \(Q\) move vertically and the string remains taut; when the length of \(S _ { 1 }\) is \(x\), the length of \(S _ { 2 }\) is ( \(3 a - x\) ) (see diagram).

1. Find the total potential energy of the system (taking the floor as the reference level for gravitational potential energy). Hence show that \(x = \frac { 4 } { 3 } a\) is a position of stable equilibrium.
2. By differentiating the energy equation, and substituting \(x = \frac { 4 } { 3 } a + y\), show that the motion is simple harmonic, and find the period.

6 \includegraphics[max width=\textwidth, alt={}, center]{ab760a4b-e0ec-4256-838f-ed6c762ff18b-3_716_483_890_790} Two small smooth pegs \(P\) and \(Q\) are fixed at a distance \(2 a\) apart on the same horizontal level, and \(A\) is the mid-point of \(P Q\). A light rod \(A B\) of length \(4 a\) is freely pivoted at \(A\) and can rotate in the vertical plane containing \(P Q\), with \(B\) below the level of \(P Q\). A particle of mass \(m\) is attached to the rod at \(B\). A light elastic string, of natural length \(2 a\) and modulus of elasticity \(\lambda\), passes round the pegs \(P\) and \(Q\) and its two ends are attached to the rod at the point \(X\), where \(A X = a\). The angle between the rod and the downward vertical is \(\theta\), where \(- \frac { 1 } { 2 } \pi < \theta < \frac { 1 } { 2 } \pi\) (see diagram). You are given that the elastic energy stored in the string is \(\lambda a ( 1 + \cos \theta )\).

1. Show that \(\theta = 0\) is a position of equilibrium, and show that the equilibrium is stable if \(\lambda < 4 m g\).
2. Given that \(\lambda = 3 m g\), show that \(\ddot { \theta } = - k \frac { g } { a } \sin \theta\), stating the value of the constant \(k\). Hence find the approximate period of small oscillations of the system about the equilibrium position \(\theta = 0\).

1 A uniform rod with centre \(C\) has mass \(2 M\) and length 4a. The rod is free to rotate in a vertical plane about a smooth fixed horizontal axis passing through a point \(A\) on the rod, where \(A C = k a\) and \(0 < k < 2\). The rod is making small oscillations about the equilibrium position with period \(T\).

1. Show that \(T = 2 \pi \sqrt { \frac { a } { 3 g } \left( \frac { 4 + 3 k ^ { 2 } } { k } \right) }\). (You may assume the standard formula \(T = 2 \pi \sqrt { \frac { I } { m g h } }\) for the period of small oscillations of a compound pendulum.)
2. Hence find the value of \(k ^ { 2 }\) for which the period of oscillations is least.

3

1. Show, by integration, that the moment of inertia of a uniform rod of mass \(m\) and length \(2 a\) about an axis through its centre and perpendicular to the rod is \(\frac { 1 } { 3 } m a ^ { 2 }\). A pendulum of length 1 m is made by attaching a uniform sphere of mass 2 kg and radius 0.1 m to the end of a uniform rod AB of mass 1.2 kg and length 0.8 m , as shown in Fig. 3. The centre of the sphere is collinear with A and B . \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{8aab7e54-a204-481b-8f09-4bf4ca4e115d-3_442_291_717_886} \captionsetup{labelformat=empty} \caption{Fig. 3}
   \end{figure}
2. Find the moment of inertia of the pendulum about an axis through A perpendicular to the rod. The pendulum can swing freely in a vertical plane about a fixed horizontal axis through A .
3. The pendulum is held with AB at an angle \(\alpha\) to the downward vertical and released from rest. At time \(t , \mathrm { AB }\) is at an angle \(\theta\) to the vertical. Find an expression for \(\dot { \theta } ^ { 2 }\) in terms of \(\theta\) and \(\alpha\).
4. Hence, or otherwise, show that, provided that \(\alpha\) is small, the pendulum performs simple harmonic motion. Calculate the period.

3 A uniform circular disc has mass \(M\) and radius \(a\). The centre of the disc is at point C .

1. Show by integration that the moment of inertia of the disc about an axis through C and perpendicular to the disc is \(\frac { 1 } { 2 } M a ^ { 2 }\). The point A on the disc is at a distance \(\frac { 1 } { 10 } a\) from its centre.
2. Show that the moment of inertia of the disc about an axis through A and perpendicular to the disc is \(0.51 M a ^ { 2 }\). The disc can rotate freely in a vertical plane about an axis through A that is horizontal and perpendicular to the disc. The disc is held slightly displaced from its stable equilibrium position and is released from rest. In the motion that follows, the angle that AC makes with the downward vertical is \(\theta\).
3. Write down the equation of motion for the disc. Assuming \(\theta\) remains sufficiently small throughout the motion, show that the disc performs approximate simple harmonic motion and determine the period of the motion. A particle of mass \(m\) is attached at a point P on the circumference of the disc, so that the centre of mass of the system is now at A .
4. Sketch the position of P in relation to A and C . Find \(m\) in terms of \(M\) and show that the moment of inertia of the system about the axis through A and perpendicular to the disc is \(0.6 M a ^ { 2 }\). The system now rotates at a constant angular speed \(\omega\) about the axis through A .
5. Find the kinetic energy of the system. Hence find the magnitude of the constant resistive couple needed to bring the system to rest in \(n\) revolutions.

4 In this question you may assume without proof the standard results in Examination Formulae and Tables (MF2) for

• the moment of inertia of a disc about an axis through its centre perpendicular to the disc,
• the position of the centre of mass of a solid uniform cone.

Fig. 4 shows a uniform cone of radius \(a\) and height \(2 a\), with its axis of symmetry on the \(x\)-axis and its vertex at the origin. A thin slice through the cone parallel to the base is at a distance \(x\) from the vertex. \begin{figure}[h] \end{figure} The slice is taken to be a thin uniform disc of mass \(m\).

1. Write down the moment of inertia of the disc about the \(x\)-axis. Hence show that the moment of inertia of the disc about the \(y\)-axis is \(\frac { 17 } { 16 } m x ^ { 2 }\).
2. Hence show by integration that the moment of inertia of the cone about the \(y\)-axis is \(\frac { 51 } { 20 } M a ^ { 2 }\), where \(M\) is the mass of the cone. [You may assume without proof the formula for the volume of a cone.] The cone is now suspended so that it can rotate freely about a fixed, horizontal axis through its vertex. The axis of symmetry of the cone moves in a vertical plane perpendicular to the axis of rotation. The cone is released from rest when its axis of symmetry is at an acute angle \(\alpha\) to the downward vertical. At time \(t\), the angle the axis of symmetry makes with the downward vertical is \(\theta\).
3. Use an energy method to show that \(\dot { \theta } ^ { 2 } = \frac { 20 g } { 17 a } ( \cos \theta - \cos \alpha )\).
4. Hence, or otherwise, show that if \(\alpha\) is small the cone performs approximate simple harmonic motion and find the period. RECOGNISING ACHIEVEMENT

4

1. Show by integration that the moment of inertia of a uniform circular lamina of radius \(a\) and mass \(m\) about an axis perpendicular to the plane of the lamina and through its centre is \(\frac { 1 } { 2 } m a ^ { 2 }\). A closed hollow cylinder has its curved surface and both ends made from the same uniform material. It has mass \(M\), radius \(a\) and height \(h\).
2. Show that the moment of inertia of the cylinder about its axis of symmetry is \(\frac { 1 } { 2 } M a ^ { 2 } \left( \frac { a + 2 h } { a + h } \right)\). For the rest of this question take the cylinder to have mass 8 kg , radius 0.5 m and height 0.3 m .
   The cylinder is at rest and can rotate freely about its axis of symmetry. It is given a tangential impulse of magnitude 55 Ns at a point on its curved surface. The impulse is perpendicular to the axis.
3. Find the angular speed of the cylinder after the impulse. A resistive couple is now applied to the cylinder for 5 seconds. The magnitude of the couple is \(2 \dot { \theta } ^ { 2 } \mathrm { Nm }\), where \(\dot { \theta }\) is the angular speed of the cylinder in rad s \({ } ^ { - 1 }\).
4. Formulate a differential equation for \(\dot { \theta }\) and hence find the angular speed of the cylinder at the end of the 5 seconds. The cylinder is now brought to rest by a constant couple of magnitude 0.03 Nm .
5. Calculate the time it takes from when this couple is applied for the cylinder to come to rest.

3. The position vector \(\mathbf { r }\) of a particle \(P\) at time \(t\) satisfies the vector differential equation $$\frac { \mathrm { d } \mathbf { r } } { \mathrm {~d} t } + 2 \mathbf { r } = 4 \mathbf { i }$$ Given that the position vector of \(P\) at time \(t = 0\) is \(2 \mathbf { j }\), find the position vector of \(P\) at time \(t\).
(Total 6 marks)

7. A uniform plane circular disc, of mass \(m\) and radius \(a\), hangs in equilibrium from a point \(B\) on its circumference. The disc is free to rotate about a fixed smooth horizontal axis which is in the plane of the disc and tangential to the disc at \(B\). A particle \(P\), of mass \(m\), is moving horizontally with speed \(u\) in a direction which is perpendicular to the plane of the disc. At time \(t = 0 , P\) strikes the disc at its centre and adheres to the disc.

1. Show that the angular speed of the disc immediately after it has been struck by \(P\) is \(\frac { 4 u } { 9 a }\).
   (6) It is given that \(u ^ { 2 } = \frac { 1 } { 10 } a g\), and that air resistance is negligible.
2. Find the angle through which the disc turns before it first comes to instantaneous rest. The disc first returns to its initial position at time \(t = T\).
3. 1. Write down an equation of motion for the disc.
   2. Hence find \(T\) in terms of \(a , g\) and \(m\), using a suitable approximation which should be justified.

6.

1. Prove, using integration, that the moment of inertia of a uniform circular disc, of mass \(m\) and radius \(a\), about an axis through its centre \(O\) perpendicular to the plane of the disc is \(\frac { 1 } { 2 } m a ^ { 2 }\). The line \(A B\) is a diameter of the disc and \(P\) is the mid-point of \(O A\). The disc is free to rotate about a fixed smooth horizontal axis \(L\). The axis lies in the plane of the disc, passes through \(P\) and is perpendicular to \(O A\). A particle of mass \(m\) is attached to the disc at \(A\) and a particle of mass \(2 m\) is attached to the disc at \(B\).
2. Show that the moment of inertia of the loaded disc about \(L\) is \(\frac { 21 } { 4 } m a ^ { 2 }\). At time \(t = 0 , P B\) makes a small angle with the downward vertical through \(P\) and the loaded disc is released from rest. By obtaining an equation of motion for the disc and using a suitable approximation,
3. find the time when the loaded disc first comes to instantaneous rest. END

4. A uniform circular disc, of mass \(m\) and radius \(r\), has a diameter \(A B\). The point \(C\) on \(A B\) is such that \(A C = \frac { 1 } { 2 } r\). The disc can rotate freely in a vertical plane about a horizontal axis through \(C\), perpendicular to the plane of the disc. The disc makes small oscillations in a vertical plane about the position of equilibrium in which \(B\) is below \(A\).

1. Show that the motion is approximately simple harmonic.
2. Show that the period of this approximate simple harmonic motion is \(\pi \sqrt { \left( \frac { 6 r } { g } \right) }\). The speed of \(B\) when it is vertically below \(A\) is \(\sqrt { \left( \frac { g r } { 54 } \right) }\). The disc comes to rest when \(C B\) makes an angle \(\alpha\) with the downward vertical.
3. Find an approximate value of \(\alpha\).
   (3)

2 A light elastic string has natural length \(a\) and modulus of elasticity \(k m g\), where \(k > 2\). One end of the string is attached to a fixed point O . A particle P of mass \(m\) is attached to the other end of the string. P is held at rest a distance \(\frac { 3 } { 2 } a\) vertically below O . At time \(t\) after P is released, its vertical distance below O is \(y\).

1. Show that, while the string is in tension, the equation of motion of P is given by the differential equation \(\frac { d ^ { 2 } y } { d t ^ { 2 } } = ( k + 1 ) g - \frac { k g } { a } y\). A student transforms the differential equation in part (a) into the standard SHM equation \(\frac { d ^ { 2 } x } { d t ^ { 2 } } = - \omega ^ { 2 } x\).
2. - Find an expression for \(x\) in terms of \(y , k\) and \(a\).

15 In an oscillating system, a particle of mass \(m \mathrm {~kg}\) moves in a horizontal line. Its displacement from its equilibrium position O at time \(t\) seconds is \(x\) metres, its velocity is \(v \mathrm {~ms} ^ { - 1 }\), and it is acted on by a force \(2 m x\) newtons acting towards O as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{b57a2590-84e8-4998-9633-902db861f23a-6_212_914_408_242} Initially, the particle is projected away from O with speed \(1 \mathrm {~ms} ^ { - 1 }\) from a point 2 m from O in the positive direction.

1. 1. Show that the motion is modelled by the differential equation \(\frac { d ^ { 2 } x } { d t ^ { 2 } } + 2 x = 0\).
   2. State the type of motion.
   3. Write down the period of the motion.
   4. Find \(x\) in terms of \(t\).
   5. Find the amplitude of the motion.
2. The motion is now damped by a force \(2 m v\) newtons.
   1. Show that \(\frac { d ^ { 2 } x } { d t ^ { 2 } } + 2 \frac { d x } { d t } + 2 x = 0\).
   2. State, giving a reason, whether the system is under-damped, critically damped or over-damped.
   3. Determine the general solution of this differential equation.
3. Finally, a variable force \(2 m \cos 2 t\) newtons is added, so that the motion is now modelled by the differential equation \(\frac { d ^ { 2 } x } { d t ^ { 2 } } + 2 \frac { d x } { d t } + 2 x = 2 \cos 2 t\).
   1. Find \(x\) in terms of \(t\). In the long term, the particle is seen to perform simple harmonic motion with a period of just over 3 seconds.
   2. Verify that this behaviour is consistent with the answer to part (c)(i).

12 A particle P of mass \(m\) is fixed to one end of a light elastic string of natural length \(l\) and modulus of elasticity 12 mg . The other end of the string is attached to a fixed point O . Particle P is held next to O and then released from rest.

1. Show that P next comes instantaneously to rest when the length of the string is \(\frac { 3 } { 2 } l\). The string first becomes taut at time \(t = 0\). At time \(t \geqslant 0\), the length of the string is \(l + x\), where \(x\) is the extension in the string.
2. Show that when the string is taut, \(x\) satisfies the differential equation $$\ddot { \mathrm { x } } + \omega ^ { 2 } \mathrm { x } = \mathrm { g } \text {, where } \omega ^ { 2 } = \frac { 12 \mathrm {~g} } { \mathrm { I } } \text {. }$$
3. By using the substitution \(x = y + \frac { g } { \omega ^ { 2 } }\), solve the differential equation to show that the time when the string first becomes slack satisfies the equation $$\cos \omega \mathrm { t } - \sqrt { \mathrm { k } } \sin \omega \mathrm { t } = 1$$ where \(k\) is an integer to be determined.

3. A light elastic string, of natural length \(l \mathrm {~m}\) and modulus of elasticity 14 N , is hanging vertically with its upper end fixed and a particle of mass \(m \mathrm {~kg}\) attached to the lower end. The particle is initially in equilibrium and air resistance is to be neglected.

1. Find, in terms of \(m , g\) and \(l\), the extension, \(e\), of the string when the particle is in equilibrium. The particle is pulled vertically downwards a further distance from its equilibrium position and released. In its subsequent motion, the string remains taut. Let \(x \mathrm {~m}\) denote the extension of the string from the equilibrium position at time \(t \mathrm {~s}\).
2. 1. Write down, in terms of \(x , m , g\) and \(l\), an expression for the tension in the string.
   2. Hence, show that the particle is moving with Simple Harmonic Motion which satisfies the differential equation, $$\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } = - \frac { 14 } { m l } x$$
   3. State the maximum distance that the particle could be pulled vertically downwards from its equilibrium position and still move with Simple Harmonic Motion. Give a reason for your answer.
3. Given that \(m = 0.5 , l = 0.7\) and that the particle is pulled to the position where \(x = 0.2\) before being released,
   1. find the maximum speed of the particle,
   2. determine the time taken for the particle to reach \(x = 0.15\) for the first time.

2. A particle \(P\) moves along the \(x\)-axis such that its position \(x\) metres, after \(t\) seconds, is given by $$x = \sin ( \pi t ) + \sqrt { 3 } \cos ( \pi t )$$

1. 1. Show that the motion of the particle \(P\) is Simple Harmonic. State the value of \(x\) at the centre of motion.
   2. Show that the period of the motion of \(P\) is 2 s and determine the amplitude. Suppose that another particle \(Q\) is introduced so that it also moves along the \(x\)-axis with Simple Harmonic Motion with centre of motion, \(O\), and period equal to that of particle \(P\). When \(t = 0\), the particle \(Q\) is at \(O\) and when it is \(2 \sqrt { 3 } \mathrm {~m}\) from \(O\) its speed is \(2 \pi \mathrm {~ms} ^ { - 1 }\).
2. Find the amplitude of particle \(Q\).
3. Determine the time when particles \(P\) and \(Q\) first meet.

6. The diagram shows a particle \(P\), of mass 4 kg , lying on a smooth horizontal surface. It is attached by two light springs to fixed points \(A\) and \(B\), where \(A B = 2.8 \mathrm {~m}\).
Spring \(A P\) has natural length 0.8 m and modulus of elasticity 60 N .
Spring \(P B\) has natural length 1.2 m and modulus of elasticity 30 N . \includegraphics[max width=\textwidth, alt={}, center]{b9c63cb4-d446-4548-be42-e30b10cb4b99-5_231_1253_612_404} When \(P\) is in equilibrium, it is at the point \(C\).

1. Show that \(A C = 1 \mathrm {~m}\).
2. The particle \(P\) is pulled horizontally and is initially held at rest at the midpoint of \(A B\). The system is then released.
   1. Show that \(P\) performs Simple Harmonic Motion about centre \(C\) and find the period of its motion.
   2. Determine the shortest time taken for \(P\) to reach a position where there is no tension in the spring \(A P\). \section*{END OF PAPER}

3. The vertical motion of a point on the surface of the water in a certain harbour may be modelled as Simple Harmonic Motion about a mean level. The diagram shows that, on a particular day, the depth of water in the harbour at low tide is 2 m and the depth of the water in the harbour at high tide is 10 m . The table below shows the times of high and low tides on this day. \includegraphics[max width=\textwidth, alt={}, center]{d7f600c5-af4a-4708-bfd9-92b37a95c634-4_405_912_621_233}

1. Write down the period and amplitude of the motion.
2. Let \(x \mathrm {~m}\) denote the height of water above mean level \(t\) hours after 5a.m. Find an expression for \(x\) in terms of \(t\).
3. The depth of water must be at least 4 m for boats to safely use the harbour. Determine the earliest time, after low tide at 5 a.m., at which boats can safely leave the harbour and hence find the latest possible time of return before the next low tide.
4. Calculate the rate at which the level of water is falling at 2 p.m.

6. The diagram shows a playground ride consisting of a seat \(P\), of mass 12 kg , attached to a vertical spring, which is fixed to a horizontal board. When the ride is at rest with nobody on it, the compression of the spring is 0.05 m . \includegraphics[max width=\textwidth, alt={}, center]{3efc4ef6-8a80-4267-8e95-733200e875c5-4_305_654_1032_667} The spring is of natural length 0.75 m and modulus of elasticity \(\lambda\).

1. Find the value of \(\lambda\). The seat \(P\) is now pushed vertically downwards a further 0.05 m and is then released from rest.
2. Show that \(P\) makes Simple Harmonic oscillations of period \(\frac { \pi } { 7 }\) and write down the amplitude of the motion.
3. Find the maximum speed of \(P\).
4. Calculate the speed of \(P\) when it is at a distance 0.03 m from the equilibrium position.
5. Find the distance of \(P\) from the equilibrium position 1.6 s after it is released.[3]
6. State one modelling assumption you have made about the seat and one modelling assumption you have made about the spring.

10. \begin{figure}[h] \end{figure} The motion of a pendulum, shown in Figure 3, is modelled by the differential equation $$\frac { \mathrm { d } ^ { 2 } \theta } { \mathrm {~d} t ^ { 2 } } + 9 \theta = \frac { 1 } { 2 } \cos 3 t$$ where \(\theta\) is the angle, in radians, that the pendulum makes with the downward vertical, \(t\) seconds after it begins to move. \begin{enumerate}[label=(\alph*)] \item

1. Show that a particular solution of the differential equation is $$\theta = \frac { 1 } { 12 } t \sin 3 t$$
2. Hence, find the general solution of the differential equation. Initially, the pendulum
   Given that, 10 seconds after it begins to move, the pendulum makes an angle of \(\alpha\) radians with the downward vertical,
3. determine, according to the model, the value of \(\alpha\) to 3 significant figures. Given that the true value of \(\alpha\) is 0.62
4. evaluate the model. The differential equation $$\frac { \mathrm { d } ^ { 2 } \theta } { \mathrm {~d} t ^ { 2 } } + 9 \theta = \frac { 1 } { 2 } \cos 3 t$$ models the motion of the pendulum as moving with forced harmonic motion.
5. Refine the differential equation so that the motion of the pendulum is simple harmonic motion.

1. A company plans to build a new fairground ride. The ride will consist of a capsule that will hold the passengers and the capsule will be attached to a tall tower. The capsule is to be released from rest from a point half way up the tower and then made to oscillate in a vertical line.

The vertical displacement, \(x\) metres, of the top of the capsule below its initial position at time \(t\) seconds is modelled by the differential equation, $$m \frac { \mathrm {~d} ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } + 4 \frac { \mathrm {~d} x } { \mathrm {~d} t } + x = 200 \cos t , \quad t \geqslant 0$$ where \(m\) is the mass of the capsule including its passengers, in thousands of kilograms.
The maximum permissible weight for the capsule, including its passengers, is 30000 N .
Taking the value of \(g\) to be \(10 \mathrm {~ms} ^ { - 2 }\) and assuming the capsule is at its maximum permissible weight,

1. 1. explain why the value of \(m\) is 3
   2. show that a particular solution to the differential equation is $$x = 40 \sin t - 20 \cos t$$
   3. hence find the general solution of the differential equation.
2. Using the model, find, to the nearest metre, the vertical distance of the top of the capsule from its initial position, 9 seconds after it is released.