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

1. A particle \(P\) moves along a straight line.

At time \(t\) minutes, the displacement, \(x\) metres, of \(P\) from a fixed point \(O\) on the line is modelled by the differential equation $$t ^ { 2 } \frac { \mathrm {~d} ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } - 2 t \frac { \mathrm {~d} x } { \mathrm {~d} t } + 2 x + 16 t ^ { 2 } x = 4 t ^ { 3 } \sin 2 t$$

1. Show that the transformation \(x =\) ty transforms equation (I) into the equation $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} t ^ { 2 } } + 16 y = 4 \sin 2 t$$
2. Hence find a general solution for the displacement of \(P\) from \(O\) at time \(t\) minutes.

1. The points \(A\) and \(B\) lie on a smooth horizontal surface with \(A B = 4.5 \mathrm {~m}\).

A light elastic string has natural length 1.5 m and modulus of elasticity 15 N . One end of the string is attached to \(A\) and the other end of the string is attached to \(B\). A particle, \(P\), of mass 0.2 kg , is attached to the stretched string so that \(A P B\) is a straight line and \(A P = 1.5 \mathrm {~m}\). The particle rests in equilibrium on the surface. The particle is now moved directly towards \(A\) and is held on the surface so \(A P B\) is a straight line with \(A P = 1 \mathrm {~m}\). The particle is released from rest.

1. Prove that \(P\) moves with simple harmonic motion.
2. Find
   1. the maximum speed of \(P\) during the motion,
   2. the maximum acceleration of \(P\) during the motion.
3. Find the total time, in each complete oscillation of \(P\), for which the speed of \(P\) is greater than \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Throughout this question, use \(\boldsymbol { g } = \mathbf { 1 0 m ~ s } ^ { \mathbf { - 2 } }\)

A light elastic string has natural length 1.25 m and modulus of elasticity 25 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 fixed point \(A\). Particle \(P\) hangs freely in equilibrium with \(P\) vertically below \(A\) The particle is then pulled vertically down to a point \(B\) and released from rest.

1. Show that, while the string is taut, \(P\) moves with simple harmonic motion with period \(\frac { \pi } { \sqrt { 10 } }\) seconds. The maximum kinetic energy of \(P\) during the subsequent motion is 2.5 J .
2. Show that \(A B = 2 \mathrm {~m}\) The particle returns to \(B\) for the first time \(T\) seconds after it was released from rest at \(B\)
3. Find the value of \(T\)

8. \begin{figure}[h] \end{figure} The fixed points \(A\) and \(B\) lie on a smooth horizontal surface with \(A B = 6 \mathrm {~m}\).
A particle \(P\) has mass 0.3 kg .
One end of a light elastic string, of natural length 2 m and modulus of elasticity 20 N , is attached to \(P\), and the other end is attached to \(A\). One end of another light elastic string, of natural length 2 m and modulus of elasticity 40 N , is attached to \(P\) and the other end is attached to \(B\). The particle \(P\) is at rest in equilibrium at the point \(E\) on the surface, as shown in Figure 7.

1. Show that \(E B = \frac { 8 } { 3 } \mathrm {~m}\). The particle \(P\) is now held at the midpoint of \(A B\) and released from rest.
2. Show that \(P\) oscillates with simple harmonic motion about the point \(E\). The time between the instant when \(P\) is released and the instant when it first returns to the point \(E\) is \(S\) seconds.
3. Find the exact value of \(S\).
4. Find the length of time during one oscillation for which the speed of \(P\) is more than \(2 \mathrm {~ms} ^ { - 1 }\)

1. A particle \(P\) moves in a straight line with simple harmonic motion about a fixed point \(O\). The magnitude of the greatest acceleration of \(P\) is \(18 \mathrm {~m} \mathrm {~s} ^ { - 2 }\)

When \(P\) is 0.3 m from \(O\), the speed of \(P\) is \(2.4 \mathrm {~ms} ^ { - 1 }\) The amplitude of the motion is \(a\) metres.

1. Show that \(a = 0.5\)
2. Find the greatest speed of \(P\). During one oscillation, the speed of \(P\) is at least \(2 \mathrm {~ms} ^ { - 1 }\) for \(S\) seconds.
3. Find the value of \(S\).

1. Two points \(A\) and \(B\) are 6 m apart on a smooth horizontal surface.

A light elastic string of natural length 2 m and modulus of elasticity 20 N , has one end attached to the point \(A\). A second light elastic string of natural length 2 m and modulus of elasticity 50 N , has one end attached to the point \(B\). A particle \(P\) of mass 3.5 kg is attached to the free end of each string.
The particle \(P\) is held at the point on \(A B\) which is 2 m from \(B\) and then released from rest.
In the subsequent motion both strings remain taut.

1. Show that \(P\) moves with simple harmonic motion about its equilibrium position.
2. Find the maximum speed of \(P\).
3. Find the length of time within each oscillation for which \(P\) is closer to \(A\) than to \(B\).

6 One end of a light inextensible string is attached to a small mass. The other end is attached to a fixed point \(O\). Initially the mass hangs at rest vertically below \(O\). The mass is then pulled to one side with the string taut and released from rest. \(\theta\) is the angle, in radians, that the string makes with the vertical through \(O\) at time \(t\) seconds and \(\theta\) may be assumed to be small. The subsequent motion of the mass can be modelled by the differential equation $$\frac { \mathrm { d } ^ { 2 } \theta } { \mathrm {~d} t ^ { 2 } } = - 4 \theta$$

1. Write down the general solution to this differential equation.
2. Initially the pendulum is released from rest at an angle of \(\theta _ { 0 }\). Find the particular solution to the equation in this case.
3. State any limitations on the model.

6. A light elastic spring \(A B\) has natural length \(2 a\) and modulus of elasticity \(2 m n ^ { 2 } a\), where \(n\) is a constant. A particle \(P\) of mass \(m\) is attached to the end \(A\) of the spring. At time \(t = 0\), the spring, with \(P\) attached, lies at rest and unstretched on a smooth horizontal plane. The other end \(B\) of the spring is then pulled along the plane in the direction \(A B\) with constant acceleration \(f\). At time \(t\) the extension of the spring is \(x\).

1. Show that $$\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } + n ^ { 2 } x = f .$$
2. Find \(x\) in terms of \(n , f\) and \(t\). Hence find
3. the maximum extension of the spring,
4. the speed of \(P\) when the spring first reaches its maximum extension.
   1. \hspace{0pt} [In this question \(\mathbf { i }\) and \(\mathbf { j }\) are unit vectors due east and due north respectively]
   A man cycles at a constant speed \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\) on level ground and finds that when his velocity is \(u \mathbf { j } \mathrm {~m} \mathrm {~s} ^ { - 1 }\) the velocity of the wind appears to be \(v ( 3 \mathbf { i } - 4 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\), where \(v\) is a positive constant. When the man cycles with velocity \(\frac { 1 } { 5 } u ( - 3 \mathbf { i } + 4 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\), the velocity of the wind appears to be \(w \mathbf { i } \mathrm {~m} \mathrm {~s} ^ { - 1 }\), where \(w\) is a positive constant. Find, in terms of \(u\), the true velocity of the wind.

13 Two light elastic strings each have one end attached to a particle \(B\) of mass \(3 c \mathrm {~kg}\), which rests on a smooth horizontal table. The other ends of the strings are attached to the fixed points \(A\) and \(C\), which are 8 metres apart. \(A B C\) is a horizontal line. \includegraphics[max width=\textwidth, alt={}, center]{44e22a98-6424-4fb1-8a37-c965773cb7b6-26_92_910_635_566} String \(A B\) has a natural length of 4 metres and a stiffness of \(5 c\) newtons per metre.
String \(B C\) has a natural length of 1 metre and a stiffness of \(c\) newtons per metre.
The particle is pulled a distance of \(\frac { 1 } { 3 }\) metre from its equilibrium position towards \(A\), and released from rest. 13

1. Show that the particle moves with simple harmonic motion.
   13
2. Find the speed of the particle when it is at a point \(P\), a distance \(\frac { 1 } { 4 }\) metre from the equilibrium position. Give your answer to two significant figures.
   [0pt] [4 marks]

1. The diagram below shows part of a game at a funfair that consists of a target moving along a straight horizontal line \(A B\). The centre of the target may be modelled as a particle moving with Simple Harmonic Motion about centre \(O\), where \(O\) is the midpoint of \(A B\). \includegraphics[max width=\textwidth, alt={}, center]{36112cfa-20c4-4ba8-b972-6b7b44e5182f-14_245_1145_525_452}

When the target is at a distance of 84 cm from \(O\), its speed is \(52 \mathrm { cms } ^ { - 1 }\) and the magnitude of its acceleration is \(1344 \mathrm { cms } ^ { - 2 }\).

1. Show that the period of the motion is \(\frac { \pi } { 2 } \mathrm {~s}\).
   

   (b) Determine the maximum speed of the target.

   Examiner

2. During a game, players fire a ball at the target. A timer is started when the target is at \(A\). Players must wait for the target to complete at least one full cycle before firing. Given that the target is hit when it is at a distance of 67 cm from \(O\), calculate the two earliest possible times taken to hit the target.
   \section*{PLEASE DO NOT WRITE ON THIS PAGE}

5 A particle of mass \(m\) is attached by a light elastic string of natural length \(l\) and modulus of elasticity \(\lambda\) to a fixed point \(A\), from which it is allowed to fall freely. The particle first comes to rest, instantaneously, at \(B\), where \(A B = 2 l\). Prove that

1. \(\lambda = 4 m g\),
2. while the string is taut, \(\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } = - \frac { 4 g } { l } x\), where \(x\) is the displacement from the equilibrium position at time \(t\),
3. the time taken between the first occasion when the string becomes taut and the next occasion when it becomes slack is $$\left[ \frac { 1 } { 2 } \pi + \sin ^ { - 1 } \left( \frac { 1 } { 3 } \right) \right] \sqrt { \frac { l } { g } }$$

8 A particle, \(P\), is moving in a straight line with simple harmonic motion about a centre \(O\). When \(P\) is at the point \(A , 2 \mathrm {~m}\) from \(O\), it has speed \(4 \sqrt { 5 } \mathrm {~m} \mathrm {~s} ^ { - 1 }\). When \(P\) is at the point \(B , \sqrt { 5 } \mathrm {~m}\) from \(O\), it has speed \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Find the amplitude and period of the motion.
2. Given that \(A\) and \(B\) are on opposite sides of \(O\), find the time taken for \(P\) to travel directly from \(A\) to \(B\).

9 A light string, of natural length 0.5 m and modulus of elasticity 4 N , has one end attached to the ceiling of a room. A particle of mass 0.2 kg is attached to the free end of the string and hangs in equilibrium.

1. Find the extension of the string when the particle is in the equilibrium position. The particle is pulled down a further 0.5 m from the equilibrium position and released from rest. At time \(t\) seconds the displacement of the particle from the equilibrium position is \(x \mathrm {~m}\).
2. Show that, while the string is taut, the equation of motion is \(\ddot { x } = - 40 x\).
3. Find the time taken for the string to become slack for the first time.
4. Show that the particle comes to instantaneous rest 0.125 m below the ceiling.

8 \includegraphics[max width=\textwidth, alt={}, center]{86cc07e7-ea69-4480-96c8-82b818445199-4_182_803_264_671} A light spring of modulus of elasticity 8 N and natural length 0.4 m has one end fixed to a smooth horizontal table at a fixed point \(L\). A particle of mass 0.2 kg is attached to the other end of the spring and pulled out horizontally to a point \(M\) on the table, so that the spring is extended by 0.2 m . The particle is then released from rest. The mid-point of \(L M\) is \(N\) and the point \(O\) is on \(L M\) such that \(L O = 0.4 \mathrm {~m}\) (see diagram).

1. Show that the particle moves in simple harmonic motion with centre \(O\) and state the exact period of its motion.
2. Find the exact time taken for the particle to move directly from \(M\) to \(N\).

14 One end of a light elastic string of natural length 0.5 m and modulus of elasticity 3 N is attached to a ceiling at a point \(P\). A particle of mass 0.3 kg is attached to the other end of the string.

1. Find the extension of the string when the particle hangs vertically in equilibrium. The particle is released from rest at \(P\) so that it falls vertically. Find
2. the maximum extension of the string,
3. the equation of motion for the particle when the string is stretched, in terms of the displacement \(x \mathrm {~m}\) below the equilibrium position,
4. the time between the string first becoming stretched and next becoming unstretched again.

13 Two light strings, each of natural length \(l\) and modulus of elasticity \(6 m g\), are attached at their ends to a particle \(P\) of mass \(m\). The other ends of the strings are attached to two fixed points \(A\) and \(B\), which are at a distance \(6 l\) apart on a smooth horizontal table. Initially \(P\) is at rest at the mid-point of \(A B\). The particle is now given a horizontal impulse in the direction perpendicular to \(A B\). At time \(t\) the displacement of \(P\) from the line \(A B\) is \(x\).

1. Show that the tension in each string is \(\frac { 6 m g } { l } \left( \sqrt { 9 l ^ { 2 } + x ^ { 2 } } - l \right)\).
2. Show that $$\ddot { x } = - \frac { 12 g x } { l } \left( 1 - \frac { l } { \sqrt { 9 l ^ { 2 } + x ^ { 2 } } } \right) .$$
3. Given that throughout the motion \(\frac { x ^ { 2 } } { l ^ { 2 } }\) is small enough to be negligible, show that the equation of motion is approximately $$\ddot { x } = - \frac { 8 g x } { l } .$$
4. Given that the initial speed of \(P\) is \(\sqrt { \frac { g l } { 200 } }\), find the time taken for the particle to travel a distance of \(\frac { 1 } { 80 } l\).

14 \includegraphics[max width=\textwidth, alt={}, center]{22640c3b-792f-4003-a4f8-78220efd73b0-5_86_1589_1297_278} A particle of mass 0.05 kg is attached to two identical light elastic strings, each of natural length 1.2 m and modulus of elasticity 0.6 N . The other ends of the strings are attached to points \(A\) and \(E\) on a smooth horizontal table. The distance \(A E\) is 2 m and points \(B , C\) and \(D\) lie between \(A\) and \(E\) so that \(A B = 0.7 \mathrm {~m} , B C = 0.1 \mathrm {~m} , C D = 0.4 \mathrm {~m}\) and \(D E = 0.8 \mathrm {~m}\) (see diagram). Initially the particle is held at \(B\) and it is then released. In the subsequent motion the displacement of the particle from \(C\), in the direction of \(A\), is denoted by \(x \mathrm {~m}\).

1. Find the equation of motion for the particle when it is between \(B\) and \(C\).
2. Find the velocity of the particle when it is at \(C\).
3. Find the total time that elapses before the particle first returns to \(B\).

13 Two light strings, each of natural length \(l\) and modulus of elasticity \(6 m g\), are attached at their ends to a particle \(P\) of mass \(m\). The other ends of the strings are attached to two fixed points \(A\) and \(B\), which are at a distance \(6 l\) apart on a smooth horizontal table. Initially \(P\) is at rest at the mid-point of \(A B\). The particle is now given a horizontal impulse in the direction perpendicular to \(A B\). At time \(t\) the displacement of \(P\) from the line \(A B\) is \(x\).

1. Show that the tension in each string is \(\frac { 6 m g } { l } \left( \sqrt { 9 l ^ { 2 } + x ^ { 2 } } - l \right)\).
2. Show that $$\ddot { x } = - \frac { 12 g x } { l } \left( 1 - \frac { l } { \sqrt { 9 l ^ { 2 } + x ^ { 2 } } } \right)$$
3. Given that throughout the motion \(\frac { x ^ { 2 } } { l ^ { 2 } }\) is small enough to be negligible, show that the equation of motion is approximately $$\ddot { x } = - \frac { 8 g x } { l } .$$
4. Given that the initial speed of \(P\) is \(\sqrt { \frac { g l } { 200 } }\), find the time taken for the particle to travel a distance of \(\frac { 1 } { 80 } l\).

13 Two light strings, each of natural length \(l\) and modulus of elasticity \(6 m g\), are attached at their ends to a particle \(P\) of mass \(m\). The other ends of the strings are attached to two fixed points \(A\) and \(B\), which are at a distance \(6 l\) apart on a smooth horizontal table. Initially \(P\) is at rest at the mid-point of \(A B\). The particle is now given a horizontal impulse in the direction perpendicular to \(A B\). At time \(t\) the displacement of \(P\) from the line \(A B\) is \(x\).

1. Show that the tension in each string is \(\frac { 6 m g } { l } \left( \sqrt { 9 l ^ { 2 } + x ^ { 2 } } - l \right)\).
2. Show that $$\ddot { x } = - \frac { 12 g x } { l } \left( 1 - \frac { 1 } { \sqrt { 9 l ^ { 2 } + x ^ { 2 } } } \right)$$
3. Given that throughout the motion \(\frac { x ^ { 2 } } { l ^ { 2 } }\) is small enough to be negligible, show that the equation of motion is approximately $$\ddot { x } = - \frac { 8 g x } { l } .$$
4. Given that the initial speed of \(P\) is \(\sqrt { \frac { g l } { 200 } }\), find the time taken for the particle to travel a distance of \(\frac { 1 } { 80 } l\).

6 A simple pendulum consists of a light inextensible string of length 1.5 m with a small bob of mass 0.2 kg at one end. When suspended from a fixed point and hanging at rest under gravity, the bob is given a horizontal speed of \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and it comes instantaneously to rest when the string makes an angle of 0.1 rad with the vertical. At time \(t\) seconds after projection the string makes an angle \(\theta\) with the vertical.

1. Show that, neglecting air resistance, $$\left( \frac { \mathrm { d } \theta } { \mathrm {~d} t } \right) ^ { 2 } = \frac { 40 } { 3 } \{ \cos \theta - \cos ( 0.1 ) \}$$
2. Find, correct to 2 significant figures,
   1. the value of \(u\),
   2. the tension in the string when \(\theta = 0.05 \mathrm { rad }\).
   3. By differentiating the above equation for \(\left( \frac { \mathrm { d } \theta } { \mathrm { d } t } \right) ^ { 2 }\), or otherwise, show that the motion of the bob can be modelled approximately by simple harmonic motion.
   4. Hence find the value of \(t\) at which the bob first comes instantaneously to rest.