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

6. (a) Determine the general solution of the differential equation $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 2 \frac { \mathrm {~d} y } { \mathrm {~d} x } + 5 y = 6 \cos x$$ (b) Find the particular solution for which \(y = 0\) and \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 0\) at \(x = 0\)

9. Resonance in an electrical circuit is modelled by the differential equation $$\frac { \mathrm { d } ^ { 2 } V } { \mathrm {~d} t ^ { 2 } } + 64 V = \cos 8 t$$ where \(V\) represents the voltage in the circuit and \(t\) represents time.

1. Find the value of \(\lambda\) for which \(\lambda\) tsin8t is a particular integral of the differential equation.
2. Find the general solution of the differential equation. Given that \(V = 0\) and \(\frac { \mathrm { d } V } { \mathrm {~d} t } = 0\) when \(t = 0\),
3. find the particular solution of the equation.
4. Describe the behaviour of \(V\) as \(t\) becomes large, according to this model.

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 \(\lambda\). The other end of the string is attached to a fixed point \(A\) on a smooth plane which is inclined at \(30 ^ { \circ }\) to the horizontal. The string lies along a line of greatest slope of the plane. The particle rests in equilibrium at the point \(B\), where \(B\) is lower than \(A\) and \(A B = \frac { 6 } { 5 } a\).

1. Show that \(\lambda = \frac { 5 } { 2 } m g\). The particle is now pulled down a line of greatest slope to the point \(C\), where \(B C = \frac { 1 } { 5 } a\), and released from rest.
2. Show that \(P\) moves with simple harmonic motion of period \(2 \pi \sqrt { \frac { 2 a } { 5 g } }\)
3. Find, in terms of \(g\), the greatest magnitude of the acceleration of \(P\) while the string is taut. The midpoint of \(B C\) is \(D\) and the string becomes slack for the first time at the point \(E\).
4. Find, in terms of \(a\) and \(g\), the time taken by \(P\) to travel directly from \(D\) to \(E\).

5. \begin{figure}[h] \end{figure} A particle \(P\) of mass \(m\) is attached to one end of a light elastic string, of natural length \(l\) and modulus of elasticity \(\lambda\). The other end of the string is attached to a fixed point \(A\) on a smooth plane inclined at angle \(\alpha\) to the horizontal, where \(\sin \alpha = \frac { 3 } { 5 }\). The particle rests in equilibrium on the plane at the point \(B\) with the string lying along a line of greatest slope of the plane, as shown in Figure 2. Given that \(A B = \frac { 6 } { 5 } l\)

1. show that \(\lambda = 3 \mathrm { mg }\) The particle is pulled down the line of greatest slope to the point \(C\), where \(B C = \frac { 1 } { 2 } l\), and released from rest.
2. Show that, while the string remains taut, \(P\) moves with simple harmonic motion about centre \(B\).
3. Find the greatest magnitude of the acceleration of \(P\) while the string remains taut. The point \(D\) is the midpoint of \(B C\). The time taken by \(P\) to move directly from \(D\) to the point where the string becomes slack for the first time is \(k \sqrt { \frac { l } { g } }\), where \(k\) is a constant.
4. Find, to 2 significant figures, the value of \(k\).

6. One end of a light elastic string, of natural length 5 l and modulus of elasticity 20 mg , is attached to a fixed point \(A\). A particle \(P\) of mass \(2 m\) is attached to the free end of the string and \(P\) hangs freely in equilibrium at the point \(B\).

1. Find the distance \(A B\).
   (3) The particle is now pulled vertically downwards from \(B\) to the point \(C\) and released from rest. In the subsequent motion the string does not become slack.
2. Show that \(P\) moves with simple harmonic motion with centre \(B\).
3. Find the period of this motion. The greatest speed of \(P\) during this motion is \(\frac { 1 } { 5 } \sqrt { g l }\)
4. Find the amplitude of this motion. The point \(D\) is the midpoint of \(B C\) and the point \(E\) is the highest point reached by \(P\).
5. Find the time taken by \(P\) to move directly from \(D\) to \(E\).

7. \begin{figure}[h] \end{figure} The fixed points \(A\) and \(B\) are 4.2 m apart on a smooth horizontal floor. One end of a light elastic spring, of natural length 1.8 m and modulus of elasticity 20 N , is attached to a particle \(P\) and the other end is attached to \(A\). One end of another light elastic spring, of natural length 0.9 m and modulus of elasticity 15 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 5.

1. Show that \(A O = 2.7 \mathrm {~m}\). The particle \(P\) now receives an impulse acting in the direction \(O B\) and moves away from \(O\) towards \(B\). In the subsequent motion \(P\) does not reach \(B\).
2. Show that \(P\) moves with simple harmonic motion about centre \(O\). The mass of \(P\) is 10 kg and the magnitude of the impulse is \(J \mathrm { Ns }\). Given that \(P\) first comes to instantaneous rest at the point \(C\) where \(A C = 2.9 \mathrm {~m}\),
3. 1. find the value of \(J\),
   2. find the time taken by \(P\) to travel a total distance of 0.5 m from when it first leaves \(O\).

3. A particle \(P\) is moving in a straight line with simple harmonic motion between two points \(A\) and \(B\), where \(A B\) is \(2 a\) metres. The point \(C\) lies on the line \(A B\) and \(A C = \frac { 1 } { 2 } a\) metres. The particle passes through \(C\) with speed \(\frac { 3 a \sqrt { 3 } } { 2 } \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Find the period of the motion. The maximum magnitude of the acceleration of \(P\) is \(45 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). Find
2. the value of \(a\),
3. the maximum speed of \(P\). The point \(D\) lies on \(A B\) and \(P\) takes a quarter of one period to travel directly from \(C\) to \(D\).
4. Find the distance CD.

5. \begin{figure}[h] \end{figure} The fixed points, \(A\) and \(B\), are a distance \(10 a\) apart, with \(B\) vertically above \(A\). One end of a light elastic string, of natural length \(2 a\) and modulus of elasticity \(2 m g\), is attached to a particle \(P\) of mass \(m\) and the other end is attached to \(A\). One end of another light elastic string, of natural length \(4 a\) and modulus of elasticity \(6 m g\), is attached to \(P\) and the other end is attached to \(B\). The particle \(P\) rests in equilibrium at the point \(C\), as shown in Figure 6.

1. Show that each string has an extension of \(2 a\).
   (5) The particle \(P\) is now pulled down vertically, so that it is a distance \(a\) below \(C\) and then released from rest.
2. Show that in the subsequent motion, \(P\) performs simple harmonic motion.
3. Find, in terms of \(a\) and \(g\), the speed of \(P\) when it is a distance \(\frac { 7 } { 2 } a\) above \(A\).

1. A particle \(P\) of mass \(m\) is attached to one end of a light elastic spring of natural length 2l. The other end of the spring is attached to a fixed point \(A\). The particle \(P\) hangs in equilibrium vertically below \(A\), at the point \(E\) where \(A E = 6 l\). The particle \(P\) is then raised a vertical distance \(2 l\) and released from rest.

Air resistance is modelled as being negligible.

1. Show that \(P\) moves with simple harmonic motion of period \(T\) where $$T = 4 \pi \sqrt { \frac { l } { g } }$$
2. Find, in terms of \(m , l\) and \(g\), the kinetic energy of \(P\) as it passes through \(E\)
3. Find, in terms of \(T\), the exact time from the instant when \(P\) is released to the instant when \(P\) has moved a distance 31 .

7. \begin{figure}[h] \end{figure} Two points \(A\) and \(B\) lie on a smooth horizontal table where \(A B = 41\).
A particle \(P\) of mass \(m\) is attached to one end of a light elastic spring of natural length I and modulus of elasticity 2 mg . The other end of the spring is attached to A . The particle P is also attached to one end of another light elastic spring of natural length I and modulus of elasticity mg . The other end of the spring is attached to B.
The particle \(P\) rests in equilibrium on the table at the point 0 , where \(A 0 = \frac { 5 } { 3 } I\), as shown in Figure 7.
The particle \(P\) is moved a distance \(\frac { 1 } { 2 } \mathrm { I }\) along the table, from 0 towards \(A\), and released from rest.

1. Show that P moves with simple harmonic motion of period T , where $$\mathrm { T } = 2 \pi \sqrt { \frac { l } { 3 g } }$$
2. Find, in terms of I and g , the speed of P as it passes through 0 .
3. Find, in terms of g , the maximum acceleration of P .
4. Find the exact time, in terms of I and g , from the instant when P is released from rest to the instant when P is first moving with speed \(\frac { 3 } { 4 } \sqrt { g l }\) \includegraphics[max width=\textwidth, alt={}, center]{631b78c4-2763-4a1e-9d30-2f301fe3af2e-20_2269_56_311_1978} \(\_\_\_\_\) VIAV SIHI NI JIIHM ION OC
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1. The fixed point \(A\) is vertically above the fixed point \(B\), with \(A B = 3 l\)

A light elastic string has natural length \(l\) and modulus of elasticity \(4 m g\) One end of the string is attached to \(A\) and the other end is attached to a particle \(P\) of mass \(m\) A second light elastic string also has natural length \(l\) and modulus of elasticity \(4 m g\) One end of this string is attached to \(P\) and the other end is attached to \(B\). Initially \(P\) rests in equilibrium at the point \(E\), where \(A E B\) is a vertical straight line.

1. Show that \(A E = \frac { 13 } { 8 } l\) The particle \(P\) is now held at the point that is a distance \(2 l\) vertically below \(A\) and released from rest. At time \(t\), the vertical displacement of \(P\) from \(E\) is \(x\), where \(x\) is measured vertically downwards.
2. Show that \(\ddot { x } = - \frac { 8 g } { l } x\)
3. Find, in terms of \(g\) and \(l\), the speed of \(P\) when it is \(\frac { 1 } { 8 } l\) below \(E\).
4. Find the length of time, in each complete oscillation, for which \(P\) is more than \(1.5 l\) from \(A\), giving your answer in terms of \(g\) and \(l\)

1. A particle \(P\) moves in a straight line with simple harmonic motion. The period of the motion is \(\frac { \pi } { 4 }\) seconds. At time \(t = 0 , P\) is at rest at the point \(A\) and the acceleration of \(P\) has magnitude \(20 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).

Find

1. the amplitude of the motion,
2. the greatest speed of \(P\) during the motion,
3. the time \(P\) takes to travel a total distance of 1.5 m after it has first left \(A\).

6. A particle of mass \(m\) is attached to one end of a light elastic string, of natural length \(6 a\) and modulus of elasticity 9 mg . The other end of the string is attached to a fixed point \(A\) on a ceiling. The particle hangs in equilibrium at the point \(B\), where \(B\) is vertically below \(A\) and \(A B = ( 6 + p ) a\).

1. Show that \(p = \frac { 2 } { 3 }\) The particle is now released from rest at a point \(C\) vertically below \(B\), where \(A C < \frac { 22 } { 3 } a\).
2. Show that the particle moves with simple harmonic motion.
3. Find the period of this motion.
4. Explain briefly the significance of the condition \(A C < \frac { 22 } { 3 } a\). The point \(D\) is vertically below \(A\) and \(A D = 8 a\). The particle is now released from rest at \(D\). The particle first comes to instantaneous rest at the point \(E\).
5. Find, in terms of \(a\), the distance \(A E\).

5. \begin{figure}[h] \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 \(D B\).

3. A particle \(P\) of mass 0.5 kg moves in a straight line with simple harmonic motion, completing 4 oscillations per second. The particle comes to instantaneous rest at the fixed points \(A\) and \(B\), where \(A B = 0.5 \mathrm {~m}\).

1. Find the maximum magnitude of the acceleration of \(P\). When \(P\) is moving at its maximum speed it receives an impulse. The direction of this impulse is opposite to the direction in which \(P\) is moving when it receives the impulse. The impulse causes \(P\) to reverse its direction of motion but \(P\) continues to move with simple harmonic motion. The centre and period of this new simple harmonic motion are the same as the centre and period of the original simple harmonic motion. The amplitude is now half the original amplitude.
2. Find the magnitude of the impulse.
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2. A particle \(P\) is moving in a straight line with simple harmonic motion about the fixed point \(O\) as centre. When \(P\) is a distance 0.02 m from \(O\), the speed of \(P\) is \(0.3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the magnitude of the acceleration of \(P\) is \(0.5 \mathrm {~m} \mathrm {~s} ^ { - 2 }\)

1. Find the period of the motion. The amplitude of the motion is \(a\) metres. Find
2. the value of \(a\),
3. the total length of time during each complete oscillation for which \(P\) is within \(\frac { 1 } { 2 } a\) metres of \(O\). metres of \(O\).

6. A light elastic string has natural length \(a\) and modulus of elasticity \(\frac { 3 } { 4 } \mathrm { mg }\). A particle \(P\) of mass \(m\) 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 at the point \(O\), vertically below \(A\).

1. Find the distance \(O A\). The particle \(P\) is now pulled vertically down to a point \(B\), where \(A B = 3 a\), and released from rest.
2. Show that, throughout the subsequent motion, \(P\) performs only simple harmonic motion, justifying your answer. The point \(C\) is vertically below \(A\), where \(A C = 2 a\).
   Find, in terms of \(a\) and \(g\),
3. the speed of \(P\) at the instant that it passes through \(C\),
4. the time taken for \(P\) to move directly from \(B\) to \(C\). \includegraphics[max width=\textwidth, alt={}, center]{ace84823-db30-463e-b24b-f0cd7df73746-17_2255_50_314_34}
   

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4. \begin{figure}[h] \end{figure} A circus performer has mass \(m\). She is attached to one end of a cable of length \(l\). The other end of the cable is attached to a fixed point \(O\) Initially she is held at rest at point \(A\) with the cable taut and at an angle of \(30 ^ { \circ }\) below the horizontal, as shown in Figure 3. The circus performer is released from \(A\) and she moves on a vertical circular path with centre \(O\) The circus performer is modelled as a particle and the cable is modelled as light and inextensible.

1. Find, in terms of \(m\) and \(g\), the tension in the cable at the instant immediately after the circus performer is released.
2. Show that, during the motion following her release, the greatest tension in the cable is 4 times the least tension in the cable.

7. \begin{figure}[h] \end{figure} The fixed points \(A\) and \(B\) are 7 m apart on a smooth horizontal surface.
A light elastic string has natural length 2 m and modulus of elasticity 4 N . One end of the string is attached to a particle \(P\) of mass 2 kg and the other end is attached to \(A\) Another light elastic string has natural length 3 m and modulus of elasticity 2 N . One end of this string 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 4.

1. Show that \(O A = 2.5 \mathrm {~m}\). The particle \(P\) now receives an impulse of magnitude 6Ns in the direction \(O B\)
2. 1. Show that \(P\) initially moves with simple harmonic motion with centre \(O\)
   2. Determine the amplitude of this simple harmonic motion. The point \(C\) lies on \(O B\). As \(P\) passes through \(C\) the string attached to \(B\) becomes slack.
3. Find the speed of \(P\) as it passes through \(C\)
4. Find the time taken for \(P\) to travel directly from \(O\) to \(C\)

1. A particle \(P\) moves in a straight line with simple harmonic motion between two fixed points \(A\) and \(B\). The particle performs 2 complete oscillations per second. The midpoint of \(A B\) is \(O\) and the midpoint of \(O A\) is \(C\)

The length of \(A B\) is 0.6 m .

1. Find the maximum speed of \(P\)
2. Find the time taken by \(P\) to move directly from \(O\) to \(C\)

7. \begin{figure}[h] \end{figure} Figure 5 shows two fixed points, \(A\) and \(B\), which are 5 m apart on a smooth horizontal floor. A particle \(P\) of mass 1.25 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 \(\lambda\) newtons, has one end attached to \(P\) and the other end attached to \(B\) Initially \(P\) rests in equilibrium at the point \(O\), where \(A O = 3 \mathrm {~m}\)

1. Show that \(\lambda = 15\) The particle is now projected along the floor towards \(B\) At time \(t\) seconds, \(P\) is a displacement \(x\) metres from \(O\) in the direction \(O B\)
2. Show that, while both strings are taut, \(P\) moves with simple harmonic motion where \(\ddot { x } = - 18 x\) The initial speed of \(P\) is \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\)
3. Find the speed of \(P\) at the instant when the string \(P B\) becomes slack. Both strings are taut for \(T\) seconds during one complete oscillation.
4. Find the value of \(T\)

1. A particle \(P\) of mass \(m\) is attached to one end of a light elastic string of natural length \(l\). The other end of the string is attached to a fixed point on a ceiling. The particle \(P\) hangs in equilibrium at a distance \(D\) below the ceiling.

The particle \(P\) is now pulled vertically downwards until it is a distance \(3 l\) below the ceiling and released from rest. Given that \(P\) comes to instantaneous rest just before it reaches the ceiling,

1. show that \(D = \frac { 5 l } { 3 }\)
2. Show that, while the elastic string is stretched, \(P\) moves with simple harmonic motion, with period \(2 \pi \sqrt { \frac { 2 l } { 3 g } }\)
3. Find, in terms of \(g\) and \(l\), the exact time from the instant when \(P\) is released to the instant when the elastic string first goes slack.

1. In a harbour, the water level rises and falls with the tides with simple harmonic motion.

On a particular day, the depths of water in the harbour at low and high tide are 4 m and 10 m respectively. Low tide occurs at 12:00 and high tide occurs at 18:20

1. Find, in \(\mathrm { mh } ^ { - 1 }\), the speed at which the water level is rising on this particular day at 13:35 A ship can only safely enter the harbour when the depth of water is at least 8.5 m .
2. Find the earliest time after 12:00 on this particular day at which it is safe for the ship to enter the harbour, giving your answer to the nearest minute.