Prove SHM and find period

7. \begin{figure}[h] \end{figure} A smooth hollow narrow tube of length \(l\) has one open end and one closed end. The tube is fixed in a vertical position with the closed end at the bottom. A light elastic spring has natural length \(l\) and modulus of elasticity \(8 m g\). The spring is inside the tube and has one end attached to a fixed point \(O\) on the closed end of the tube. The other end of the spring is attached to a particle \(P\) of mass \(m\). The particle rests in equilibrium at a distance \(e\) below the top of the tube, as shown in Figure 4.

1. Find \(e\) in terms of \(l\). The particle \(P\) is now held inside the tube at a distance \(\frac { 1 } { 2 } l\) below the top of the tube and released from rest at time \(t = 0\)
2. Prove that \(P\) moves with simple harmonic motion of period \(2 \pi \sqrt { \left( \frac { l } { 8 g } \right) }\). The particle \(P\) passes through the open top of the tube with speed \(u\).
3. Find \(u\) in terms of \(g\) and \(l\).
4. Find the time taken for \(P\) to first attain a speed of \(\sqrt { \left( \frac { 9 g l } { 32 } \right) }\).

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 \(k\).

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 .

3. \begin{figure}[h] \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 \(E\).

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)

6. A light elastic string, of natural length \(4 a\) and modulus of elasticity \(8 m g\), 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 \(A O\).
   (2) The particle is now pulled down to a point \(C\) vertically below \(O\), where \(O C = d\). It is released from rest. In the subsequent motion the string does not become slack.
2. Show that \(P\) moves with simple harmonic motion of period \(\pi \sqrt { \left( \frac { 2 a } { g } \right) }\). The greatest speed of \(P\) during this motion is \(\frac { 1 } { 2 } \sqrt { } ( g a )\).
3. 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,
4. describe briefly the subsequent motion of \(P\).
   (2)

5. A particle \(P\) of mass 0.8 kg is attached to one end \(A\) of a light elastic spring \(O A\), 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 \(O B = 85 \mathrm {~cm}\), and is released from rest.

1. Prove that the motion of \(P\) is simple harmonic with period \(\frac { 2 \pi } { 5 }\) s.
2. Find the greatest magnitude of the acceleration of \(P\) during the motion. Two seconds after being released from rest, \(P\) passes through the point \(C\).
3. Find, to 2 significant figures, the speed of \(P\) as it passes through \(C\).
4. State the direction in which \(P\) is moving 2 s after being released.

4. \begin{figure}[h] \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.

Two particles \(A\) and \(B\), of equal mass \(m\), are connected by a light elastic string of natural length \(a\) and modulus of elasticity \(4 m g\). Particle \(A\) rests on a rough horizontal table at a distance \(a\) from the edge of the table. The string passes over a small smooth pulley \(P\) fixed at the edge of the table. At time \(t = 0 , B\) is released from rest at \(P\) and falls vertically. At time \(t , B\) has fallen a distance \(x\), without \(A\) slipping (see diagram). Show that $$\ddot { x } = - \frac { g } { a } ( 4 x - a ) .$$ Deduce that, while \(A\) does not slip, \(B\) moves in simple harmonic motion and identify the centre of the motion. Given that the coefficient of friction between \(A\) and the table is \(\frac { 1 } { 3 }\), find the value of \(x\) when \(A\) starts to slip, and the corresponding value of \(t\), expressing this answer in the form \(k \sqrt { } \left( \frac { a } { g } \right)\). Give the value of \(k\) correct to 3 decimal places.

3
\includegraphics[max width=\textwidth, alt={}, center]{f6887893-66c5-40df-ba8d-9439a5c268eb-3_351_314_255_918} A spring balance is modelled by a vertical light elastic spring \(A B\), of natural length 0.25 m and modulus of elasticity \(\lambda \mathrm { N }\). The bottom end \(B\) of the spring is fixed, and the top end \(A\) is attached to a small tray of mass 0.1 kg which is free to move vertically (see diagram). When in the equilibrium position, \(A B = 0.24 \mathrm {~m}\). Show that \(\lambda = 25\). The tray is pushed down by 0.02 m to the point \(C\) and released from rest. At time \(t\) seconds after release the displacement of the tray from the equilibrium position is \(x \mathrm {~m}\). Show that $$\ddot { x } = - 1000 x .$$ Find the time taken for the tray to move a distance of 0.03 m from \(C\).

3
\includegraphics[max width=\textwidth, alt={}, center]{b486decd-75b8-44bd-889f-2472f1163871-2_570_419_1539_863} A uniform disc, of mass 2 kg and radius 0.2 m , is free to rotate in a vertical plane about a smooth horizontal axis through its centre. One end of a light inextensible string is attached to a point on the rim of the disc and the string is wound round the rim. The other end of the string is attached to a small block of mass 4 kg , which hangs freely (see diagram). The system is released from rest. During the subsequent motion, the block experiences a constant resistance to its motion, of magnitude \(R \mathrm {~N}\). Given that the angular speed of the disc after it has turned through 2 radians is \(5 \mathrm { rad } \mathrm { s } ^ { - 1 }\), find \(R\) and the tension in the string.
[0pt] [9]

4
\includegraphics[max width=\textwidth, alt={}, center]{9b520e69-a14e-47e5-97d7-998f5145844b-06_465_663_262_742} A small ring \(P\) of weight \(W\) is free to slide on a rough horizontal wire, one end of which is attached to a vertical wall at \(Q\). The end \(A\) of a thin uniform \(\operatorname { rod } A B\) of length \(2 a\) and weight \(\frac { 5 } { 2 } W\) is freely hinged to the wall at the point \(A\) which is a distance \(a\) vertically below \(Q\). A light elastic string of natural length \(2 a\) has one end attached to the ring \(P\) and the other end attached to the rod at \(B\). The string is at right angles to the rod and \(A , B , P\) and \(Q\) lie in a vertical plane. The system is in limiting equilibrium with \(A B\) making an angle \(\theta\) with the horizontal, where \(\sin \theta = \frac { 3 } { 5 }\) (see diagram).

1. Find the tension in the string in terms of \(W\).
2. Find the coefficient of friction between the ring and the wire.
3. Find the magnitude of the resultant force on the rod at the hinge in terms of \(W\).
4. Find the modulus of elasticity of the string in terms of \(W\).
   \includegraphics[max width=\textwidth, alt={}, center]{9b520e69-a14e-47e5-97d7-998f5145844b-08_862_698_260_721} A uniform picture frame of mass \(m\) is made by removing a rectangular lamina \(E F G H\) in which \(E F = 4 a\) and \(F G = 2 a\) from a larger rectangular lamina \(A B C D\) in which \(A B = 6 a\) and \(B C = 4 a\). The side \(E F\) is parallel to the side \(A B\). The point of intersection of the diagonals \(A C\) and \(B D\) coincides with the point of intersection of the diagonals \(E G\) and \(F H\). One end of a light inextensible string of length \(10 a\) is attached to \(A\) and the other end is attached to \(B\). The frame is suspended from the mid-point \(O\) of the string. A small object of mass \(\frac { 11 } { 12 } m\) is fixed to the mid-point of \(A B\) (see diagram).

4
\includegraphics[max width=\textwidth, alt={}, center]{1651d08b-b20f-4f2e-9f47-0a1a5d0a839a-06_465_663_262_742} A small ring \(P\) of weight \(W\) is free to slide on a rough horizontal wire, one end of which is attached to a vertical wall at \(Q\). The end \(A\) of a thin uniform \(\operatorname { rod } A B\) of length \(2 a\) and weight \(\frac { 5 } { 2 } W\) is freely hinged to the wall at the point \(A\) which is a distance \(a\) vertically below \(Q\). A light elastic string of natural length \(2 a\) has one end attached to the ring \(P\) and the other end attached to the rod at \(B\). The string is at right angles to the rod and \(A , B , P\) and \(Q\) lie in a vertical plane. The system is in limiting equilibrium with \(A B\) making an angle \(\theta\) with the horizontal, where \(\sin \theta = \frac { 3 } { 5 }\) (see diagram).

1. Find the tension in the string in terms of \(W\).
2. Find the coefficient of friction between the ring and the wire.
3. Find the magnitude of the resultant force on the rod at the hinge in terms of \(W\).
4. Find the modulus of elasticity of the string in terms of \(W\).
   \includegraphics[max width=\textwidth, alt={}, center]{1651d08b-b20f-4f2e-9f47-0a1a5d0a839a-08_862_698_260_721} A uniform picture frame of mass \(m\) is made by removing a rectangular lamina \(E F G H\) in which \(E F = 4 a\) and \(F G = 2 a\) from a larger rectangular lamina \(A B C D\) in which \(A B = 6 a\) and \(B C = 4 a\). The side \(E F\) is parallel to the side \(A B\). The point of intersection of the diagonals \(A C\) and \(B D\) coincides with the point of intersection of the diagonals \(E G\) and \(F H\). One end of a light inextensible string of length \(10 a\) is attached to \(A\) and the other end is attached to \(B\). The frame is suspended from the mid-point \(O\) of the string. A small object of mass \(\frac { 11 } { 12 } m\) is fixed to the mid-point of \(A B\) (see diagram).

4
\includegraphics[max width=\textwidth, alt={}, center]{2ab1a594-6c37-4c78-b53c-33c13bf6eb21-06_465_663_262_742} A small ring \(P\) of weight \(W\) is free to slide on a rough horizontal wire, one end of which is attached to a vertical wall at \(Q\). The end \(A\) of a thin uniform \(\operatorname { rod } A B\) of length \(2 a\) and weight \(\frac { 5 } { 2 } W\) is freely hinged to the wall at the point \(A\) which is a distance \(a\) vertically below \(Q\). A light elastic string of natural length \(2 a\) has one end attached to the ring \(P\) and the other end attached to the rod at \(B\). The string is at right angles to the rod and \(A , B , P\) and \(Q\) lie in a vertical plane. The system is in limiting equilibrium with \(A B\) making an angle \(\theta\) with the horizontal, where \(\sin \theta = \frac { 3 } { 5 }\) (see diagram).

1. Find the tension in the string in terms of \(W\).
2. Find the coefficient of friction between the ring and the wire.
3. Find the magnitude of the resultant force on the rod at the hinge in terms of \(W\).
4. Find the modulus of elasticity of the string in terms of \(W\).
   \includegraphics[max width=\textwidth, alt={}, center]{2ab1a594-6c37-4c78-b53c-33c13bf6eb21-08_862_698_260_721} A uniform picture frame of mass \(m\) is made by removing a rectangular lamina \(E F G H\) in which \(E F = 4 a\) and \(F G = 2 a\) from a larger rectangular lamina \(A B C D\) in which \(A B = 6 a\) and \(B C = 4 a\). The side \(E F\) is parallel to the side \(A B\). The point of intersection of the diagonals \(A C\) and \(B D\) coincides with the point of intersection of the diagonals \(E G\) and \(F H\). One end of a light inextensible string of length \(10 a\) is attached to \(A\) and the other end is attached to \(B\). The frame is suspended from the mid-point \(O\) of the string. A small object of mass \(\frac { 11 } { 12 } m\) is fixed to the mid-point of \(A B\) (see diagram).

One end of a light elastic spring, of natural length 0.8 m and modulus of elasticity 40 N , is attached to a fixed point \(O\). The spring hangs vertically, at rest, with particles of masses 2 kg and \(M \mathrm {~kg}\) attached to its free end. The \(M \mathrm {~kg}\) particle becomes detached from the spring, and as a result the 2 kg particle begins to move upwards.

1. Show that the 2 kg particle performs simple harmonic motion about its equilibrium position with period \(\frac { 2 } { 5 } \pi \mathrm {~s}\). State the distance below \(O\) of the centre of the oscillations.
   The speed of the 2 kg particle is \(0.4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when its displacement from the centre of oscillation is 0.06 m .
2. Find the amplitude of the motion.
3. Deduce the value of \(M\).

14 A particle, \(P\), of mass \(M\) is released from rest and moves along a horizontal straight line through a point \(O\). When \(P\) is at a displacement of \(x\) metres from \(O\), moving with a speed \(v \mathrm {~ms} ^ { - 1 }\), a force of magnitude \(| 8 M x |\) acts on the particle directed towards \(O\). A resistive force, of magnitude \(4 M v\), also acts on \(P\). 14

1. Initially \(P\) is held at rest at a displacement of 1 metre from \(O\). Describe completely the motion of \(P\) after it is released. Fully justify your answer.
   [0pt] [8 marks]
   14
2. It is decided to alter the resistive force so that the motion of \(P\) is critically damped. Determine the magnitude of the resistive force that will produce critically damped motion.
   [0pt] [4 marks]

6. 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.
   \(P\) is displaced to a point 0.5 m vertically below its equilibrium position and released from rest.
2. Show that the subsequent motion of \(P\) is simple harmonic, with period 1.68 s .
3. Calculate the maximum speed of \(P\) during its motion.
4. 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.

7. A particle of mass \(m \mathrm {~kg}\) is attached to one end of an elastic string of natural length \(l \mathrm {~m}\) and modulus of elasticity \(\lambda \mathrm { N }\). The other end of the string is attached to a fixed point \(O\). The particle hangs in equilibrium at a point \(C\).

1. 1. Prove that if the particle is slightly displaced in a vertical direction, it will perform simple harmonic motion about \(C\).
   2. Find the period, \(T \mathrm {~s}\), of the motion in terms of \(l , m\) and \(\lambda\).
   3. Explain the significance of the term 'slightly' as used in (i) above. 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 \(T _ { 1 } \mathrm {~s}\).
2. 1. Write down an expression for \(T _ { 1 }\) in terms of \(l , m , M\) and \(\lambda\).
   2. Hence show that \(T _ { 1 } ^ { 2 } - T ^ { 2 } = \frac { 4 \pi ^ { 2 } d } { g }\).

7. A particle \(P\) of mass \(m \mathrm {~kg}\) is fixed to one end of a light elastic string of modulus \(m g \mathrm {~N}\) and natural length \(l \mathrm {~m}\). 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 \(O X = \frac { 5 l } { 4 } \mathrm {~m}\).

1. Find the coefficient of friction between \(P\) and the table.
   \(P\) is now given a small displacement \(x \mathrm {~m}\) horizontally along \(O X\), away from \(O\). While \(P\) is in motion, the frictional resistance remains constant at its limiting value.
2. Show that as long as the string remains taut, \(P\) performs simple harmonic motion with \(X\) as the centre. If \(P\) is held at the point where the extension in the string is \(l m\) and then released,
3. show that the string becomes slack after a time \(\left( \frac { \pi } { 2 } + \arcsin \left( \frac { 1 } { 3 } \right) \right) \sqrt { \frac { l } { g } } \mathrm {~s}\).
4. Determine the speed of \(P\) when it reaches \(O\).

6. 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-
\includegraphics[max width=\textwidth, alt={}, center]{627b3411-07ba-4ee4-a672-93a64eeb90b3-2_200_318_923_1668}
point of the cross-bar, both springs extend by 30 cm to the position \(A\), as shown. Each spring has natural length \(l \mathrm {~m}\) and modulus of elasticity \(\lambda \mathrm { N }\).

1. Show that \(\lambda = 392 l\). The 24 kg mass is left on the bar and the bar is then displaced downwards by a further 20 cm .
2. Prove that the system comprising the bar and the mass now performs simple harmonic motion with the centre of oscillation at the level \(A\).
3. Calculate the number of oscillations made per second in this motion.
4. Find the maximum acceleration which the mass experiences during the motion.

4 A particle \(P\) of mass 0.2 kg is suspended from a fixed point \(O\) by a light elastic string of natural length 0.7 m and modulus of elasticity \(3.5 \mathrm {~N} . P\) is at the equilibrium position when it is projected vertically downwards with speed \(1.6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). At time \(t \mathrm {~s}\) after being set in motion \(P\) is \(x \mathrm {~m}\) below the equilibrium position and has velocity \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Show that the equilibrium position of \(P\) is 1.092 m below \(O\).
2. Prove that \(P\) moves with simple harmonic motion, and calculate the amplitude.
3. Calculate \(x\) and \(v\) when \(t = 0.4\).

5 A particle \(P\) of mass 0.05 kg is suspended from a fixed point \(O\) by a light elastic string of natural length 0.5 m and modulus of elasticity 2.45 N .

1. Show that the equilibrium position of \(P\) is 0.6 m below \(O\).
   \(P\) is held at rest at a point 0.675 m vertically below \(O\) and then released. At time \(t \mathrm {~s}\) after \(P\) is released, its downward displacement from the equilibrium position is \(x \mathrm {~m}\).
2. Show that \(\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } = - 98 x\).
3. Find the value of \(x\) and the magnitude and direction of the velocity of \(P\) when \(t = 0.2\).

6 A particle \(P\) of mass 0.1 kg moves in a straight line on a smooth horizontal surface. A force of \(( 0.36 - 0.144 x ) \mathrm { N }\) acts on \(P\) in the direction from \(O\) to \(P\), where \(x \mathrm {~m}\) is the displacement of \(P\) from a point \(O\) on the surface at time \(t \mathrm {~s}\).

1. By using the substitution \(x = y + 2.5\), or otherwise, show that \(P\) moves with simple harmonic motion of period 5.24 s , correct to 3 significant figures. The maximum value of \(x\) during the motion is 3 .
2. Write down the amplitude of \(P\) 's motion and find the two possible values of \(x\) for which \(P\) 's speed is \(0.48 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
3. On each of the first two occasions when \(P\) has speed \(0.48 \mathrm {~m} \mathrm {~s} ^ { - 1 } , P\) is moving towards \(O\). Find the time interval between
   (a) these first two occasions,
   (b) the second and third occasions when \(P\) has speed \(0.48 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).