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

2 The tip of a sewing-machine needle oscillates vertically in simple harmonic motion through a distance of 2.10 cm . It takes 2.25 s to perform 100 complete oscillations. Find, in \(\mathrm { m } \mathrm { s } ^ { - 1 }\), the maximum speed of the tip of the needle. Show that the speed of the tip when it is at a distance of 0.5 cm from a position of instantaneous rest is \(2.50 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), correct to 3 significant figures.

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.

1 A particle \(P\), of mass 0.2 kg , moves in simple harmonic motion along a straight line under the action of a resultant force of magnitude \(F \mathrm {~N}\). The distance between the end-points of the motion is 0.6 m , and the period of the motion is 0.5 s . Find the greatest value of \(F\) during the motion.

5 \includegraphics[max width=\textwidth, alt={}, center]{f8dd2aee-4ed5-4588-aa03-5dd56d9e7529-3_378_625_1272_758} A light elastic band, of total natural length \(a\) and modulus of elasticity \(\frac { 1 } { 2 } m g\), is stretched over two small smooth pins fixed at the same horizontal level and at a distance \(a\) apart. A particle of mass \(m\) is attached to the lower part of the band and when the particle is in equilibrium the sloping parts of the band each make an angle \(\beta\) with the vertical (see diagram). Express the tension in the band in terms of \(m , g\) and \(\beta\), and hence show that \(\beta = \frac { 1 } { 4 } \pi\). The particle is given a velocity of magnitude \(\sqrt { } ( a g )\) vertically downwards. At time \(t\) the displacement of the particle from its equilibrium position is \(x\). Show that, neglecting air resistance, $$\ddot { x } = - \frac { 2 g } { a } x .$$ Show that the particle passes through the level of the pins in the subsequent motion, and find the time taken to reach this level for the first time.

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 \(C\).

1 A particle oscillates in simple harmonic motion with centre \(O\). When its distance from \(O\) is 3 m its speed is \(16 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), and when its distance from \(O\) is 4 m its speed is \(12 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find the period and amplitude of the motion.

2 A particle of mass \(m\) is attached to the mid-point of a light elastic string. The string is stretched between two points \(A\) and \(B\) on a smooth horizontal surface, where \(A B = 2 a\). The string has modulus of elasticity \(\lambda\) and natural length \(2 l\), where \(l < a\). The particle is in motion on the surface along a line passing through the mid-point of \(A B\) and perpendicular to \(A B\). When the displacement of the particle from \(A B\) is \(x\), the tension in the string is \(T\). Given that \(x\) is small enough for \(x ^ { 2 }\) to be neglected, show that $$T = \frac { \lambda } { l } ( a - l )$$ The particle is slightly disturbed from its equilibrium position. Show that it will perform approximate simple harmonic motion and find the period of the motion.

One end of a light elastic string is attached to a fixed point \(O\). A particle of mass \(m\) is attached to the other end of the string and hangs freely under gravity. In the equilibrium position, the extension of the string is \(d\). Show that the period of small vertical oscillations about the equilibrium position is \(2 \pi \sqrt { } \left( \frac { d } { g } \right)\). The particle is now pulled down and released from rest at a distance \(2 d\) below the equilibrium position. Given that the particle does not reach \(O\) in the subsequent motion, show that the time taken until the particle first comes to instantaneous rest is \(\left( \sqrt { } 3 + \frac { 2 } { 3 } \pi \right) \sqrt { } \left( \frac { d } { g } \right)\).

2 \includegraphics[max width=\textwidth, alt={}, center]{71a3b842-9d31-4c25-b894-ca6d1f47d84b-2_293_875_525_635} Two light elastic strings, each of natural length \(a\) and modulus of elasticity \(2 m g\), are attached to a particle \(P\) of mass \(m\). The strings join the particle to the points \(A\) and \(B\) which are fixed and at a distance \(4 a\) apart on a smooth horizontal surface. The particle is at rest at the mid-point \(O\) of \(A B\). The particle is now displaced a small distance in a direction perpendicular to \(A B\), on the surface, and released from rest. At time \(t\), the displacement of \(P\) from \(O\) is \(x\) (see diagram). Show that $$\ddot { x } = - \frac { 4 g x } { a } \left( 1 - \frac { 1 } { 2 } \left( 1 + \frac { x ^ { 2 } } { 4 a ^ { 2 } } \right) ^ { - \frac { 1 } { 2 } } \right) .$$ Given that \(\frac { x } { a }\) is so small that \(\left( \frac { x } { a } \right) ^ { 2 }\) and higher powers may be neglected, show that the motion of \(P\) is approximately simple harmonic and state the period of the motion.

3 The point \(O\) is on the fixed line \(l\). Points \(A\) and \(B\) on \(l\) are such that \(O A = 6 \mathrm {~m}\) and \(O B = 8 \mathrm {~m}\), with \(O\) between \(A\) and \(B\). A particle \(P\) oscillates on \(l\) in simple harmonic motion with centre \(O\). When \(P\) is at \(A\) its speed is \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), and when \(P\) is at \(B\) its speed is \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Show that the amplitude of the motion is 10 m and find the period of the motion. Find the time taken by \(P\) to travel directly from \(A\) to \(B\), through \(O\).

5 \includegraphics[max width=\textwidth, alt={}, center]{71a3b842-9d31-4c25-b894-ca6d1f47d84b-3_319_794_255_678} A uniform rod \(A B\), of mass \(m\) and length \(6 a\), is rigidly attached at \(B\) to a point on the circumference of a uniform circular lamina of mass \(m\), radius \(2 a\) and centre \(O\). The lamina and the rod are in the same vertical plane, and \(A B O\) is a straight line (see diagram). Show that the moment of inertia of the system about an axis \(l\) through \(A\) perpendicular to the plane of the lamina is \(78 m a ^ { 2 }\). A particle of mass \(2 m\) is now attached at \(B\) and the system is free to rotate in a vertical plane about the fixed axis \(l\) which is horizontal. Initially \(A B\) is horizontal, with \(O\) moving downwards and the system having angular velocity \(\frac { 3 } { 5 } \sqrt { } \left( \frac { g } { a } \right)\). At time \(t , A B\) makes an angle \(\theta\) with the downward vertical through \(A\).

1. Find, in terms of \(a , g\) and \(\theta\), an expression for \(\frac { \mathrm { d } ^ { 2 } \theta } { \mathrm {~d} t ^ { 2 } }\).
2. Find the angular velocity of the system when \(B\) is vertically below \(A\).

4 \includegraphics[max width=\textwidth, alt={}, center]{137d2806-f45c-4121-8ee9-bf89580e1cca-3_906_1538_248_301} The end \(A\) of a uniform \(\operatorname { rod } A B\), of mass \(4 m\) and length \(3 a\), is rigidly attached to a point on a uniform spherical shell, of mass \(\lambda m\) and radius \(3 a\). The end \(B\) of the rod is rigidly attached to a point on a uniform ring. The ring has centre \(O\), mass \(4 m\) and radius \(\frac { 1 } { 2 } a\). The ring and the rod are in the same vertical plane. The line \(O B A\), extended, passes through the centre of the spherical shell. \(B C\) is a diameter of the ring (see diagram). Show that the moment of inertia of this system, about a fixed horizontal axis through \(C\) perpendicular to the plane of the ring, is \(( 30 + 55 \lambda ) m a ^ { 2 }\). Given that the system performs small oscillations of period \(2 \pi \sqrt { } \left( \frac { 5 a } { g } \right)\) about this axis, find the value of \(\lambda\).

4 A particle \(P\) of mass \(m\) moves along part of a horizontal straight line \(A B\). The mid-point of \(A B\) is \(O\), and \(A B = 4 a\). At time \(t , A P = 2 a + x\). The particle \(P\) is acted on by two horizontal forces. One force has magnitude \(m g \left( \frac { 2 a + x } { 2 a } \right) ^ { - \frac { 1 } { 2 } }\) and acts towards \(A\); the other force has magnitude \(m g \left( \frac { 2 a - x } { 2 a } \right)\) and acts towards \(B\). Show that, provided \(\frac { x } { a }\) remains small, \(P\) moves in approximate simple harmonic motion with centre \(O\), and state the period of this motion. At time \(t = 0 , P\) is released from rest at the point where \(x = \frac { 1 } { 20 } a\). Show that the speed of \(P\) when \(x = \frac { 1 } { 40 } a\) is \(\frac { 1 } { 80 } \sqrt { } ( 3 a g )\), and find the value of \(t\) when \(P\) reaches this point for the first time.

2 The point \(O\) is on the fixed line \(l\). Points \(A\) and \(B\) on \(l\) are such that \(O A = 0.5 \mathrm {~m}\) and \(O B = 0.75 \mathrm {~m}\), with \(A\) between \(O\) and \(B\). A particle \(P\) of mass \(m\) oscillates on \(l\) in simple harmonic motion with centre \(O\). The ratio of the kinetic energy of \(P\) when it is at \(A\) to its kinetic energy when it is at \(B\) is \(12 : 11\). Find the amplitude of the motion. Given that the greatest speed of \(P\) is \(0.6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), find the time taken by \(P\) to travel directly from \(A\) to \(B\).

5 A uniform rectangular thin sheet of glass \(A B C D\), in which \(A B = 8 a\) and \(B C = 6 a\), has mass \(\frac { 3 } { 5 } M\). Each of the edges \(A B , B C , C D\) and \(D A\) has a thin strip of metal attached to it, as a border to the glass. The strips along \(A B\) and \(C D\) each have mass \(M\), and the strips along \(B C\) and \(D A\) each have mass \(\frac { 1 } { 3 } M\). Show that the moment of inertia of the whole object (glass and metal strips) about an axis through \(A\) perpendicular to the plane of the object is \(128 M a ^ { 2 }\). The object is free to rotate about this axis, which is fixed and smooth. The object hangs in equilibrium with \(C\) vertically below \(A\). It is displaced through a small angle and released from rest. Show that it will move in approximate simple harmonic motion and state the period of the motion.

2 A particle \(P\) moves on a straight line \(A O B\) in simple harmonic motion. The centre of the motion is \(O\), and \(P\) is instantaneously at rest at \(A\) and \(B\). The point \(C\) is on the line \(A O B\), between \(A\) and \(O\), and \(C O = 10 \mathrm {~m}\). When \(P\) is at \(C\), the magnitude of its acceleration is \(0.625 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) and it is moving towards \(O\) with speed \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find

1. the period of the motion, in terms of \(\pi\),
2. the amplitude of the motion. The point \(M\) is the mid-point of \(O B\). Find the time that \(P\) takes to travel directly from \(C\) to \(M\).

3 A particle moves on a straight line \(A O B\) in simple harmonic motion, where \(A B = 2 a \mathrm {~m}\). The centre of the motion is \(O\) and the particle is instantaneously at rest at \(A\) and \(B\). The point \(M\) is the mid-point of \(O B\). The particle passes through \(M\) moving towards \(O\) and next achieves its maximum speed one second later. Find the period of the motion. Find the distance of the particle from \(O\) when its speed is equal to one half of its maximum speed. At an instant 2.5 seconds after the particle passes through \(M\) moving towards \(O\), the distance of the particle from \(O\) is \(\sqrt { } 2 \mathrm {~m}\). Find, in metres, the amplitude of the motion.

5 \includegraphics[max width=\textwidth, alt={}, center]{833c338f-53c1-436e-a772-0cdaf17fa72d-3_316_949_1320_598} The end \(B\) of a uniform rod \(A B\), of mass \(3 M\) and length \(4 a\), is rigidly attached to a point on the circumference of a uniform disc. The disc has centre \(O\), mass \(2 M\) and radius \(a\), and \(A B O\) is a straight line. The disc and the rod are in the same vertical plane. A particle \(P\), of mass \(M\), is attached to the rod at a distance \(k a\) from \(A\), where \(k\) is a positive constant (see diagram). Show that the moment of inertia of this system, about a fixed horizontal axis \(l\) through \(A\) perpendicular to the plane of the disc, is \(\left( 67 + k ^ { 2 } \right) M a ^ { 2 }\). The system is free to rotate about \(l\) and performs small oscillations of period \(4 \pi \sqrt { } \left( \frac { a } { g } \right)\). Find the possible values of \(k\).

3 A particle \(P\) is performing simple harmonic motion with amplitude 0.25 m . During each complete oscillation, \(P\) moves with a speed that is less than or equal to half of its maximum speed for \(\frac { 4 } { 3 }\) seconds. Find the period of the motion and the maximum speed of \(P\).

5 \includegraphics[max width=\textwidth, alt={}, center]{3e224c82-68df-427e-a59b-7dc2bfd716a2-3_727_517_258_813} A thin uniform \(\operatorname { rod } A B\) has mass \(\frac { 3 } { 4 } m\) and length \(3 a\). The end \(A\) of the rod is rigidly attached to a point on the circumference of a uniform disc with centre \(C\), mass \(m\) and radius \(a\). The end \(B\) of the rod is rigidly attached to a point on the circumference of a uniform disc with centre \(D\), mass \(4 m\) and radius \(2 a\). The discs and the rod are in the same plane and \(C A B D\) is a straight line. The mid-point of \(C D\) is \(O\). The object consisting of the two discs and the rod is free to rotate about a fixed smooth horizontal axis \(l\), through \(O\) in the plane of the object and perpendicular to the rod (see diagram). Show that the moment of inertia of the object about \(l\) is \(50 m a ^ { 2 }\). The object hangs in equilibrium with \(D\) vertically below \(C\). It is displaced through a small angle and released from rest, so that it makes small oscillations about the horizontal axis \(l\). Show that it will move in approximate simple harmonic motion and state the period of the motion.

2 A particle \(P\) moves on a straight line in simple harmonic motion. The centre of the motion is \(O\). The points \(A\) and \(B\) are on the line, on opposite sides of \(O\), with \(O A = 1.6 \mathrm {~m}\) and \(O B = 1.2 \mathrm {~m}\). The ratio of the speed of \(P\) at \(A\) to its speed at \(B\) is \(3 : 4\).

1. Find the amplitude of the motion.
   The maximum speed of \(P\) during its motion is \(\frac { 1 } { 3 } \pi \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
2. Find the period of the motion.
3. Find the time taken for \(P\) to travel directly from \(A\) to \(B\).

3 A particle \(P\) moves on the positive \(x\)-axis in simple harmonic motion. The centre of the motion is a distance \(d \mathrm {~m}\) from the origin \(O\), where \(0 < d < 6.5\). The points \(A\) and \(B\) are on the positive \(x\)-axis, with \(O A = 6.5 \mathrm {~m}\) and \(O B = 7.5 \mathrm {~m}\). The magnitude of the acceleration of \(P\) when it is at \(B\) is twice the magnitude of the acceleration of \(P\) when it is at \(A\).

1. Find \(d\).
   The period of the motion is \(\pi \mathrm { s }\) and the maximum acceleration of \(P\) during the motion is \(10 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
2. Find the speed of \(P\) when it is 7 m from \(O\).
3. Find the time taken by \(P\) to travel directly from \(A\) to \(B\).

2 A particle \(P\) moves on a straight line in simple harmonic motion. The centre of the motion is \(O\). The points \(A\) and \(B\) are on the line on opposite sides of \(O\) such that \(O A = 3.5 \mathrm {~m}\) and \(O B = 1 \mathrm {~m}\). The speed of \(P\) when it is at \(B\) is twice its speed when it is at \(A\). The maximum acceleration of \(P\) is \(1 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).

1. Find the speed of \(P\) when it is at \(O\). \includegraphics[max width=\textwidth, alt={}, center]{2aaf3493-6509-4668-91a2-9f4708bbbb58-04_64_1566_492_328}
2. Find the time taken by \(P\) to travel directly from \(A\) to \(B\).

5 \includegraphics[max width=\textwidth, alt={}, center]{2aaf3493-6509-4668-91a2-9f4708bbbb58-10_809_778_258_680} A thin uniform \(\operatorname { rod } A B\) has mass \(k M\) and length \(2 a\). The end \(A\) of the rod is rigidly attached to the surface of a uniform hollow sphere with centre \(O\), mass \(k M\) and radius \(2 a\). The end \(B\) of the rod is rigidly attached to the circumference of a uniform ring with centre \(C\), mass \(M\) and radius \(a\). The points \(C , B , A , O\) lie in a straight line. The horizontal axis \(L\) passes through the mid-point of the rod and is perpendicular to the rod and in the plane of the ring (see diagram). The object consisting of the rod, the ring and the hollow sphere can rotate freely about \(L\).

1. Show that the moment of inertia of the object about \(L\) is \(\frac { 3 } { 2 } ( 8 k + 3 ) M a ^ { 2 }\).
   The object performs small oscillations about \(L\), with the ring above the sphere as shown in the diagram.
2. Find the set of possible values of \(k\) and the period of these oscillations in terms of \(k\).

A light spring has natural length \(a\) and modulus of elasticity \(k m g\). The spring lies on a smooth horizontal surface with one end attached to a fixed point \(O\). A particle \(P\) of mass \(m\) is attached to the other end of the spring. The system is in equilibrium with \(O P = a\). The particle is projected towards \(O\) with speed \(u\) and comes to instantaneous rest when \(O P = \frac { 3 } { 4 } a\).

1. Use an energy method to show that \(k = \frac { 16 u ^ { 2 } } { a g }\).
2. Show that \(P\) performs simple harmonic motion and find the period of this motion, giving your answer in terms of \(u\) and \(a\).
3. Find, in terms of \(u\) and \(a\), the time that elapses before \(P\) first loses \(25 \%\) of its initial kinetic energy.