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Waves and oscillations pdf

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Waves and oscillations are enormously important for current research, yet other three stand out as particularly helpful: Vibrations and Waves, by A. P. French ( . Waves and Oscillations. Periodic & Oscillatory Motion: The motion in which repeats after a regular interval of time is called periodic motion. 1. The periodic. The topics for the second part of our physics class this quarter will be oscillations and waves. We will start with periodic motion for the first two lectures, with our.


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PDF | 4+ hours read | This note is presented to the undergraduate students who are interested in the oscillations and waves. One of the. Chap 15Ha-Oscillations-Revised 10/13/ 2. Oscillations and Waves. • Simple Harmonic Motion. • Energy in SHM. • Some Oscillating Systems. • Damped. The purpose of this book is to present a comprehensive study of waves and oscillations in different fields of Physics. The book explains the basic concepts of .

Fluctuation in intensity is due to motion of the ionospheric layer. Determine the subsequent displacements of both masses. MN 2 sin hb2khg PQ b Find the phase and group velocities for gravity waves of frequency 1 Hz in a liquid of depth 0. Neglecting the inertia and friction of the pulley, find a the amplitude of the resulting oscillation, b its centre point of oscillation, and c the expressions for the potential energy and the kinetic energy of the system at a distance y downward from the centre point of oscillation. The higher frequency solution of 2.

It includes new material on electron waves in solids using the Kronig-Penney model to show how their allowed energies are limited to Brillouin zones, The role of phonons is also discussed. An Optical Transform is used to demonstrate the modern method of lens testing. In the last two chapters the sections on chaos and solitons have been reduced but their essential contents remain.

As with earlier editions, the book has a large number of problems together with hints on how to solve them. The Physics of Vibrations and Waves, 6th Edition will prove invaluable for students taking a first full course in the subject across a variety of disciplines particularly physics, engineering and mathematics.

Please check your email for instructions on resetting your password. If you do not receive an email within 10 minutes, your email address may not be registered, and you may need to create a new Wiley Online Library account.

If the address matches an existing account you will receive an email with instructions to retrieve your username. Skip to Main Content. First published: Print ISBN: About this book The main theme of this highly successful book is that the transmission of energy by wave propogation is fundamental to almost every branch of physics. Reviews "This is an excellent textbook, full of interesting material clearly explained, and fully worthy of being studied by future contributors Author Bios H.

He taught physics at Imperial College, London where his research interests were in shock waves and magnetohydrodynamics. A simple pendulum is suspended from a peg on a vertical wall.

The pendulum is pulled away from the wall to a horizontal position Fig. The ball hits the wall, the coefficient of restitution being 2 5. Let x, y be the coordinates of the initial position A of the bob Fig.

Two identical balls A and B each of mass 0. The spring-mass system is constrained to move inside a rigid smooth pipe bent in the form of a circle as shown in Fig. The pipe is fixed in a horizontal plane. The centres of the balls can move in a circle of radius 0. If the earth were a homogeneous sphere of radius R and a straight hole were bored in it through its centre, show that a body dropped into the hole will execute SHM.

Find its time period. Solution Suppose AB is a straight hole Fig. A body of mass m is dropped into the hole. At any instant of time the body is at C at a distance x from the centre of the earth. A cylindrical piston of mass M slides smoothly inside a long cylinder closed at one end, enclosing a certain mass of gas. The cylinder is kept with its axis horizontal.

If the piston is disturbed from its equilibrium position, it oscillates simple harmonically. Show that the period of oscillation will be Fig. The piston is moved isothermally from C to D through a distance x Fig. The gas inside the cylinder will be compressed and it will try to push the piston to its original position. An ideal gas is enclosed in a vertical cylindrical container and supports a freely moving piston of mass M.

The piston and the cylinder have equal cross-sectional area A. Atmospheric pressure is P0 and when the piston is in equilibrium, the volume of the gas is V0. The piston is now displaced slightly from the equilibrium position.

Assuming that the system is completely isolated from its surroundings, show that the piston executes simple harmonic motion and find the frequency of oscillation. Let the initial pressure and the volume of the gas be P and V0 respectively. Each liquid occupies one-fourth the circumference of the tube. Find the time period of these oscillations. Ten kg of mercury are poured into a glass U tube [Fig. Calculate a the effective spring constant of motion, and b the time period.

Ignore frictional and surface tension effects. Find the frequency of oscil-lations. A point charge — q is placed on the axis of the ring at a distance x from the centre of the ring and released from rest. Show that the motion of the charged particle is approximately simple harmonic. Find the frequency of oscillations. The net force on the point charge along the x-direction Fig.

An object of 98 N weight suspended from the end of a vertical spring of negligible mass stretches the spring by 0. Solution a Let D and O represent the position of the end of the spring before and after the object is put Fig. Position O is the equilibrium position of the object.

The positive z-axis is downward with origin at the equilibrium position O. When the elongation of the spring is 0. When the elongation is 0. Thus, when the object is released at F, there A particle of mass 3 units moves along the x-axis attracted toward origin by a force whose magnitude is numerically equal to 27x.

This is the required differential equation. Find the amplitude, frequency and the phase angle in each case. The amplitude and frequency are the same as in part a. The only difference is in the phase angle. A pail of water, at the end of a rope of length r, is whirled in a horizontal circle at constant speed v.

A distant ground-level spot light casts a shadow of the pail onto a vertical wall which is perpendicular to the spotlight beam.

Solution The figure gives a top view of the set-up Fig. Two particles oscillate in simple harmonic motion along a common straight line segment of length A. Each particle has a period of 1. Two particles execute SHM of the same amplitude and frequency along the same line.

They pass one another when going in opposite directions each time their displacement is half their amplitude. Hence F is conservative. What is the frequency and amplitude of vibration of the motion?

A spring of mass M and spring constant k is hanged from a rigid support. A mass m is suspended at the lower end of the spring. If the mass is pulled down and released then it will execute SHM.

Solution Let l be the length of the coiled wire of the spring. Let the suspended mass m be at a distance z from the equilibrium position. Solution The equation of motion for a simple pendulum, if small vibrations are not assumed, is: Find the period of small oscillations that the particle performs about the equilibrium position. A bead of mass m slides on a frictionless wire of nearly parabolic shape Fig.

Let the point P be the point at the bottom of the wire. Show that the bead will oscillate about P if displaced slightly from P and released. Any conservative system will oscillate with SHM about a minimum in its potential energy curve provided the oscillation amplitude is small enough. The potential energy of a particle of mass m is given by a b — x2 x where a and b are positive constants. Find the minimum of V x and expand V x about the point of minimum of V x. Find the period of small oscillations that the particle performs about the position of minimum of V x.

A thin rod of length 10 cm and mass g is suspended at its midpoint from a long wire. Its period Ta of angular SHM is measured to be 2 s.

An irregular object, which we call object X, is then hung from the same wire, and its period Tx is found be 3 s. What is the rotational inertia of the object X about its suspension axis? A uniform disc of radius R and mass M is attached to the end of a uniform rigid rod of length L and mass m. When the disc is suspended from a pivot as shown in Fig.

Now, external torque comes both from the rod and the disc: A thin rod of length L and area of cross-section S is pivoted at its lowest point P inside a stationary, homogeneous and non-viscous liquid Fig. The rod is free to rotate in a vertical plane about a horizontal axis passing through P.

The density d1 of the material of the rod is smaller than the density d2 of the liquid. Show that the motion of the rod is simple harmonic and determine its angular frequency in terms of the given parameters.

In the slightly displaced position two forces are acting on the rod: I dt The motion of the rod is simple harmonic. A thin light beam of uniform cross-section A is clamped at one end and loaded at free end by placing a mass M.

Find an expression for the time period of vibration of the loaded light cantilever. Solution We shall assume that the bar is not subjected to any tension and the amplitude of motion is so small that the rotatory effect can be neglected. The x-axis is taken along the length of the bar and the transverse vibration is taking place in the y-direction. The filaments above PQ are extended whereas the filaments below PQ are contracted.

From Fig. We neglect the weight of the beam. We are interested in finding the depression at any point F x, y of the cantilever.

Waves and Oscillations

This equals the resisting moment YIg d2 y dx 2. A rectangular light beam of breadth b, thickness d and length l is clamped at one end and loaded at free end by placing a mass M. PQ is the neutral line. We consider the strip ST of thickness dx at a distance x from the neutral axis PQ.

A light beam of circular cross-section of radius a and length l is clamped at one end and loaded at free end by placing a mass M. We would like to make an LC circuit that oscillates at Hz. If we have a 2 H inductor, what value of capacitance should we use? If the capacitor is initially charged to 5 V, what will be the peak charge on the capacitor? What is the total energy in the circuit? Solution The total energy in the circuit is the sum of the magnetic and electric energy: Two particles of mass m each are tied at the ends of a light string of length 2a.

The whole system is kept on a frictionless horizontal surface with the string held tight so that each mass is at a distance a from the centre P as shown in the m m Figure 1. Now the mid-point of the string is pulled vertia P a cally upwards with a small but constant force F. As a result, Fig. For which of the following data, the measurement of g will be most accurate? The amplitude should be small for SHM of the simple pendulum. Correct choice: It goes further up by a height h so that the K.

The spring has its natural length at this position. If the sphere is slightly pushed and released, it starts performing simple harmonic motion.

Find the frequency of these oscillations. Now, the sphere is pushed downward slightly by a distance x inside the liquid Fig. A particle of mass m moves on the x-axis as follows: A spring of force constant k is cut into two pieces such that one piece is n times the length of the other. Find the force constant of the long piece. The long piece has n such springs which are in series. Then a At points away from the origin, the particle is in unstable equilibrium.

Minimum P. A particle of mass m is executing oscillations about the origin on the x-axis. Two blocks A and B each of mass m are connected by a massless spring of natural length L and spring constant K. The blocks are initially resting on a smooth horizontal floor with the spring at its natural length as shown in Fig. Then a the kinetic energy of the A—B system at maximum compression of the spring, is zero, b the kinetic energy of the A—B system at maximum compression of the spring is mV 2 4.

A point moves with SHM. Find its amplitude and time period. Find its acceleration when it is at the greatest distance from the centre. A particle is moving with SHM in a straight line. When the distances of the particle from the equilibrium position are x1 and x2, the corresponding values of the velocity are u1 and u2.

A particle of mass 0. Find the maximum velocity and its total energy. A particle moves with SHM. A particle is executing SHM. Show that, average K.

A particle moves with simple harmonic motion in a straight line. Find a the time period of the motion, b the amplitude of the motion. A loudspeaker produces a musical sound by the oscillation of a diaphragm. If the amplitude of oscillation is limited to 9.

A small body is undergoing SHM of amplitude A. While going to the right from the equilibrium position, it takes 0. Find the period of the motion. A block is on a piston that is moving vertically with SHM. The piston in the cylindrical head of a locomotive has a stroke of 0. A 40 g mass hangs at the end of a spring. When 25 g more is added to the end of the spring, it stretches 7.

Two bodies M and N of equal masses are suspended from two separate massless springs of spring constants k1 and k2 respectively. The block is pulled a distance 10 cm from its equilibrium position and released from rest. A cubical block vibrates horizontally in SHM with an amplitude of 4. If a smaller block sitting on it is not to slide, what is the minimum value that the coefficient of static friction between the two blocks can have?

The vibration frequencies of atoms in solids at normal temperatures are of the order of Hz. Suppose that a single silver atom vibrates with this frequency and that all the other atoms are at rest. Compute the effective spring constant. One mole of silver has a mass of g and contains 6.

Suppose that in Fig. It strikes the spring and becomes attached to it. After sliding 50 cm along the flat table from the point of release the mass comes to rest. How large a friction force opposes the motion? A mass of g placed at the lower end of a vertical spring stretches it 20 cm. When it is in equilibrium the mass is hit upward and due to this it goes up a distance of 8 cm before coming down. Find a the magnitude of the velocity imparted to the mass when it is hit, b the period of motion.

With a g mass at its end a spring executes SHM with a frequency of 1 Hz.

How much work is done in stretching the spring 10 cm from its unstretched length? When cocked the spring is compressed 2 cm. How high can the gun shoot a 4 g projectile? A block of mass M, at rest on a horizontal frictionless table, is attached to a rigid support by a spring of spring constant k. A bullet of mass m and velocity v strikes the block as shown in Fig. The bullet remains embedded in the block. A g mass at the end of a Hookean spring vibrates up and down in such a way that it is 2 cm above the table top at its lowest point and 12 cm above the table top at its highest point.

Its period is 5s. Find a the spring constant, b the amplitude of vibration, c the speed and acceleration of the mass when it is 10 cm above the table top.

A thin metallic wire of length L and area of cross-section A is suspended from free end which stretches it through a distance l. Show that the vertical oscillations of the system are simple harmonic in nature and its time period is given by T There are two spring systems a and b of Fig.

A kg block is suspended from each system. If the block is constrained to move in the vertical direction only, and is displaced 0. Find the frequency with which the motor vibrates vertically. A spring of force constant k is cut into three equal parts, the force constant of each part will be When the block is passing through its equilibrium position, an object of mass m is put on it and the two move together.

A body of mass m on a horizontal frictionless plane is attached to two horizontal springs of spring constants k1 and k2 and equal relaxed lengths L. Now the free ends of the springs are pulled apart and fastened to two fixed walls a distance 3L apart.

Find the elongations of the springs k1 and k2 at the equilibrium position of the body and the time period of small longitudinal oscillations about the equilibrium position. Neglecting the mass of the spring find the period of small longitudinal oscillations of the body. Assume that the gravitational force is absent.

A uniform spring whose unstretched length is L has a force constant k. What are the corresponding force constants k1 and k2 in terms of n and k? Two bodies of masses m1 and m2 are interconnected by a weightless spring of stiffness k and placed on a smooth horizontal surface. The bodies are drawn closer to each other and released simultaneously. A particle executes SHM with an amplitude A.

At what displacement will the K. A body of mass 0. Relaxed length of each spring is 1m. The mass m is displaced from the initial position O to the point D, the mid-point of BC O m and then released from rest. What will be the kinetic energy of m if it returns to the point O?

What will be the speed of the body at O? Compare the period of the simple pendulum at the surface of the earth to that at the surface of the moon. Show that the radius R of the earth is given by Th. If the pendulum executes small oscillations about the equilibrium position, what will be its time period of oscillation? Find the time period of small oscillation of the bob in the liquid. Solve problem 27 Fig. The mass and diameter of a planet are twice those of the earth.

The other end is tied to a massless spring of spring constant k. A mass m hangs freely from the free end of the spring. A simple pendulum of mass M is suspended by a thread of length l when a bullet of mass m hits the bob horizontally and sticks in it. It is pushed down slightly and released. Find the period of oscillations if the cylinder has weight W and cross-sectional area A. A vertical U-tube of uniform cross-section contains a liquid of total mass M.

The mass of the liquid per unit length is m. When disturbed the liquid oscillates back and forth from arm to arm. Calculate the time period if the liquid on one side is depressed and then released. Compute the effective spring constant of the motion. A point negative charge — q of mass m lies midway between the fixed charges. Show that the motion of the negative charged particle is approximately simple harmonic.

Calculate the time period of oscillations. A simple pendulum consists of a small sphere of mass m suspended by a thread of length l. The sphere carries a positive charge q. The pendulum is placed in a uniform electric field of strength E directed vertically upwards. With what period the pendulum oscillates if the electrostatic force acting on the sphere is less than the gravitational force? Assume that the oscillations are small I. Find the spring constant and the position of the particle at time 1 s.

A particle vibrates about the origin of the coordinates along the y-axis with a frequency of 15 Hz and an amplitude of 3. Find its equation of motion. A particle of mass m moves along the x-axis, attracted toward the origin O by a force proportional to the distance from O.

Initially the particle is at distance x0 from O and is given a velocity of magnitude v0 a away from O b toward O. Find the position at any time, the amplitude and maximum speed in each case.

An object of mass 2 kg moves with SHM on the x-axis. A point particle of mass 0. The potential energy of a particle of mass m is given by V x Find the location of the equilibrium point s , the nature of the equilibrium, and the period of small oscillations that the particle performs about the equilibrium position.

An engineer wants to find the moment of inertia of an odd-shaped object about an axis passing through its centre of mass. The object is supported with a wire through its centre of mass along the desired axis. The engineer observes that this torsional pendulum oscillates through 20 complete cycles in 50s. What value of moment of inertia is obtained?

A 90 kg solid sphere with a 10 cm radius is suspended by a vertical wire attached to the ceiling of a room. A torque of 0. What is the period of oscillation when the sphere is released from this position? Compare the time periods of vibrations of two loaded light cantilevers made of the same material and having the same length and weight at the free end with the only difference that while one has a circular cross-section of radius a, the other has a square cross-section, each side of which is equal to a.

A long horizontal wire AB, which is free to move in a vertical plane and carries a steady current of 20 A, is in equilibrium at a height of 0. Show that when AB is slightly depressed, it executes simple harmonic motion. Find the period of oscillations. You have a 2. What range of capacitance should your variable capacitor cover? An object of mass 0. Find the amplitude of oscillations. T1 is the time period of a simple pendulum. Now the time period is T2. T1 LMHints: Consider a linear homogeneous differential equation of degree n: Suppose that a driving force F1 t produces an oscillation x1 t and another driving force F2 t produces an oscillation x2 t [when F2 t is the only driving force].

The equation of the path is obtained by eliminating t from these two equations. The position of the particle in the xy plane is given by r Find the resultant motion. Find the resultant motion due to superposition of a large number of simple harmonic motions of same amplitude and same frequency along the x-axis but differing progressively in phase.

The resultant amplitude may be obtained by the vector polygon method Fig. The polygon will then become an arc of a circle and the chord joining the first and the last points of the arc will represent C the amplitude of the resultant vibration Fig. When the last component vibration is at B, the first and the B last component vibrations are in phase, the polygon becomes O a complete circle and the amplitude of the resultant vibration Fig.

Thus, due to superposition of two simple A harmonic vibrations at right angles to each other, the displacement of the particle will be along a curve given by Eqn. The particle vibrates along the straight line AC Fig.

The direction of rotation clockwise or anticlockwise of the particle may be obtained form the x- and y-motions of the particle when t is increased gradually. Thus in this case the particle moves in the anticlockwise direction. As the phase difference is changed gradually, the shape of the loop also changes gradually. Two vibrations of frequencies in the ratio 1: Find the equation of the figure traced by the particle. It should be noted from Eqn. In general, if the frequencies are in the ratio 1: N, the curve will have N loops.

What deductions may be made about the frequency of the other tuning fork? Two-dimensional harmonic oscillation: The mass m is free to move in the xy plane Fig. It is connected to the rigid walls by two unstretched massless springs of spring constant, k1 oriented along the x-axis and by the unstretched massless springs of spring constant k2 oriented along the y-axis.

The relaxed length of each spring is a. Discuss the motion of the mass in the xy plane in the small oscillation approximation. Solution In the general configuration [Fig. Thus we find that the x-component of the return force is entirely due to the two springs of lengths l1 and l2: Thus we get two uncoupled differential equations for the mass m along the x- and y-directions: The spherical pendulum: Consider a simple pendulum of length l.

Find the motion of the bob for small oscillations x and y are small. Solution Suppose A is the position of the bob at any instant of time Fig. From A we drop a perpendicular AB on the z-axis. These equation can be solved independently: Find the angular frequencies for the normal modes of oscillations and the normal coordinates. Solution For normal modes of oscillations, both the degrees of freedom, namely x1 and x2 oscillate with the same frequency and they oscillate in phase or out of phase with one another: Substituting Eqns.

From Eqn. Normal coordinates: If the differential equations are coupled, we have to search for new variables which satisfy uncoupled differential equations. The new variables are then called normal coordinates.

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Suppose X and Y are obtained from the linear combinations of x1 and x2 so that we may write Longitudinal oscillations of two coupled masses: Two bodies of masses m1 and m2 are attached to each other and to two fixed points by three identical light springs of relaxed length a.

The whole arrangement rests on a smooth horizontal table. Find the angular frequencies of the normal modes for longitudinal oscillations of small amplitude. Describe the motions of the two bodies for each normal mode. Solution Let the spring constant of each spring be k.

We consider the forces acting on m1 and m2 when m1 is displaced from its equilibrium position by x1 and m2 is displaced by x2 from its equilibrium position Fig. The first spring is stretched by amount x1 and the force a m1 a m2 a a b x1 x2 Fig. The second spring is stretched by x2 — x1. The third spring is compressed by amount x2. The lower frequency solution of 2. The two bodies oscillate in phase. Mode 2: The higher frequency solution of 2.

The two bodies oscillate in antiphase. The central spring has the same length as it had at equilibrium, so that the central spring exerts no force on either mass. When the left hand mass goes to the right, the right hand mass also goes to the right; when the left hand mass goes to the left, the right hand mass also goes to the left, the length of the central spring remaining unchanged always. When the left hand mass goes to the right, the right hand mass goes to the left [the central spring is compressed].

When the left hand mass goes to the left, the right hand mass to the right [the central spring is elongated]. The two masses move in opposite directions antiphase. When the central spring is compressed, the side springs are elongated, and when the central spring is elongated, the side springs are compressed.

By adding Eqns. Two bodies of masses m1 and m2 are attached to each other and to two fixed points by three identical light springs of relaxed length a0. Find the angular frequencies of the normal modes for transverse oscillations. Solution At equilibrium let the length of each spring be a and the spring constant be k. Let y1 and y2 be the vertical displacements of the masses m1 and m2 from the initial positions at any instant of time Fig.

Waves and Oscillations, Second Edition

Thus the normal mode frequencies are given by Eqn. Two identical simple pendulums having the same length l and same bob mass m are suspended by strings of negligible mass. The bobs are connected by a spring of relaxed length a. At the equilibrium position the spring has its relaxed length. Assuming small-oscillation amplitudes in the vertical plane, find the normal modes of oscillations of the system.

Solution At the general configuration Fig. If the spring were not present, we have for the x-component of motion of the spherical pendulums. How does the modulation amplitude vary with time?

Waves and Oscillations

Number of beats per second is twice the modulation frequency. Thus A2mod t oscillates about its average value at twice the modulation frequency i. Two tuning forks when sounded together give 4 beats per second. One is unison with a length of 96 cm of a sonometer wire under certain tension and the other with Find the frequencies of these forks.

Two identical simple pendulums a and b with the same string length l and the same bob mass m are coupled by a massless spring of spring constant k which is attached to the bobs. The spring is unstretched when both the strings of the pendulums are vertical. Find the expressions for the energy of bob a and that of bob b at any time.

Assume weak coupling between the oscillators. Solution For solutions of i and ii see problem 13 of Chapter 2. The solutions of Eqns 2. From Eqns. We have from Eqns.

The pendulum a is under the influence of two harmonic modes of equal amplitudes with slightly different frequencies. Thus the motion a exhibits beats. The pendulum b also exhibits beats. Initially the bob b is at zero and the bob a has displacement 2A. Gradually the oscillation amplitude of pendulum a decreases and that of pendulum b increases, until pendulum a comes to rest and pendulum b oscillates with the amplitude and energy that pendulum a started out with.

Oscillations pdf and waves

Here we neglect the frictional forces. The vibration energy is transferred completely from one pendulum to the other and the process continues. The vibration energy slowly flows back and forth between a and b. One complete round trip for the energy from a to b and back to a is a beat. The beat period is the time for the round trip. For the pendulum a the oscillation amplitude Amod t is almost constant over one cycle of the fast oscillation.

If the spring is weak it does not have significant amount of stored energy. The total energy is twice the average value of the KE: When the pendulum a has the maximum energy, the pendulum b has zero energy, and vice versa. The total energy of the system E is constant and it flows back and forth between the two pendulums at the beat frequency.

A detector receives the signals from the two stations simultaneously. Three simple harmonic motions in the same direction having the same amplitude a and same period are superimposed. Two masses m1 and m2 connected by a light spring of natural length l is compressed completely and tied by a light thread.

At this position the thread snaps. Find the positions of two masses as functions of time for t m 0. Solution Suppose x1 and x2 are the positions of two masses m1 and m2 at time t.

The acceleration is not 2p always directed towards a focus. Correct Choice: Find the amplitude of the resultant motion. Find a the trajectory equation y x of the point and the direction of its motion along r the trajectory, b the acceleration a of the point as a function of its radius vector r r relative to the origin of the co-ordinates. A particle of mass m moves in two dimensions under the following potential energy function: It is found that the figure completes its cycle in 10 seconds. If the frequency of A slightly greater than that of B is Hz, calculate the frequency of B.

It takes 5s to go through a cycle of changes. On loading slightly the fork of higher frequency, the period of cycle is raised to 10s.

If the frequency of the lower fork is Hz, what is the frequency of the other fork, before and after loading?

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On slightly loading A with wax, the figures go through cycle in 5 s. If the frequency of B is Hz, what is the frequency of A before and after loading? Two bodies of masses m and 2m are attached to each other and to two fixed points by three identical light springs along a straight line. Two bodies of masses m and 3m are attached to each other to two fixed points by three identical light springs of relaxed length a0 along the x-axis.

Two masses m1 and m2, initially at rest on a frictionless surface are connected by a 3 spring of force constant k and natural length a. The spring is compressed to of its 4 natural length and released from rest. Determine the subsequent displacements of both masses.

Two masses m1 and m2 are connected by springs of spring constants k1 and k2 and natural lengths L1 and L2 Fig.

The point P is fixed and O1 and O2 mark the equilibrium positions of the springs. Assume no friction or external forces. A uniform bar AB of length L and mass m is supported at its end by identical springs with spring constant k. Motion is set by depressing the end B by a small distance a and releasing it from rest Fig. Solve the problem of motion, identifying normal modes and frequencies.

Consider the motion of a three-particle system in which the particles all lie in a straight line. The two end particles, each of mass m, are bound to the central particle of mass M through the springs of stiffness k as shown in Fig. If two tuning forks A and B are sounded together they produce 5 beats per second. When A is slightly loaded with wax they produce 3 beats per second when sounded together.

Find the original frequency of A if the frequency of B is Hz. You are given four tuning forks. The fork with the lowest frequency vibrates at Hz. By using two tuning forks at a time, the following beat frequencies are heard: What are the possible frequencies of the other tuning forks? You are given five tuning forks, each of which has a different frequency. By trying every pair of tuning forks find a maximum number of different beat frequencies b minimum number of different beat frequencies.

Two identical piano wires have a fundamental frequency of Hz when kept under the same tension. The Damped Harmonic Oscillator 3 3. Such a motion is called damped harmonic motion. Suppose a particle of mass m is subject to a restoring force proportional to the distance from a fixed point on the x-axis and a damping force proportional to the velocity. Equation 3. This is equivalent to Eqn. Obtain an expression for the displacement of the damped harmonic oscillator where the damping force is proportional to the velocity.

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Discuss the effect of the damping on the displacement and frequency of the oscillator. Solution The differential equation of the damped harmonic motion is given by Eqn. Case I: Case II: Its amplitude R exp —bt decreases exponentially with time. Suppose that the values of x in both directions corresponding 2 2 to these times are x0, x1, x2, x3 etc. Thus the damping coefficient b can be found from an experimental measurement of consecutive amplitudes. We can regard equation 3. If the oscillator is damped, the mechanical energy is not constant but decreases with time.

For a damped oscillator the amplitude is R exp — bt and the mechanical energy is 1 kR2 is the initial mechanical energy. Case III: One solution of Eqn. A particle of mass 3 moves along the x-axis attracted toward origin by a force whose magnitude is numerically equal to 12x. The particle is also subjected to a damping force whose magnitude is numerically equal to 12 times the instantaneous speed.

A particle of mass 1 g moves along the x-axis under the influence of two forces: Assuming that the particle starts from rest at a distance 10 cm from the origin, a set up the differential equation of motion of the particle, b find the position of the particle at any time, c determine the amplitude, period and frequency of the damped oscillation, and d find the logarithmic decrement of the problem.

A system of unit mass whose natural angular frequency in the absence of damping is 4 rad s—1 is subject to a damping force which is proportional to the velocity of the system, the constant of proportionality being 10 s—1.

Find the smallest value of v that will produce negative displacement. The general solution for x is [see Eqn. In order to make x negative it is necessary that 5. In a damped oscillatory motion an object oscillates with a frequency of 1 Hz and its amplitude of vibration is halved in 5 s. Find the differential equation for the oscillation. Find also the logarithmic decrement of the problem.

At the end of the recoil, a damping dashpot is arranged in such a way that the launcher returns to the firing position without any oscillation critical damping. Solution a We use the principle of conservation of energy for the rocket launcher and the recoil spring: Solve the problem of simple pendulum if a damping force proportional to the instantaneous tangential velocity is taken into account.

Now we may discuss in the light of Eqn. Three cases arise. In cases 1 and 2 the pendulum bob gradually returns to the equilibrium position without oscillation. Assuming that the effect of viscosity can be described by a force proportional to velocity, determine the constant of proportionality.

The effect of buoyancy is neglected. Determine the equilibrium extension of the spring. Show that the motion is under damped and find its period of oscillation. Find the time in which the amplitude of oscillation falls by a factor of e. Solution a When the mass is lifted vertically it moves with a constant velocity.

Thus the net force acting on it is zero: A spring supports a mass of 5 g which performs damped oscillations. It is found to have successive maxima of 2. An automobile suspension system is critically damped and its period of oscillation with no damping is one second. Find the quality factor Q for the damped oscillations of Problem 5.

Show that the object does not change its direction and the kinetic energy of the object keeps on decreasing. So the object does not change its direction. Solution The force experienced by the body is 10—2 m2. Thus, we have to choose the —ve sign.

A particle of 2 g moves along the x-axis under the influence of two forces: Find the time in which its amplitude of vibration is halved. In a damped oscillatory motion an object oscillates with a frequency of 2 Hz and its amplitude of vibration is halved in 2 s. The weight is then pulled down 1 m and released. Find the position of the body at any time.

The natural frequency of a mass vibrating on a spring is 20 Hz, while its frequency with damping is 16 Hz. Find the logarithmic decrement.

A body of mass 10 g is suspended by a spring of stiffness 0. After approximately how many oscillations will the amplitude of the system be halved? A block is suspended by a spring and a dashpot with a strong damping action. Show that if the block is displaced downwards and given a downward velocity, it will never pass through its equilibrium position again.

With weak damping imposed it is found that the amplitudes of two consecutive oscillations in the same direction are 5 cm and 0. Find the new period of the system.

The frequency of a damped oscillator is one-half the frequency of the same oscillator with no damping. Find the ratio of the maxima of successive oscillations. Find the time when the displacement becomes greatest in the negative x-direction and the value of the negative displacement.

A single loop circuit consists of a 7. Initially the capacitor has a charge of 6. Calculate the charge on the capacitor after 10 and complete cycles of oscillations. A bell rings at a frequency of Hz. Its amplitude of vibration is halved in 10s. Find the quality factor of the bell. Forced Vibrations and Resonance 4 4.

A particular solution of Eqn. After a long time when x1 becomes negligible the motion of the mass m is given by Eqn. The vibrations or oscillations represented by x2 are called forced vibrations or forced oscillations. Thus from Eqn. The resonator consists of either a spherical or a cylindrical air cavity with a small neck Fig. The dimension of the cavity is small in comparison Fig.

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In case of spherical cavity the volume of the cavity is fixed whereas the volume is variable in case of cylindrical cavity. The natural frequency of vibration of Helmholtz resonator is given by v S The natural frequency of the resonator can be changed by changing the volume V of the resonator. When the sound wave of frequency resonant with the natural frequency of the resonator is incident on it, the resonator will produce sharp response. The frequency of the vibrating body is then equal to the natural frequency of the resonator given by Eqn.

Set up the differential equation of motion and find the steady-state solution. Solution The differential Eqn. The solution x1 is the displacement of the damped harmonic oscillator. Substituting in Eqn. The complete solution of Eqn. These beats are transient, as the natural vibrations become small after a short interval of time.

Obtain the expression for the velocity of the mass m when it is in the steady state forced vibration of problem 1. This phenomenon is known as velocity resonance. The velocity amplitude at resonance is f. It is maximum when the denominator or the square of the denominator is a minimum. Note that the angular frequency p of the periodic impressed force at amplitude resonance is slightly smaller than that at velocity resonance. In the steady state forced vibration describe how the phase of the driven system changes with the frequency of the driving system.

There is no difference of phase between the driven system and the impressed force. Show that in the steady state forced vibration the rate of dissipation of energy due to frictional force is equal to the rate of supply of energy by the driving force in each cycle.

Suppose at any instant the force F sin pt moves through a distance dx in time dt. In the steady state forced vibration the displacement of the particle is sinusoidal and 2 the mechanical energy remains fixed at the steady value 1 2 kA.

Determine the root-mean-square rms values of displacement, velocity and acceleration for a damped forced harmonic oscillator operating at steady state.

A machine of total mass 90 kg is supported by a spring resting on the floor and its motion is constrained to be in the vertical direction only. The machine contains an eccentrically mounted shaft which, when rotating at an angular frequency p, produces a vertical force on the system of Fp2 sin pt where F is a constant.

It is found that resonance occurs at r. Find the amplitude of vibration in the steady state when the driving frequency is a r. Find also the quality factor Q at resonance. Assume that the gravity has a negligible effect on the motion. Two bodies of masses m1 and m2 connected by a spring of spring constant k, can move along a horizontal line axis of the spring. Solution Let x1 and x2 be the respective displacements of the masses m1 and m2 from their equilibrium positions.

The extension of the spring is x2 — x1.