Important Announcement
PubHTML5 Scheduled Server Maintenance on (GMT) Sunday, June 26th, 2:00 am - 8:00 am.
PubHTML5 site will be inoperative during the times indicated!

Home Explore Ch-2 Waves and Motion._KB

Ch-2 Waves and Motion._KB

Published by jaykathrotiya33, 2020-06-17 05:05:40

Description: Ch-2 Waves and Motion._KB

Search

Read the Text Version

L. J. Institute of Engineering and Technology Chapter 2: Waves & Motion Subject: Physics (Group-1) Subject code: 3110011 F.Y. - B.E. Kamaldeep Bhatia

Waves and Motion PERIODIC & OSCILLATORY MOTION The motion in which repeats after a regular interval of time is called periodic motion. 1. The periodic motion in which there is existence of a restoring force and the body moves along the same path to and fro about a definite point called equilibrium position/mean position, is called oscillatory motion. 2. In all type of oscillatory motion one thing is common i.e each body (performing oscillatory motion) is subjected to a restoring force that increases with increase in displacement from mean position. TYPES OF OSCILLATORY MOTION It is of two types such as linear oscillation and circular oscillation. Example of linear oscillation:- 1. Oscillation of mass spring system. 2. Oscillation of fluid column in a U-tube. 3. Oscillation of floating cylinder. 4. Oscillation of body dropped in a tunnel along earth diameter. 5. Oscillation of strings of musical instruments. Example of circular oscillation:- 1. Oscillation of simple pendulum. 2. Oscillation of solid sphere in a cylinder (If solid sphere rolls without slipping). 3. Oscillation of a circular ring suspended on a nail. 4. Oscillation of balance wheel of a clock. 5. Rotation of the earth around the sun. OSCILLATORY SYSTEM 1. The system in which the object exhibit to & fro motion about the equilibrium position with a restoring force is called oscillatory system. 2. Oscillatory system is of two types such as mechanical and non- mechanical system. Mechanical oscillatory system:- In this type of system body itself changes its position. For mechanical oscillation two things are especially responsible i.e Inertia & Restoring force. 1

E.g oscillation of mass spring system, oscillation of fluid-column in a U-tube, oscillation of simple pendulum, rotation of earth around the sun, oscillation of body dropped in a tunnel along earth diameter, oscillation of floating cylinder, oscillation of a circular ring suspended on a nail, oscillation of atoms and ions of solids, vibration of swings…etc. Non-mechanical oscillatory system:- In this type of system, body itself doesn’t change its position but its physical property varies periodically. e.g:-The electric current in an oscillatory circuit, the lamp of a body which is heated and cooled periodically, the pressure in a gas through a medium in which sound propagates, the electric and magnetic waves propagates undergoes oscillatory change. The characteristic feature of vibratory motion is its periodicity, i.e., there is movement or displacement that repeats itself over and over again. In this chapter we will study a number of aspects of periodic motion. We will begin with the case of a simple harmonic motion, which is the simplest of all periodic motion. SIMPLE HARMONIC MOTION We define Simple Harmonic Motion (SHM) as a motion in which the acceleration of the body (or force on the body) is directly proportional to its displacement from a fixed point and is always directed towards the fixed point. SHM possesses the following characteristics. (i) The motion is periodic. (ii) When displaced from the fixed point or the mean position, a restoring force acts on the particle tending to bring it to the mean position. (iii) Restoring force on the particle is directly proportional to its displacement. It should be noted that the study of SHM is of practical interest. A vast variety of deformation of physical systems involving stretching, compression, bending or twisting (or combinations involving all of these) result in restoring forces proportional to displacement and hence, leads to SHM. 2

Fig.1.1 Simple harmonic motion Now we will consider an example of a SHM: Let a particle 'A' move along the circumference of a circle with a constant speed V ( = ������������ ) where 'r' is the radius of the circle and '������′ is its angular speed. Let the center of the circle be ‘o’ and a perpendicular AP be drawn from the particle on the diameter YY of the circle (Fig. 1.1). Then as the particle moves along the circumference of the circle, the point P, the foot of the perpendicular vibrates along the diameter. Since the motion of A is uniform, the motion of P is periodic. At any instant, the distance OP from O is called the displacement. If the particle moves from X to A in time‘t’ then AÔX = P���̂���O = ωt = θ i.e., OP = y =r sin ωt Velocity = v = ������������ = rω cosωt (1.1) (1.2) ������������ = rω(1 − ������������������2ωt)1⁄2 = (1 − ������2/������2)1⁄2 Acceleration = ������2������ = -rω2������������������ωt ������������2 = -ω2������ 3

Thus, acceleration is directly proportional to displacement and directed towards a fixed point. Hence, the above example corresponds to SHM. It is instructive to learn how velocity and acceleration in a SHM vary with time. We notice that when the displacement is maximum (+r or -r), the velocity = 0, because now the point 'P' must change its direction. But when y is maximum (+ r or - r), the acceleration is also maximum (-ω2������ or +ω2������ respectively) and is directed opposite to the displacement. When y = 0, the velocity is maximum (rω or - rω) and the acceleration is zero. The time period T (the time required to complete one oscillation) is given by the following relation, Angular velocity = Angle described in one revolution Time taken for one revolution ω = 2π T or T = 2π ω Substituting the value of ω from (1.2) we have T = 2π √Acceleration / Displacement T = 2π (1.3) √Acceleration per unit displacement = 2π√Displacement Acceleration The frequency 'n' is given by 1/T. The idea of phase is very important in SHM. Phase difference between two SHMS indicates how much the two motions are out of step with each other or by how much angle or how much time one is ahead of the other. In general, displacement is given by, y = r sin (ω t + ∅) Clearly at r=0, y =rsin∅. ′∅′ is called the initial phase (Fig. 1.2). 4

A0 is the position at t = 0 Fig.1.2 Simple harmonic motion Now let us calculate the total energy associated with the particle executing SHM. When a body undergoes SHM, its total energy consists of potential energy and kinetic energy. The velocity and consequently kinetic energy is maximum at the mean position. Potential energy is zero at mean position and is maximum at the extreme position. Let us calculate the potential energy. Potential energy = Force × Distance The work done in moving through dy is mω2ydy A. ∴ P. E = ∫ mω2ydy = mω2y2 2 = mω2r2sin2������������ 2 K. E = mv2 = mω2r2cos2������������ 22 T. E = mr2ω2 (cos2������������ + sin2������������) 2 5

= mr2ω2 = mV2 (1.4) 22 FREE VIBRATIONS Let us consider a body of mass 'm' executing SHM. Equation of motion is of the form ������2������ ������ ������������2 = −Kx Or ������2������ = −ω2������ (1.5) ������������2 where ω2= K/m. ������ is the natural angular frequency of the simple harmonic oscillator. ‘K’ is often called the spring constant. This ensures that the acceleration of the particle is always directed towards a fixed point on the line and proportional to the displacement from that point. For a solution of equation (1.5) consider, so that ������ = ������������������������ And ������������ = ������������������������������ ������������ d2������ = b2������������������������ ������t2 This satisfies the equation (1.5) if b2������������������������ + ω2������������������������ = 0 Or ������������������(b2 + ω2) = 0 Or b2 + ω2 = 0 ∴ ������ = ±������������ Thus equation (1.5) is satisfied by ������ = ������������������������ and ������ = ������−������������������. A lincar combination of these two also satisfy the equation ∴ ������ = ������1������������������������ + ������2������−������������������ As this contains two arbitrary constants ������1 and ������2. it is the most general solution. ������1 and ������2 may be complex quantities. This solution may be written as X = ������1 (cos ������t+i sin ������t) + ������2 (cos ������������- i sin ������t) = (������1 + ������2) cos ������t +i (������1 - ������2) sin ������t 6

= A cos������������ + B sin������������ = C sin (������������ + ������) (1.6) = C sin ������������ cos ������ + C cos ������������ sin ������ A= C sin ������ and B = C cos ������ ∴ C = √������2 + ������2 and ������ = tan-1 A/B The values of C and ������ depend upon the initial conditions. Clearly C is the amplitude.The value of x repeats when‘t’ changes by 2π/������ because x = C sin{������(t + 2π/ω) + ������} = C sin (������������ + ������ + 2π) = C sin (������������ + ������ ) ∴Periodic time = 2π/������ Or Frequency = (������ /2π ) Thus, in the case of free vibrations, the maximum amplitude of vibration is constant with time (Fig. 1.3). Consequently, the total energy of the harmonic oscillator remains constant, i.e., it does not change with time. Fig.1.3 Undamped oscillation DAMPED HARMONIC MOTION In the previous section we have seen that the amplitude and hence the total energy of the SHM remains constant. SHM which persists indefinitely without loss of amplitude is called free or undamped. However, observation of the free oscillations of a real physical system reveals that the energy of the oscillator gradually decreases with time and the oscillator eventually comes to rest. For example, the 7

amplitude of pendulum oscillating in air decreases with time and it ultimately stops. The vibrations of a tuning fork die away with the passage of time. This happens because in actual systems, the friction (or damping) is always present. Friction resists motion. To a first approximation, when the resistance is due to motion through fluid such as air, the frictional force is proportional to velocity. Hence, the equation of motion of a body subject to friction is of the form: ������ ������2������ = −Kx − L ∙ ������������ where K and L are constants. ������������2 ������������ L is often known as resistive force constant. This equation can be rewritten as ������2������ + 2������ ������������ + ω2������ = 0 (1.7) ������������2 ������������ ������ = 2������ ������������������ ������ = ω2 Where ������ ������ ‘2k’ represents the frictional force per unit mass per unit velocity. Trying as before x=A������������������ as a solution we have ������������ = ������������������������������ and ������2������ = ������2A������������������ so ������������ ������������2 that the value satisfies equation (1.7) if ������2+ 2kb +ω2 = 0 i.e., if b = -k ± √������2 − ω2 Hence, the general solution is ������ = ������1������(−������ + √������2−ω2)������ + ������2������(−������− √������2−ω2)������ (1.8) The actual form of this solution depends upon the relative magnitudes of k and ������. 8

(1) K > ������ Over damped motion (2) K= ������ critically damped motion Fig.1.4 Over damped and critically damped motion Case (i) If k> ������, √������2 − ω2 is real (but less than k). Hence, x consists in this case of two terms, both dying off exponentially and there is no oscillation (Fig. 1.4). This type of motion is called as overdamped or dead beat. In practice such a kind of motion is represented by a pendulum moving in thick oil or an overdamped moving coil galvanometer. Case (ii) If k= ������ , then this solution fails and we do not get any information (i.e., on substituting the solution which is of the form (A1 +������2) ������−������������ in Eq. (1.7), we get back the condition k= ������ ). Let us consider that √������2 − ω2 is not zero but that is equal to a small quantity 'h' i.e, √������2 − ω2 = ℎ →0. Now equation (1.8) reduces to ������ = ������1������(−������ +ℎ)������ + ������2������(−������−ℎ)������ = ������−������������ {������1������ℎ������ + ������2������−ℎ������} = ������−������������ {������1(1 + ℎ������ + … ) + ������2(1 − ℎ������ + … ) } = ������−������������ {(������1 + ������2 ) + ℎ������ (������1 − ������2 ) + … … } = ������−������������ (������ + ������������) (1.9) where P = ������1 + ������2 Q =(������1 − ������2 )ℎ Equation (1.9) represents one form of the solution. It is clear from equation (1.9) that as 'r' increases, the factor (P+Q) increases but the factor e-kt decreases. Thus, 9

the displacement 'x' first increases due to factor (P+ Qt) but at the same time reversal occurs due to the term ������−������������ and displacement approaches zero as t increases. Further, in this case exponent is ������−������������ whereas in the first case, it was more than ������−������������. Hence, in this case the particle tends to move to equilibrium much more rapidly than in case (i) Fig. (1.4). Such a motion is called critically damped motion. Note that in this case also there is no oscillation. Case (iii) When k< ������, √������2 − ω2 is imaginary and may be written √������2 − ω2 = ip or ω2−������2 = ������2 . so that the solution becomes x = ������1 {������(−������+������������)������} + ������2 {������(−������−������������)������} = ������−������������{������1������������������������ + ������2������−������������������ } This may be put in the form (as in section 1.3). x = C������������������ sin (pt + ������) (1.10) Comparing this with equation (1.6) it is seen that it has two effects. (1) The amplitude is no longer C, but dies away according to the factor ������−������������. The decay of 'x' is shown in Fig.(1.5b.) It depends on the damping coefficient 'k'. It is clear that the amplitude C������−������������ decreases to l/e = 0.368 of its original value C in a time t = ������ given by k������ =1 or ������ = 1/k. ������ is called the relaxation time. It is a measure of how rapidly the motion is damped by friction. The higher the value of 'k' the shorter is the relaxation time. Further since the amplitude decays as ������−������������, the energy which is proportional to the square of the amplitude 10

Fig.1.5 (b) Underdamped oscillations (c) Energy of an underdamped oscillator decays as ������−2������������, E=E0������−2������������ (Fig. 1.5c). The time taken to decay to 1/e of its value is called Γ. It is equal to = 1/2k = ������ /2. (ii) The frequency, formerly 2������/������, is now 2������/������ where p = √ω2 − ������2 This is called underdamped motion. The period is now 2������/√ω2 − ������2. FORCED VIBRATION Now let us study the motion of a particle which has damped SHM, but at the same time subjected to a periodic driving force so that its vibrations do not decay with time. The equation of motion then becomes ������ ������2������ = −Kx − L ∙ ������������ + F0sinqt ������������2 ������������ This can be written as ������������2���������2���+2k������������������������ +ω2x = F sinqt (1.11) Where ������ = 2������ , ������ = ω2 ������������������ ������0 = ������ ������ ������ ������ Here, ‘F’ is the external force per unit mass. As a particular solution of the equation (1.11) try 11

x= A sin (qt -������) so that ������������ = ������������ cos(qt − ������) ������������ ������2������ and ������������2 = −������������2 sin (qt − ������) writing F sin qt as F sin ((qt - ������ ) + ������) = F sin (qt - ������) cos ������ + F cos (qt - ������) sin ������ we have −������������2 sin (qt − ������) + 2kAq cos (qt - ������) + A ω2sin (qt - ������) = F sin (qt - ������) cos ������ + F cos (qt - ������) sin ������ A (ω2 − q2) = Fcos ������ 2kAq = F sin ������ Squaring and adding the above two expressions we have F2 = A2 (ω2 − q2)2 + 4k2A2q2 or ������ = ������ 4k2q2}1⁄2 (1.12) {(ω2−q2)2 + (1.13) and tan ������ = 2������������ ω2−q2 Equations (1.12) and (1.13) give the amplitude and phase of the forced vibration. Depending on the relative values of q and ������, the following cases are possible. Case (i) When driving frequency is low i.e., q << ω. In this case ������ = ������ 4k2q2}1⁄2 {(ω2−q2)2 + = ������ ⋍ ������ = ������������������������������������������������ ω2 {ω4[1−ωq22]2 + ω4 [4kω24q2]}1⁄2 And ������ = tan-1ω22���−������q��� 2 = ������������������−1 2������������ = ������������������−1(0) = 0 ω2(1− q2/ω2) This shows that the amplitude of vibration is independent of frequency of external force. This amplitude depends on the magnitude of the applied force F. The force and displacement are always in phase. Case (ii) When q= ������, the frequency of the external force matches with that of the natural frequency of motion, then we have A= ������ 2������������ ������ = ������������������−1 2������������ = ������������������−1(∞) = ������/2 0 12

Fig.1.6 Plot of amplitude and phase as a function frequency and damping The amplitude is clearly a function of the applied force and damping force. The displacement lags behind the force by ������/2. 13

Case (iii) When q>>������, the frequency of the force is greater than the natural frequency. We have ������ = {(q4+ ������ = ������ 4k2q2)}1⁄2 q2 and ������ = ������������������−1 {ω22���−������q��� 2} = ������������������−1 {−2������} = ������������������−1(−0) = ������ ������ Thus, in this case the amplitude goes on decreasing with applied frequency and phase difference tends to ������ The plot of amplitude and phase for the above three cases are shown in Fig. 1.6. An important parameter which characterizes an underdamped oscillator is the Quality factor or the Q-factor. Q-factor is defined as 2������ times the ratio of energy stored in the oscillator to the energy dissipated per cycle. It can be shown that Q = ������/2k. Thus high value of Q implies a low underdamping and low value of Q implies high underdamping. In an underdamped oscillator with a high Q, the oscillations decay slowly with time since the energy dissipated per cycle is low. Opposite is the case for an underdamped oscillator with a low Q. A plot of A and ε for various values of Q is shown in Fig. 1.7 AMPLITUDE RESONANCE The amplitude A of the forced vibration varies with the frequency of the force. We have from equation (1.12) ������ = ������ {(ω2 − q2)2 + 4k2q2}1⁄2 A is greatest when the denominator is least i.e, when ������ {(ω2 − q2)2 + 4k2q2}1⁄2=0 ������������ i.e., -4(ω2 − q2)q + 8k2������ = 0 i.e., q2 = ω2 − 2k2 Thus, the amplitude is maximum when the forcing frequency is (1.14) (ω2−2k2)1⁄2 2������ 14

This is not the natural frequency of the system, (ω2−k2)1⁄2 nor the value this would 2������ have in the absence of damping, /2������ . But a frequency slightly lower than either. Substituting ������ = (ω2 − 2k2)1⁄2 in the expression for A (from equation 1.12) we get ������ ������ = 2������(q2 + k2)1⁄2 For low damping (i.e., for small k) ������ = ������ 2������������ Hence, greater the maximum amplitude, lower the value of the damping constant. Fig.1.7 Plot of amplitude and phase as a function of quality factor(Q) 15

SHARPNESS OF RESONANCE We have seen that the amplitude of the forced vibration is maximum when the frequency of the applied force has a value q = √ω2 − 2������2 . If the frequency changes from this value, the amplitude falls. When the fall in amplitude for a small departure from the resonance condition is very large, the resonance is said to be sharp. On the other hand if the fall in amplitude is small, the resonance is termed as flat. Thus, the term sharpness of resonance means the rate of fall in amplitude with the change of forcing frequency on each side of the resonance frequency. Figure 1.7 shows the variation of amplitude with forcing frequency at different amounts of damping or Q-factor. Clearly smaller the damping, sharper the resonance or larger the damping flatter the resonance. As can be seen from the figure, larger the Q, smaller the bandwidth and sharper the resonance i.e., the amplitude of the oscillator is more sensitive to the frequency of the driving force. Infact Q-factor is a measure of the sharpness of resonance. It is also defined as the ratio of the resonant frequency to the bandwidth (i.e., full width at half the maximum amplitude) VELOCITY RESONANCE AND ENERGY INTAKE The solution to equation (1.11) is x =A sin (qt -������) so that ������������ = ������������ ������������������(qt − ������) ������������ Thus, velocity ������������ is maximum when ������������ cos (qt - ������) = l, i.e., when sin (qt - ������) =0 or x= 0. This maximum value of velocity, Aq is known as the velocity amplitude. Using the equation (1.12) we have ������ = ������������ {(ω2 − q2)2 + 4k2q2}1⁄2 = ������ 1⁄2 {[ω2���−��� q2]2 + 4k2 } 16

The above expression is greatest when the denominator is least i.e., when ω2 =q2 and its value is F/2k. Note that the condition for maximum velocity amplitude is different from the condition for maximum displacement (equation 1.14). Velocity resonance occurs when the frequency of the applied force is equal to the natural frequency, which the system would have in the absence of damping. The oscillation so produced is sometimes spoken of as a maintained oscillation to distinguish it from a free or forced oscillation. The energy of the system is proportional to the square of the velocity in the position of zero displacement. Thus, in the present instance, the maxımum energy per unit mass is 1 ������ 2 1 F2 2 [2������] = 8 k2 The maximum energy is therefore, inversely proportional to the square of the damping constant. In each cycle work is done against damping. Work done per unit mass = ∫ 2������ (������������) ������������, the integral being taken over a cycle ������������ As ������������ = ������������ cos(qt – ������) ������������ ������������ = ������������ ������������������(qt − ������)������������ Energy dissipated per cycle per unit mass = 2������A2q2 ∫02������⁄������ ������������������2(qt − ������)������������ (Note that (2������/������) is the time for one cycle) = 2������A2q2 ������/������ = 2������A2������������ ∴ Energy dissipated per unit time per unit mass =A2q2������ (Note that the number of cycles per unit time = q/2 ������) Substituting for A, for equation (1.12) this becomes F2q2������ (ω2 − q2)2 + 4k2q2 17

This is the energy that is taken from the driving force per unit time. It is greatest when q = ������ and then becomes F2/4k per unit mass per unit time. This power absorbed per unit mass can also be plotted as a function of Q. It is shown in Fig. 1.8. Note that for higher Q (ie, lower k), power absorption is sharper. SUMMARY OF ANALYTICAL RESULTS WITH FORCED VIBRATION AND RESONANCE (i) A system without damping in which restoring acceleration at unit displacement is ω2, has a natural frequency ������ /2������. (ii) (a) If the above system is subject to a damping force 2k per unit mass per unit velocity, the amplitude dies away in proportion to e-kt and the Fig.1.8 Mean power observed by a forced oscillator as a function of frequency for different values of quality factor (Q) frequency becomes p/2 ������ = (ω2−q2)1⁄2 . This is for the case when k< ������. 2������ (b) When k>> ������, the motion is damped and there is no SHM. For k > ������ it is overdamped and for k = ������, it is critically damped. (iii) When a forcing frequency q/2 ������ is applied to the system, vibration of this frequency is present, the amplitude showing no variation with time. 18

(iv) The amplitude of the forced vibration of frequency q/2������ is greatest when q2=ω2 − 2k2. (v) The velocity in the mean position of the forced oscillation is greatest when q = ������ and the energy taken in to maintain this oscillation is then maximum. This is the normal condition of resonance. (vi) The phase of the forced vibration lags behind the driving force, the lag being ������/ 2 in condition (v). (vii) The maximum velocity at resonance and the energy intake per unit time are inversely proportional to the damping constant k and k2 respectively 19

A sound wave traveling through air is a classic example of a longitudinal wave. As a sound wave moves from the lips of a speaker to the ear of a listener, particles of air vibrate back and forth in the same direction and the opposite direction of energy transport. Each individual particle pushes on its neighboring particle so as to push it forward. The collision of particle #1 with its neighbor serves to restore particle #1 to its original position and displace particle #2 in a forward direction. This back and forth motion of particles in the direction of energy transport creates regions within the medium where the particles are pressed together and other regions where the particles are spread apart. Longitudinal waves can always be quickly identified by the presence of such regions. This process continues along the chain of particles until the sound wave reaches the ear of the listener. Waves traveling through a solid medium can be either transverse waves or longitudinal waves. Yet waves traveling through the bulk of a fluid (such as a liquid or a gas) are always longitudinal waves. Transverse waves require a relatively rigid medium in order to transmit their energy. As one particle begins to move it must be able to exert a pull on its nearest neighbor. If the medium is not rigid as is the case with fluids, the particles will slide past each other. This sliding action that is characteristic of liquids and gases prevents one particle from displacing its neighbor in a direction perpendicular to the energy transport. It is for this reason that only longitudinal waves are observed moving through the bulk of liquids such as our oceans. Earthquakes are capable of producing both transverse and longitudinal waves that travel through the solid structures of the Earth. When seismologists began to study earthquake waves they noticed that only longitudinal waves were capable of traveling through the core of the Earth. For this reason, geologists believe that the Earth's core consists of a liquid - most likely molten iron. While waves that travel within the depths of the ocean are longitudinal waves, the waves that travel along the surface of the oceans are referred to as surface waves. A surface wave is a wave in which particles of the medium undergo a circular motion. Surface waves are neither longitudinal nor transverse. In longitudinal and transverse waves, all the particles in the entire bulk of the medium move in a parallel and a perpendicular direction (respectively) relative to the direction of energy transport. In a surface wave, it is only the particles at the surface of the medium that undergo the 21

circular motion. The motion of particles tends to decrease as one proceeds further from the surface. 22


Like this book? You can publish your book online for free in a few minutes!
Create your own flipbook