Page 1 : 14.1 PERIODIC MOTION, 1. What is periodic motion ? Give some of its examples., Periodic motion. Any motion that repeats itself over, , and over again at regular intervals of time is called periodic, or harmonic motion. The smallest interval of time after, which the motion is repeated is called its time period., The time period is denoted by T and its SI unit is, , second., Examples of periodic motion :, , (i) The motion of any planet around the sun in an, elliptical orbit is periodic. The period of, revolution of Mercury is 87.97 days., , (i) The motion of the moon around the earth is, , . Periodic. Its time period is 27.3 days., , (i!) The motion of Halley’s comet around the sun is, , CHAPTER, , OSCILLATIONS, , repeats itself over and over again about a mean position, such that it remains confined within well defined limits, , (known as extreme positions) on either side of the, mean position., Examples of oscillatory motion :, (i) The swinging motion of the pendulum of a, , wall clock., , (ii) The oscillations of a mass suspended from a, spring., , (iii) The motion of the piston of an automobile, engine., , (iv) The vibrations of the string of a guitar., , (v) When a freely suspended bar magnet is displaced from its equilibrium position along, north- south line and released, it executes, oscillatory motion., , Periodic. It appears on the earth after every 44 3 PERIODIC MOTION VS., , 76 years., (it) Themotion of the hands of a clock is periodic., (©) The heart beats of a human being are periodic., , OSCILLATORY MOTION, , 3. Every oscillatory motion is necessarily periodic but, , The periodic time is about 08 second for a every periodic motion need not be oscillatory. Justify., , Normal person., , l, *2 OSCILLATORY OR HARMONIC MOTION, exempt is oscillatory motion ? Give some of its, , Oscillatory anys, epeated “ory motion. If a body moves back and forth, Oscllat,, , or vil mi, ~~. er vibratory or harmonic motion, Such a motion, , about, , motions. Eve, periodic because it is repeated at regular intervals of, time. In addition, it is bounded about one mean, position. But_every periodic motion _need_not_be, oscillatory. For example, the earth completes one, revolution around the sun in 1 year but it is not a to, and fro motion about any mean position, Hence its, motion is periodic but not oscillatory., , Sinean position, its motion is said to be, , Distinction between periodic and oscillatory, oscillatory motion is _necessaril, , (14.1), , a, , Scanned by CamScanner

Page 2 : 14,2, , 14.4 PERIODIC FUNCTIONS AND, FOURIER ANALYSIS, , 4. With suitable examples, explain the meaning of a, periodic function. Construct two infinite sets of periodic, functions with period T. Hence state Fourier theorem., , Periodic function, Any function that repeats itself at, regular intervals of its argument is called _a_ periodic, function. Consider the function f (0) satisfying the, property,, , F(O+T)=f (8) 4, , This indicates that the value of the function f, remains same when the argument is increased or, decreased by an integral multiple of T for all values of, 6. A function f satisfying this property is said to be, periodic having a period T. For example, trigonometric, functions like sin @ and cos @ are periodic with a period, of 2z radians, because, , sin(®@+27)=sin@ “, cos(8+22)=cos8, , If the independent variable @ stands for some, , dimensional quantity such as time #, then we can, construct periodic functions with period T as follows:, , ~, , th We can check the periodicity by replacing tby + T., us, , . 2nt 2nt, t)=sin = cos =, f, (= sin T and g, (t)=cos T, , _ 27 . (2, AttsTesin 4 T)=sn(2 +29), , 2nt, , =sin = f(t), , Similarly, 8 (t+T) = 8, (#), , It can be easily seen that functions with period T / n,, where n =1,2,3,.....also repeat their values after a time, T. Hence it is possible to construct two infinite sets of, Periodic functions such as, , a fy (t)= sin 220, , v, , , , n=1,2,3,4,....,, , 2nnt, , S bn (t) = cos, , , , n=0,1,2,3,4,, , In the set of cosine functions we, , haves, constant function 8 (t=1 ave included the, , The constant function 1 is Periodic for any value of, T and hence does not alter the periodicity of g(t), Fourier theorem. This theorem states th whi, 0 i hat if, function F (t) with peri empresa ie, , eeaergmeemnnat ressed as the uni, combination of Fane, , , , , PHYSICS-XI, , and 8 (t)= cos at =cos 224 vw :, _ Figure 14.1. shows how these functions vary with, time t., +1, *, eg 2T, = TR T ‘ar om, m1, (a), +1, tI LY |, So, to, TOT |, -1 ;, (b), Fig. 14.1 Periodic functions which are harmonic. wt, Obviously, these functions vary between a mani ,, value + 1and minimum value - 1 passing throug! :, , ant ant 6nt, F(t)=by +b, cos + bp cos——- +b, C08 a, , 2nt _ Ant _ 6nt, +4, sin = +a sin + ay ain., , =by + b, cos wt + b, cos 2mt + b; cos 3at +... i, +a, sin wt + a sin 2at + a, sin 3a +, , or F(t)=by+ 2b, cosnot + Za, sinn wt, 1H n, , where w=2n/T., , The coefficients Uy, by, By, 1 Ay May Oy ones ate Called, Fourier coefficients. These coefficients can be deter., mined uniquely by a mathematical method called, Fourier analysis. Suppose all the Fourier coefficients, , , , except a, and b, are zero, then _, se “Ont 2nt, F()=a sin—— + by cos ay, This equation is a special periodic_motion called, simple harmonic motion (S.H.M.)., , 14.5 PERIODIC, HARMONIC AND, NON-HARMONIC FUNCTIONS, , 5. Distinguish between periodic, harmonic and, non-harmonic functions. Give examples of each., , Periodic, harmonic and non-harmonic functions., Any function that repeats itself at regular intervals of its, argument is called a periodic function. The following sine, and cosine functions are periodic with period T:, , fW)=sinotesin2et, , , , A, , Scanned by CamScanner

Page 3 : OSCILL, inbetween. The periodic functions which can be, py a sine or cosine curve are called Narmonic f, , All harmonic functions are necessarily periodic but all, eriodic functions are not harmonic, T) iodic functions, zwhich cannot be represented by single net ~, , represented, ‘unctions,, , , , , c sine function, are called non-harmonic functions. Fig. 14.2 shows, some periodic functions which tepeat themselves in a, , period T but are not harmonic,, , F(t), , , , F(t), , Fit), , , , Fig. 14.2 Some non-harmonic periodic functions,, Any non-harmonic periodic function can be, constructed from two or more harmonic functions., One such function is :“F (t) =a, sin wt + a, sin2ot, Tt can be easily checked that the functions tan wt, , and cot «tare periodic with period T = x/ wwhile sec wt, and cosec wt are periodic with period T =2n/« Thus, , tn {o(++2)h = tan (ot + m)=tan wt, o, , se of + =) = sec (wt + 2m) =sec wt, o, , _ But such functions take values between zero and, infinity, So these functions cannot be used to represent, displacement functions in periodic motions because, , isplacement always takes a finite value in any, Physical situation,, , eC Based on, aL eee ET eS, , Concepts Used, , , , , , , Tutredatets(aa(slet=), , 1 A function which can be represented by a single, Sine or cosine function is a harmonic function, , otherwise non-harmonic., , a Sadi, A Petiodic function can be expressed as the sum, , | Sf sine and’ cosiné functions of different time, a ds with suitable coefficients,, , io., , , , Scanned by CamScanner

Page 4 : 14.4 PHYSICS-XI, , This equation defines S.HLM., Here_k.j, , “roblems For Practice cea constant called force ae or apis factor and is,, . pd ime represent “defined as the restoring force pee per unit displace., Which of the following functions of time represent’ def : "8 sf paren toner n—LISPIACe:, (a) Sie harmonie motion, (b) periodic but not simple went, The SL unit of k is Nm ~. The negative sign in the, Iranoricand(¢) mon peritic motion ? Find the period Of above equation shows that the restoring force F always, vtion. Here wis a positive real constant, acts in the opposite direction of the displacement x,, , , , , , , , each periodic mo, , 1. sin wl + cos wf. (Ans. Simple harmonic) Now, according to Newton’s second law of motion,, , 2 sin xt + 2eos 2at + 3sin Sat. Fema, (Ans. Periodic but not simple harmonic) as ma =-kx, , 3 cos (Zot + 2/3). (Ans. Simple harmonic) on 7 __k : ie, (eee), , 4. sin? wt. (Ans. Periodic but not simple harmonic) m , 3. cos at + 2sin? af. Hence simple harmonic motion may also be, (Ans. Periodic but not simple harmonic) defined as follows :, , HINTS A particle is said to possess simple harmonic motion ifit, 1. sin ot + cos of = V2sin (ot + x/ 4), T= 2n/o moves to and fro about a mean position under an acceleration, , ‘which is directly proportional to its displacement from the, , 2) erm represents S.H.M. . = —, oe Rens mean position and is always directed towards that position,, k apteaconfamenne, , s 7 2n, Penodotamiaty Pa Examples of simple harmonic motion :, Period of 2cos 2nt = 2m Is=T/2 (i) Oscillations of a loaded spring., 2n (ii) Vibrations of a tuning fork., Period of 3sin 3xt = 2 aan T/3 (iii) Vibrations of the balance wheel of a watch., ha, , (iv) Oscillations of a freely suspended magnet ina, , The sum is not simple harmonic but periodic with uniform magnetic field., , T=2s,, 3. cos (2mt + z/3) represents S.H.M. with, T=2n/2o=n/a, 4. sin? wt =1/2-(1/2) cos 2wt., The function does not represent S.H.M. but is, periodic with T = 2n/2m=n/a, 5. cos wt + 2sin? wt =cos wt + 1-cos 20t, =1+ cos wt —cos 2et, , 7. State some important features of simple harmonic, motion., (ome important features of S.H.M., (i) The motion of the particle is periodic., , (ii) It is the oscillatory motion of simplest kind in, which the particle oscillates back and forth, about its mean position with constant, amplitude and fixed frequency., , Cos ut represents S.H.M. with T = 2n/ a (iii) Restoring force acting on the particle is propor, , Cos 2ut represents S.H.M. with period tional to its displacement from the mean position., =2n/2=n/@=T/2 ' (iv) The force acting on the particle always opposes, , The combined function does not represent $.H.M. the increase in its displacement., , but is periodic with T = 2n/«, (v) A simple harmonic motion can always be, , expressed in terms of a single harmonic, function of sine or cosine., , 14.6 SIMPLE HARMONIC MOTION, , 6. What is meant by simple ha: i tion ? Gi, sone examples, ome motion ? Give 1.4 7 DIFFERENTIAL EQUATION FOR S.H.M., , _ Simple harmonic motion. A particle is said to execute 8. Write down the differential equation for S.HM., simple harmonic motion if it moves to and fro about ainean Give its solution. Hence obtain expression for time period, position under the action of a restori 1g forc of S.H.M., directl its disp rr seis, Loree tion 2 ts dis nean Differential equation of S.H.M. In S.H.M,, the, , iL ays directed towards the mean position. restoring force acting on the particle is proportional to, the displacement of the oscillating body from the its displacement. Thus, mean position is small, then ., , , , , , , , , , , . F=-kx, Restoring fore i, ; = * ae eee . The negative sign shows that F and x are opp, ofS ff sitely directed. Here k is spring factor or force constant, , Scanned by CamScanner

Page 5 : °, SCILLATIONS 145, , emit 14.8 SOME IMPORTANT TERMS, ae CONNECTED WITH S.H.M., 9. Define the terms harmonic oscillator, displacement,, amplitude, cycle, time period, frequency, angular frequency,, phase and epoch with reference to oscillatory motion., , . 2,, where 17 iS the mass of the particle and d’x, 12 is its, a, , acceleration., , , , . #E oie or dx k Some important terms connected with S.H.M., + =-ky — =--y i ; 5, ate dt? a (i) Harmonic oscillator. A particle executing simple, , , , k 5 ax harmonic motion is called harmonic oscillator., Put = — =o, then —> =-@x (ii) Disp. 5, i de 1 isplacement. The distance of the_oscillating, , particle from its mean position at any instant i, , displacement. It is denoted by x. ~~ ~~, , 7 ; ; . There can be other kind of displacement variables., This is the differential equation of S.H.M. Here @ These can be voltage variations in time across a capacitor, , is the angular frequency. Clearly, x should be such a i ana. circuit, pressure variations in time in the propa, a, or qe een! $, , , , , , , , , , , , , , , Function whose second derivative is equal to the 8ation of a sound wave, the changing electric and 2, function itself multiplied with a negative constant. So a_ magnetic fields in the propagation of a light wave, etc. q, possible solution of equation (1) may be of the form (iif) Amplitude. The maximum displacement of the, #, 4 x= Acos (at + $9) oscillating particle on either side of its mean position is 7, Then Ht _@ Asin(at + 4) galled ib pitta Itis denoted by A. Thus x,,,, =+ A. |, dt (iv) Oscillation or cycle. One complete back and forth :, ax motion of a particl ing and ending at the same point is If, and Gr Aces (ats by) =- ox called a cycle or oscillation o: ‘ion. :, ax (v) Time period. The time taken by a particle to q, or —+ wx =0 ? complete one oscillation is called its time period. Or, it is the, at smallest time interval after which the oscillatory motion f, which is same as equation (1). Hence the solution of the repeats. It is denoted by T. i], equation (1) is (vi) Frequency. It is defined as the number of oscillations ;, x= Acos(t+$) (2) completed per. unit time by a particle. Itis denoted by v (nu). 8, It gives displacement of the harmonic oscillator at Frequency is equal to the reciprocal of time period., any instant ¢. That is,, Here A is the amplitude of the oscillating particle. v= ;, $=ot + 4, is the phi f th illating particle., pis th & ial i phase Of thie onc "aang et Clearly, the unit of frequency is (second)? ors”) Itis i], pis the initial phase (at t =0) or epoch. an also expressed as cycles per second (cps) or hertz (Hz). i, Time period of S.H.M. If we replace ¢ by t + a in SI unit of frequency =s~! = cps= Hz. q, @quation (2), we get (vii) Angular frequency. It is the quanti ined by. (, 2n multiplying frequency v by a factor of 2x. It is denoted |, x= Acos of t+2% +4, |, ® Y bya {, “s Thi =2mv= on, fo ~ Acos(at +2n+ §9)= A cos (at + ¢) on US, @O= a \|, - motion repeats after time interval - Hence a SI unit of angular frequency = rad st |, isthe time period of SH.M. (iif) Phase. The phase of a vibrating particle at-any, pa2m__2n »_ k}] instant gives the state ofthe particle as regards its positon, eG Jem [: oo ‘| "and the direction of motion at that instant. It is equal to the, 4 argument of sine or cosine function occurring in the |, " Ts2n ie =2 Inertia factor displacement equation of the S.H.M. Suppose a simple, Ex t Spring factor vv harmonic equation is represented by {, Ingen i = t+ |, Rector, ral, m is called inertia factor and k the spring a= theo {of 40) |, Then phase of the particle is : o=ot + % |, ee, , Scanned by CamScanner