Simple Harmonic Motion (SHM)

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1 Phsics Sipe Haronic Motion (SHM)

2 abe of Content. Periodic otion.. Osciator or Vibrator otion. 3. Haronic and Non-haronic osciation. 4. Soe iportant definitions. 5. Sipe haronic otion. 6. Dispaceent in S.H.M. 7. Veocit in S.H.M. 8. Acceeration in S.H.M. 9. Coparative stud of dispaceent, veocit and acceeration. 0. Ener in S.H.M.. ie period and Frequenc of S.H.M.. Differentia equation of S.H.M. 3. Sipe penduu. 4. Sprin penduu. 5. Various foruae of S.H.M. 6. Iportant facts and foruae. 7. Free, daped, forced and aintained osciation.

3 . Periodic Motion. A otion, which repeat itsef over and over aain after a reuar interva of tie is caed a periodic otion and the fixed interva of tie after which the otion is repeated is caed period of the otion. Exapes: (i) Revoution of earth around the sun (period one ear) (ii) Rotation of earth about its poar axis (period one da) (iii) Motion of hour s hand of a coc (period -hour) (iv) Motion of inute s hand of a coc (period -hour) (v) Motion of second s hand of a coc (period -inute) (vi) Motion of oon around the earth (period 7.3 das). Osciator or Vibrator Motion. Osciator or vibrator otion is that otion in which a bod oves to and fro or bac and forth repeated about a fixed point in a definite interva of tie. In such a otion, the bod is confined within we-defined iits on either side of ean position. Osciator otion is aso caed as haronic otion. Exape: (i) he otion of the penduu of a wa coc. (ii) he otion of a oad attached to a sprin, when it is pued and then reeased. (iii) he otion of iquid contained in U- tube when it is copressed once in one ib and eft to itsef. (iv) A oaded piece of wood foatin over the surface of a iquid when pressed down and then reeased executes osciator otion.

4 3. Haronic and Non-haronic Osciation. Haronic osciation is that osciation which can be expressed in ters of sine haronic function (i.e. sine or cosine function). Exape: a sin t or acos t Non-haronic osciation is that osciation which cannot be expressed in ters of sine haronic function. It is a cobination of two or ore than two haronic osciations. Exape: a sin t b sin t 4. Soe Iportant Definitions. () ie period: It is the east interva of tie after which the periodic otion of a bod repeats itsef. S.I. units of tie period is second. () Frequenc: It is defined as the nuber of periodic otions executed b bod per second. S.I unit of frequenc is hertz (Hz). (3) Anuar Frequenc: Anuar frequenc of a bod executin periodic otion is equa to product of frequenc of the bod with factor. Anuar frequenc = n S.I. units of is Hz [S.I.] aso represents anuar veocit. In that case unit wi be rad/sec. (4) Dispaceent: In enera, the nae dispaceent is iven to a phsica quantit which underoes a chane with tie in a periodic otion. Exapes: (i) In an osciation of a oaded sprin, dispaceent variabe is its deviation fro the ean position. (ii) Durin the propaation of sound wave in air, the dispaceent variabe is the oca chane in pressure (iii) Durin the propaation of eectroanetic waves, the dispaceent variabes are eectric and anetic fieds, which var periodica. (5) Phase: phase of a vibratin partice at an instant is a phsica quantit, which copete express the position and direction of otion, of the partice at that instant with respect to its ean position. In osciator otion the phase of a vibratin partice is the aruent of sine or cosine function invoved to represent the eneraized equation of otion of the vibratin partice. 3

5 asin asin( t 0 ) Here, t 0 = phase of vibratin partice. (i) Initia phase or epoch: It is the phase of a vibratin partice at t = 0. In t 0, when t = 0; 0 here, 0 is the ane of epoch. (ii) Sae phase : wo vibratin partice are said to be in sae phase, if the phase difference between the is an even utipe of or path difference is an even utipe of ( / ) or tie interva is an even utipe of ( / ) because tie period is equivaent to rad or wave enth () (iii) Opposite phase: When the two vibratin partices cross their respective ean positions at the sae tie ovin in opposite directions, then the phase difference between the two vibratin partices is 80o Opposite phase eans the phase difference between the partices is an odd utipe of (sa, 3, 5, 3,,...) 7..) or the path difference is an odd utipe of (sa or the tie interva is an odd utipe of ( / ). (iv) Phase difference: If two partices perfors S.H.M and their equation are a sin( t ) and a sin( t ) hen phase difference ( t ) ( t ) 5. Sipe Haronic Motion. Sipe haronic otion is a specia tpe of periodic otion, in which a partice oves to and fro repeated about a ean position under a restorin force which is awas directed towards the ean position and whose anitude at an instant is direct proportiona to the dispaceent of the partice fro the ean position at that instant. Restorin force Dispaceent of the partice fro ean position. F x F = x Where is nown as force constant. Its S.I. unit is Newton/eter and diension is [M ]. 4

6 6. Dispaceent in S.H.M. he dispaceent of a partice executin S.H.M. at an instant is defined as the distance of partice fro the ean position at that instant. As we now that sipe haronic otion is defined as the projection of unifor circuar otion on an diaeter of circe of reference. If the projection is taen on -axis. hen fro the fiure a sin t a sin t X N O Y a P =t M X a sin n t a sin( t ) Y Where a = Apitude, = Anuar frequenc, t = Instantaneous tie, = ie period, n = Frequenc and = Initia phase of partice If the projection of P is taen on X-axis then equations of S.H.M. can be iven as x acos( t ) x a cos t x acos(n t ) Iportant points (i) a sin t position. (ii) a cos t position. When the tie is noted fro the instant when the vibratin partice is at ean When the tie is noted fro the instant when the vibratin partice is at extree (iii) a sin( t ) When the vibratin partice is phase eadin or ain fro the ean position. (iv) Direction of dispaceent is awas awa fro the equiibriu position, partice either is ovin awa fro or is coin towards the equiibriu position. (v) If t is iven or phase ( ) is iven, we can cacuate the dispaceent of the partice. 5

7 t If 4 (or ) then fro equation a sin t a sin, we et 4 a sin a t Siiar if (or ) then we et 0 7. Veocit in S.H.M. Veocit of the partice executin S.H.M. at an instant, is defined as the tie rate of chane of its dispaceent at that instant. In case of S.H.M. when otion is considered fro the equiibriu position a sin t So v d dt a cos t v a cos t (i) or v a sin t [As sin t = /a] or v a..(ii) Iportant points (i) In S.H.M. veocit is axiu at equiibriu position. Fro equation (i) v a ax when cos t = i.e. = t = 0 Fro equation (ii) v a ax when 0 (ii) In S.H.M. veocit is iniu at extree position. Fro equation (i) v 0 in when cos t t = 0 i.e. Fro equation (ii) v in 0 when = a 6

8 (iii) Direction of veocit is either towards or awa fro ean position dependin on the position of partice. 8. Acceeration in S.H.M. he acceeration of the partice executin S.H.M. at an instant, is defined as the rate of chane of its dv d A ( a cos t) veocit at that instant. So acceeration dt dt A a sin t (i) A (ii) [As a sin t ] Iportant points (i) In S.H.M. as appied. Acceeration is not constant. So equations of transator otion cannot be (ii) In S.H.M. acceeration is axiu at extree position. Fro equation (i) A ax a when sin t axiu t i.e. At 4 t or Fro equation (ii) Aax a when a (iii) In S.H.M. acceeration is iniu at ean position Fro equation (i) A in 0 when sin t 0 t i.e. At t 0 or or t Fro equation (ii) A in 0 when 0 (iv) Acceeration is awas directed towards the ean position and so is awas opposite to dispaceent i.e. A 7

9 9. Coparative Stud of Dispaceent, Veocit and Acceeration. Dispaceent a sin t Veocit Acceeration v a cos t a sin( t ) A a sin t a sin( t ) Fro the above equations and raphs we can concude that. (i) A the three quantities dispaceent, veocit and acceeration show haronic variation with tie havin sae period. +a 0 Dispaceent a = a sin t +a v a +a a 0 0 a ie Veocit v = a cos t Acceeration A = a cos t (ii) he veocit apitude is ties the dispaceent apitude (iii) he acceeration apitude is ties the dispaceent apitude (iv) In S.H.M. the veocit is ahead of dispaceent b a phase ane / (v) In S.H.M. the acceeration is ahead of veocit b a phase ane / (vi) he acceeration is ahead of dispaceent b a phase ane of (vii) Various phsica quantities in S.H.M. at different position: Phsica quantities Equiibriu position ( = 0) Extree Position ( = a) Dispaceent a sin t Miniu (Zero) Maxiu (a) Veocit v a Maxiu (a) Miniu (Zero) Acceeration A Miniu (Zero) Maxiu ( a ) 8

10 0. Ener in S.H.M. A partice executin S.H.M. possesses two tpes of ener: Potentia ener and Kinetic ener () Potentia ener: his is an account of the dispaceent of the partice fro its ean position. he restorin force F aainst which wor has to be done So U dw 0 x Fdx 0 d Potentia Ener U U a sin t [As / ] [As a sin t ] Iportant points (i) Potentia ener axiu and equa to tota ener at extree positions U ax a a When a ; t / ; t / 4 (ii) Potentia ener is iniu at ean position U in 0 When 0 ; t 0 t 0 () Kinetic ener: his is because of the veocit of the partice Kinetic Ener K v K a cos t [As v a cos t ] K ( a ) v a [As ] (i) Kinetic ener is axiu at ean position and equa to tota ener at ean position. K ax a When 0 ; t 0 ; t 0 9

11 Ener (ii) Kinetic ener is iniu at extree position. K in 0 When a ; t / 4, t / (3) ota ener: ota echanica ener = Kinetic ener + Potentia ener E a ( a ) ota ener is not a position function i.e. it awas reains constant. (4) Ener position raph: Kinetic ener (K) ( a ) Ener Potentia Ener (U) = U ota Ener (E) = a = a = 0 K =+ a It is cear fro the raph that (i) Kinetic ener is axiu at ean position and iniu at extree position (ii) Potentia ener is axiu at extree position and iniu at ean position (iii) ota ener awas reains constant. (5) Kinetic Ener Potentia Ener K a cos t a ( cos t) E( cos' t) 4 sin U a t a ( cos t) E( cos' t) 4 Where ' and E a i.e. in S.H.M., inetic ener and potentia ener var periodica with doube the frequenc of S.H.M. (i.e. with tie period ' / ) E Fro the raph we note that potentia ener or inetic ener copetes two vibrations in a tie durin which S.H.M. copetes one vibration. hus the 0 ie 0

12 frequenc of potentia ener or inetic ener doube than that of S.H.M.. ie Period and Frequenc of S.H.M. For S.H.M. restorin force is proportiona to the dispaceent F or F (i) where is a force constant. For S.H.M. acceeration of the bod A (ii) Restorin force on the bod F A (iii) Fro (i) and (iii) ( ) ie period or Frequenc (n) In different tpes of S.H.M. the quantities and wi o on tain different fors and naes. In enera is caed inertia factor and is caed sprin factor. hus Inertia factor Sprin factor or n Sprin factor Inertia factor In inear S.H.M. the sprin factor stands for force per unit dispaceent and inertia factor for ass of the bod executin S.H.M. and in Anuar S.H.M. stands for restorin torque per unit anuar dispaceent and inertia factor for oent of inertia of the bod executin S.H.M. For inear S.H.M. Force/Dispaceent Dispaceent Acceerati on Dispaceent Acceeration A

13 or n Acceerati on Dispacee nt A. Differentia Equation of S.H.M. For S.H.M. (inear) Acceeration (Dispaceent) A or or A d dt or d dt 0 [As ] For anuar S.H.M. c and d 0 dt Where c I [As c = Restorin torque constant and I = Moent of inertia] 3. Sipe Penduu. An idea sipe penduu consists of a heav point ass bod suspended b a weihtess, inextensibe and perfect fexibe strin fro a riid support about which it is free to S osciate.but in reait neither point ass nor weihtess strin exist, so we can never construct a sipe penduu strict accordin to the definition. Let ass of the bob = Lenth of sipe penduu = Dispaceent of ass fro ean position (OP) = x When the bob is dispaced to position P, throuh a sa ane fro the vertica. Restorin force actin on the bob O P sin cos

14 F sin or F (When is sa sin ~ Arc Lenth OP = x = ) or x F F x (Sprin factor) So tie period Inertia factor Sprin factor / Iportant points (i) he period of sipe penduu is independent of apitude as on as its otion is sipe haronic. But if is not sa, sin then otion wi not reain sipe haronic but wi becoe osciator. In this situation if 0 is the apitude of otion. ie period sin (ii) ie period of sipe penduu is aso independent of ass of the bob. his is wh (a) If the soid bob is repaced b a hoow sphere of sae radius but different ass, tie period reains unchaned. (b) If a ir is swinin in a swin and another sits with her, the tie period reains unchaned. (iii) ie period where is the distance between point of suspension and center of ass of bob and is caed effective enth. (a) When a sittin ir on a swinin swin stands up, her center of ass wi o up and so and hence wi decrease. (b) If a hoe is ade at the botto of a hoow sphere fu of water and water coes out sow throuh the hoe and tie period is recorded ti the sphere is ept, initia and fina the center of ass wi be at the center of the sphere. However, as water drains off the sphere, the center of ass of the sste wi first ove down and then wi coe up. Due to this and hence first increase, reaches a axiu and then decreases ti it becoes equa to its initia vaue. 3

15 (iv) If the enth of the penduu is coparabe to the radius of earth then R (a) If R, then R so (b) If R( ) / / R so R inutes and it is the axiu tie period which an osciatin sipe penduu can have (c) If R so R hour (v) If the bob of sipe penduu is suspended b a wire then effective enth of penduu wi increase with the rise of teperature due to which the tie period wi increase. 0 ( ) enth of wire) (If is the rise in teperature, 0 initia enth of wire, = fina 0 0 ( ) / So 0 i.e. (vi) If bob a sipe penduu of densit is ade to osciate in soe fuid of densit (where <) then tie period of sipe penduu ets increased. As thrust wi oppose its weiht therefore ' thrust or V ' V i.e. ' ' ' ' (vii) If a bob of ass carries a positive chare q and penduu is paced in a unifor eectric fied of strenth E directed vertica upwards. In iven condition net down ward acceeration ' qe

16 So qe If the direction of fied is vertica downward then tie period qe (viii) Penduu in a ift: If the penduu is suspended fro the ceiin of the ift. (a) If the ift is at rest or ovin down ward /upward with constant veocit. and n (b) If the ift is ovin upward with constant acceeration a a and n a ie period decreases and frequenc increases (c) If the ift is ovin down ward with constant acceeration a a and n a ie period increase and frequenc decreases (d) If the ift is ovin down ward with acceeration a and n = 0 It eans there wi be no osciation in a penduu. Siiar is the case in a sateite and at the center of earth where effective acceeration becoes zero and penduu wi stop. (ix) he tie period of sipe penduu whose point of suspension ovin horizonta with acceeration a 5

17 ( a / ) and tan ( a / ) a (x) If sipe penduu suspended in a car that is ovin with constant speed v around a circe of radius r. a v r (xi) Second s Penduu: It is that sipe penduu whose tie period of vibrations is two seconds. Puttin = sec and 9.8 / sec in we et = 99.3 c Hence enth of second s penduu is 99.3 c or near eter on earth surface. oon For the oon the enth of the second s penduu wi be /6 eter [As 6 Earth ] (xii) In the absence of resistive force the wor done b a sipe penduu in one copete osciation is zero. (xiii) Wor done in ivin an anuar dispaceent to the penduu fro its ean position. W U ( cos) (xiv) Kinetic ener of the bob at ean position = wor done or potentia ener at extree KE ean ( cos) (xv) Various raph for sipe penduu 6

18 4. Sprin Penduu. A point ass suspended fro a ass ess sprin or paced on a frictioness horizonta pane attached with sprin (fi.) constitutes a inear haronic sprin penduu ie period inertia factor sprin factor n and Frequenc Iportant points (i) ie period of a sprin penduu depends on the ass suspended n or i.e. reater the ass reater wi be the inertia and so esser wi be the frequenc of osciation and reater wi be the tie period. (ii) he tie period depends on the force constant of the sprin or n (iii) ie of a sprin penduu is independent of acceeration due to ravit. hat is wh a coc based on sprin penduu wi eep proper tie everwhere on a hi or oon or in a sateite and tie period of a sprin penduu wi not chane inside a iquid if dapin effects are neected. (iv) If the sprin has a ass M and ass is suspended fro it, effective ass is iven b M e ff 3 So that eff (v) If two asses of ass and are connected b a sprin and ade to osciate on horizonta surface, the reduced ass r is iven b r 7

19 So that r (vi) If a sprin penduu, osciatin in a vertica pane is ade to osciate on a horizonta surface, (or on incined pane) tie period wi reain unchaned. However, equiibriu position for a sprin in a horizonta pain is the position of natura enth of sprin as weiht is baanced b reaction. Whie in case of vertica otion equiibriu position wi be L 0 with 0 L R 0 L + 0 (vii) If the stretch in a vertica oaded sprin is 0 0 then for equiibriu of ass, 0 i.e. So that 0 ie period does not depends on because aon with, o wi aso chane in such a wa that 0 reains constant (viii) Series cobination: If n sprins of different force constant are connected in series havin force constant,, 3... respective then eff 3... If a sprin have sae sprin constant then e ff n 3 Series cobination 8

20 (ix) Parae cobination: If the sprins are connected in parae then eff 3. If a sprin have sae sprin constant then 3 e ff n (x) If the sprin of force constant is divided in to n equa parts then sprin constant of each part wi becoe n and if these n parts connected in parae then Parae cobination e ff n (xi) he sprin constant is inverse proportiona to the sprin enth. As Extension Lenth of sprin hat eans if the enth of sprin is haved then its force constant becoes doube. (xii) When a sprin of enth is cut in two pieces of enth and such that n. ( n ) If the constant of a sprin is then Sprin constant of first part n Sprin constant of second part ( n ) and ratio of sprin constant n 9

21 5. Various Foruae of S.H.M. S.H.M. of a iquid in U tube If a iquid of densit contained in a vertica U tube perfors S.H.M. in its two ibs. hen tie period L h where L = ota enth of iquid coun, h = Heiht of undisturbed iquid in each ib (L=h) S.H.M. of a bar anet in a anetic fied I MB I = Moent of inertia of anet M = Manetic oent of anet B = Manetic fied intensit h S F F N S.H.M. of a foatin cinder If is the enth of cinder dippin in iquid then tie period S.H.M. of ba in the nec of an air chaber A V E = ass of the ba V = voue of air- chaber S.H.M. of a sa ba roin down in hei-spherica bow R r A = area of cross section of nec E = Bu oduus for Air S.H.M. of a bod suspended fro a wire L YA = ass of the bod L 0

22 R = radius of the bow r =radius of the ba S.H.M. of a piston in a cinder R L = enth of the wire Y = oun s oduus of wire A = area of cross section of wire S.H.M of a cubica boc Mh PA M L M = ass of the piston A = area of cross section h = heiht of cinder P = pressure in a cinder M = ass of the boc L = enth of side of cube = oduus of riidit L S.H.M. of a bod in a tunne du aon an chord of earth S.H.M. of bod in the tunne du aon the diaeter of earth R = 84.6 inutes R R = 84.6 inutes R R = radius of the earth = 6400 = acceeration due to ravit = 9.8/s at earth s surface S.H.M. of a conica penduu O S.H.M. of L-C circuit L cos L = enth of strin L LC L = coefficient of sef-inductance C = capacit of condenser = ane of strin fro the vertica = acceeration due to ravit

23 6. Iportant Facts and Foruae. () When a bod is suspended fro two iht sprins separate. he tie period of vertica osciations are and respective. 4 and 4 When these two sprins are connected in series and the sae ass is attached at ower end and then for series cobination B substitutin the vaues of, ie period of the sste When these two sprins are connected in parae and the sae ass is attached at ower end and then for parae cobination B substitutin the vaues of, ie period of the sste () he penduu coc runs sow due to increase in its tie period whereas it becoes fast due to decrease in tie period. (3) If infinite sprin with force constant,,4,8... effective force constant of the sprin wi be /. respective are connected in series. he (4) If et a sin t and b cos t are two S.H.M. then b the superiposition of these two S.H.M. we a sin t b cos t A sin( t ) his is aso the equation of S.H.M.

24 Where A a b and tan ( b / a) (5) If a partice perfors S.H.M. whose veocit is v at a x at distance x distance fro ean position and veocit v v x v x ; x v x v a v x v v v x ; v ax v x x v x x 7. Free, Daped, Forced and Maintained Osciation. () Free osciation (i) he osciation of a partice with fundaenta frequenc under the infuence of restorin force are defined as free osciations (ii) he apitude, frequenc and ener of osciation reains constant (iii) Frequenc of free osciation is caed natura frequenc because it depends upon the nature and structure of the bod. () Daped osciation +a 0 a t (i) he osciation of a bod whose apitude oes on decreasin with tie are defined as daped osciation (ii) In these osciation the apitude of osciation decreases exponentia due to dapin forces ie frictiona force, viscous force, hstersis etc. (iii) Due to decrease in apitude the ener of the osciator aso oes on decreasin exponentia (3) Forced osciation + A 0 A t (i) he osciation in which a bod osciates under the infuence of an externa periodic force are nown as forced osciation (ii) he apitude of osciator decrease due to dapin forces but on account of the ener ained fro the externa source it reains constant. 3

25 (iii) Resonance: When the frequenc of externa force is equa to the natura frequenc of the osciator. hen this state is nown as the state of resonance. And this frequenc is nown as resonant frequenc. (4) Maintained osciation he osciation in which the oss of osciator is copensated b the suppin ener fro an externa source are nown as aintained osciation. 4

OSCILLATIONS. Syllabus :

OSCILLATIONS. Syllabus : Einstein Casses, Unit No. 0, 0, Vardhan Rin Road Paza, Vias Puri Extn., Outer Rin Road New Dehi 0 08, Ph. : 96905, 857 PO OSCILLATIONS Sabus : Periodic otion period, dispaceent as a function of tie. Period