14 - OSCILLATIONS Page 1

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1 14 - OSCILLATIONS Page Perioic an Osciator otion Motion of a sste at reguar interva of tie on a efinite path about a efinite point is known as a perioic otion, e.g., unifor circuar otion of a partice. To an fro otion of a sste on a inear path is cae an osciator otion, e.g., otion of the bob of a sipe penuu. 14. Sipe haronic otion This is the sipest tpe of perioic otion which can be unerstoo b consiering the foowing exape. Suppose a bo of ass is suspene at the ower en of a assess eastic spring obeing Hooke s aw which is fixe to a rigi support in the vertica position. The spring eongates b ength Δ an attains equiibriu as shown in Fig. ( b ) Here two forces act on the bo. ( 1 ) Its weight, g, ownwars an ( ) the restoring force eveope in the spring, k Δ, upwars, where k = force constant of the spring. For equiibriu, g = k Δ ( 1 ) The spring is constraine to ove in the vertica irection on. Now, suppose the bo is given soe energ in its equiibriu conition an it unergoes ispaceent in the upwar irection as shown in Fig. ( c ). Two forces act on the bo in this ispace conition aso. ( 1 ) Its weight, g, ownwars an ( ) the restoring force eveope in the spring, k ( Δ - ), upwars. The resutant force acting on the bo in this conition is given b F = - g + k ( Δ - ) ( ) Fro equations ( 1 ) an ( ), F = - k

2 14 - OSCILLATIONS Page Dispaceent: The istance of the bo at an instant fro the equiibriu point is known as its ispaceent at that instant. The ispaceents aong the positive Y-axis are taken as positive an those on the negative Y-axis are taken as negative. In the equation, F = - k, F is negative when is positive an vice versa. Thus, the resutant force acting on the bo is proportiona to the ispaceent an is irecte opposite to the ispaceent, i.e., towars the equiibriu point. Differentia equation of sipe haronic otion ( SHM ) Accoring to Newton s secon aw of otion, v F = a = = = - k ( for spring-tpe osciator as above ) k = - = - k ω0 ( taking = ω0 ) + ω 0 = 0 This is the ifferentia equation of SHM. To obtain the soution of the above ifferentia equation is to obtain as a function of t such that on twice ifferentiating w.r.t. t, we get back the sae function with a negative sign. Both the sine an the cosine functions possess such a propert. Hence, taking = A 1 sin ω 0 t + A cos ω 0 t as a possibe soution an ifferentiating twice w.r.t. t, = A 1 ω 0 cos ω 0 t - A ω 0 sin ω 0 t an = - A 1 ω 0 sin ω0 t - A ω 0 cos ω0 t = - ω 0 ( A1 sin ω 0 t + A cos ω 0 t ) = - ω 0 Thus, t = A 1 sin ω 0 t + A cos ω 0 t is the soution of the ifferentia equation an is known as its genera soution, where t is the ispaceent of sipe haronic osciator ( SHO ) at tie t. Taking A 1 = A cos φ an A = A sin φ, t = A cos φ sin ω 0 t + A sin φ cos ω 0 t t = A sin ( ω 0 t + φ ) is the soution of the ifferentia equation.

3 14 - OSCILLATIONS Page 3 A an φ are the constants of the equation whose vaues epen upon the initia position an initia veocit of the sste. The equation gives ispaceent as a sinusoia function which is perioic. Hence, the otion of the object represente b this equation is perioic on a inear path about = 0 between = - A an = A. Such a otion is known as sipe haronic otion ( SHM ). Definition of SHM: The perioic otion of a bo about a fixe point, on a inear path, uner the infuence of the force acting towars the fixe point an proportiona to the ispaceent of the bo fro the fixe point, is cae a sipe haronic otion. The bo perforing SHM is known as a sipe haronic osciator ( SHO ) Apitue, Perio, Frequenc, Anguar frequenc, Phase Apitue: The axiu ispaceent of the bo executing SHM on either sie of the ean position is cae the apitue of the SHO. Phase: θ = ω 0 t + φ is the phase at tie t of SHO perforing SHM accoring to the equation t = A sin ( ω 0 t + φ ). At t = 0, θ = φ which is known as initia phase, epoch or phase constant of the given SHM. The position an irection of otion of SHO at an tie can be known fro its phase. Perio: Dispaceent of an SHO at instant t is t = A sin ( ω 0 t + φ ). As the perio of sine function is π raian, we have t = A sin ( ω 0 t + φ + π ) = A sin T = π ω0 ω 0 t π + + φ ω 0 is the perio or the tie taken to copete one osciation b the osciator. Putting ω 0 = k, T = π. k This is the perio for an SHM. In the case of spring-bock sste, heavier the ass ore the perio an sower the osciations. Aso, if the spring is har, its force constant k is arge, the perio is ess an osciations are faster. Frequenc an Anguar frequenc: Obvious, f 0 = 1 T The nuber of osciations perfore b the osciator in 1 secon is known as the frequenc f 0 of the osciator. Its unit is s - 1 or hertz ( Hz ) in honour of the scientist Hertz. ω 0 = π f 0 = π T is the anguar frequenc of the osciator. Its unit is ra / s.

4 14.4 Unifor circuar otion an SHM 14 - OSCILLATIONS Page 4 Consier a partice oving with a constant anguar spee ω 0 in an anticockwise irection on a circuar path having centre O an raius A as shown in the figure. At tie t = 0, its anguar position w.r.t. the reference ine OX is POX = φ. At tie t = t, having unergone anguar ispaceent ω 0 t reaching Q fro P, its anguar position is QOX = ω 0 t + φ. The co-orinates of point Q are x = A cos ( ω 0 t + φ ) an ( 1 ) = A sin ( ω 0 t + φ ). ( ) As the partice oves on the circuar path, its feet of perpenicuars on X- an Y- axes ove as per the equations ( 1 ) an ( ) an their otion is sipe haronic. Thus, a given SHM can be escribe as the projecte otion of a partice, known as the reference partice, perforing an appropriate unifor circuar otion on the iaeter of the circe known as the reference circe. The raius of the reference circe is equa to the apitue of the corresponing SHO an the anguar spee of the reference partice is equa to the anguar frequenc of the SHO. Aso, the anguar position of the reference partice w.r.t. the reference ine at an tie is equa to the phase of the SHO at that tie. Cobining two SHMs with phase ifference of π / an sae apitue resuts in unifor circuar otion an if the apitues are ifferent, the otion is on an eiptica path. Cobining SHMs in ifferent was, ifferent tpes of otion can be obtaine Dispaceent, veocit an acceeration of SHO Dispaceent: The equation for the ispaceent of SHO is = A sin ( ω 0 t + φ ). Veocit: Differentiating with respect to tie, we get veocit, v = = A ω0 cos ( ω 0 t + φ ) ( 1 ) 1 0 A 0 = ± A ω 0 - sin ( ω t + φ ) = ± ω 0 - A sin ( ω t + φ ) = ± ω 0 A -

5 14 - OSCILLATIONS Page 5 Veocit of SHO, v, is positive when it is oving aong positive -irection an negative when it is oving aong negative -irection. At = 0 ( equiibriu point ), v = ± A ω 0 ( which is axiu veocit ). At = ± A ( en points ), v = 0. The veocit of SHO an its corresponing reference partice are the sae ever tie the SHO is at the equiibriu point. Acceeration: Differentiating equation ( 1 ) with respect to tie, we get acceeration, a = v = = - A ω 0 sin ( ω0 t + φ ) = - ω 0 At = 0 ( equiibriu point ), a = 0. At = ± A ( en points ), a = ω 0 A. The acceeration of SHO an its corresponing reference partice are the sae ever tie the SHO is at either of the en points. Note: The veocit of the SHO can aso be foun b taking the coponent of inear veocit Aω 0 of the reference partice in the corresponing irection ( here Y-axis ), i.e., A ω 0 cos θ as shown in the figure. Siiar the coponent of acceeration A ω 0 of the reference partice in the corresponing irection ( here Y-axis ) is A ω 0 sin θ which is the agnitue of acceeration of the SHO Sipe penuu A sste of a sa assive bo suspene b a ight, inextensibe string fro a rigi ( fixe ) support an capabe of osciating in one vertica pane on is known as a sipe penuu. Mass of the penuu,, is suppose to be concentrate at the centre of the suspene bo cae bob of the penuu ( figure on the next page ). The istance of the centre of the bob fro the point of suspension A is cae the ength ( ) of the sipe penuu. At soe instant, the bob of the penuu is at B an the string akes an ange θ with the vertica. The penuu osciates on the circuar arc of raius in a vertica pane as shown in the figure.

6 Two forces act on the bob of the penuu. ( 1 ) Weight of the bob = g, in the ownwar irection an ( ) tension in the string T, in the irection BA OSCILLATIONS Page 6 The torque about A ue to T is zero as its ine of action passes through A. The torque ue to weight, g, about A is τ = g = - g sin θ But, τ = I α = ω α an α = 0 θ = θ = - g sin θ ( 1 ) For sa θ ( in raian ), inear ispaceent of the bob on the curve path is x an sin θ θ = x Putting this vaue of sin θ in equation ( 1 ), we get ( x / ) = - g x x = - g x. This is the ifferentia equation of SHM. g 4 π = ω0 = T T = π g This is the expression of the perio of the sipe penuu. Its vaue oes not epen on the ass of the bob of the penuu. The perio of the spring-bock tpe of SHO oes not change when taken to a ifferent panet as the vaues of an k appearing in the expression of its perio o not change. The perio of sipe penuu increases on a panet where the vaue of g is ess an the penuu cock taken there oses tie, whereas its perio ecreases on the panet where the vaue of g is ore an the penuu cock gains tie when taken to that panet.

7 14 - OSCILLATIONS Page Dape osciations SHM is an iea situation. In fact, there is awas a resistive force offere b the eiu. e.g., air resistance in case of osciating penuu an interna frictiona forces as in the case of a vibrating tuning fork. Energ ost in oing work against the resistive an frictiona forces is ost issipate in 1 the for of heat. The echanica energ of SHO is E = k A, where A is the apitue of its osciations. This shows that the apitue of the osciator ecreases graua ue to issipation of its energ. Such osciations are cae ape osciations. It is experienta foun that the resistive force acting on the osciator opposing its otion is irect proportiona to the veocit for sa veocities. F v = - bv, where b is a constant cae the aping coefficient. Its unit is N-s /. Two forces act on the ape osciator. ( 1 ) Restoring force = - k an ( ) resistive force = - bv = - b Accoring to Newton s secon aw, + b + = - k - b k = 0 ( 1 ) This is the ifferentia equation for ape osciations. Its soution b t is t = A e - sin ( ω t + φ ), where anguar frequenc of ape osciations, for k > b ω = k - b Here, A an φ are the constants of the soution an their vaues epen on the initia conitions. b t A t = A e - is the apitue of the ape osciator at tie t which ecreases exponentia with tie. The graph of ispaceent, t tie t is shown in the figure where the broken ines show the ecrease in the apitue with tie.

8 14 - OSCILLATIONS Page 8 Putting A t = A b t e -, the expression for echanica energ of ape osciator is E t = 1 b t k A e - b for sa aping << 1 which shows that the echanica k energ aso ecreases exponentia with tie Natura osciations, Force osciations an Resonance The osciations of an osciator in the absence of resistive forces are known as natura osciations an their frequenc as natura frequenc ( f 0 ), e.g., the natura anguar frequenc of the sipe penuu is ω 0 = g. An osciator can have ore than one natura frequenc. In reait, the apitues of osciations ecrease exponentia with tie ue to aping forces. To sustain natura osciations, soe externa perioic force ust be appie to the osciator. The osciations uner the infuence of soe externa perioic force are known as force osciations. The ifferentia equation of force osciations uner the externa perioic force, F 0 sin ωt, where ω is the frequenc of the externa force is given b = - k - b + F0 sin ωt + b + k F = 0 sin ωt + r + ω0 = a0 sin ωt ( putting b k F = r, = ω0 an 0 = a0 ). This is the ifferentia equation of force osciation in the presence of aping an its soution is given as = A sin ( ω t + α ), where A = 1 [ ] ( ω 0 - ω a 0 ) + r ω an α = tan - 1 ω 0. v0 The apitue of the osciator is axiu when the vaue of ( ω 0 - ω ) + r ω is iniu. It can be prove atheatica that this iniu vaue is reache when r ω = ω 0 -. This phenoenon is known as resonance. The vaue of ω for which resonance occurs an the apitue becoes axiu is known as the resonant anguar frequenc.

9 14 - OSCILLATIONS Page 9 ω The curves for apitue for ifferent vaues of b are shown in the figure on the ω 0 next page. The apitue becoes infinite for b = 0 which is an iea conition. For ifferent curves, the apitue is not axiu ω when = 1, but it is axiu when it ω 0 is cose to 1 for sa aping. Mechanica sstes a have ore than one resonant frequencies. When the frequenc of the externa perioic force is cose to the natura frequenc, the sste osciates with a ver arge apitue an it a break or coapse. This is the reason wh soiers are instructe to arch out of pace on the brige. Whie esigning a brige, care is taken so that its natura frequenc is not cose to the frequenc of the externa force ue to gusts of win Coupe osciations The figure shows two penuus connecte b an eastic spring. Obvious, the cannot osciate inepenent of each other. The are cae coupe osciators ( ore appropriate coupe penua ) an their osciations are known as coupe osciations. The constituent partices of sois aso unergo coupe osciations. Osciations of coupe osciators are copex an not awas sipe haronic, i.e., their ispaceents x 1 an x cannot be expresse in the for of sine or cosine functions. But b suitabe transforation of the co-orinate sste, the can be expresse in the for of equations of SHM as uner. X 1 = A sin ( ω 1 t + φ 1 ) ( 1 ) an X = B sin ( ω t + φ ) ( ), where X 1 = x 1 + x an X = x 1 - x. ω 1 an ω are nora frequencies an osciations given b X 1 an X with these frequencies are the nora oes of vibrations of the coupe osciators. This osciator has two nora oes as on two co-orinates are present. With proper seection of initia conitions, the coupe osciator can be osciate in an one of these two oes. If at t = 0, x 1 = x, i.e., both the osciators are given equa ispaceents in the sae irection, then B = 0 fro equation ( ).

10 14 - OSCILLATIONS Page 10 g The coupe osciator wi osciate with anguar frequenc ω 1 = accoring to equation ( 1 ). As shown in the figure ( previous page ), both the osciators unergo equa ispaceents in the sae irection in the sae tie. Hence the ength of the spring oes not change. So in this oe, the osciators osciate inepenent of each other as if the spring is not present. Next, if at t = 0, x 1 = - x, i.e., both the osciators are given equa ispaceents in utua opposite irections an reease, then A = 0 fro equation ( 1 ). The coupe osciator wi osciate with anguar frequenc ω = g + k accoring to equation ( ). Both these tpes of osciations are the nora oes of osciations of the given coupe osciator. If the initia conitions were ifferent fro the above two conitions, then the osciations of each osciator wou be copex. However, in such a situation, the ispaceents of both the osciators can be represente as a inear cobination of the above two equations as the function of tie.

OSCILLATIONS. dt x = (1) Where = k m

OSCILLATIONS. dt x = (1) Where = k m OSCILLATIONS Periodic Motion. Any otion, which repeats itsef at reguar interva of tie, is caed a periodic otion. Eg: 1) Rotation of earth around sun. 2) Vibrations of a sipe penduu. 3) Rotation of eectron