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

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1 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 around nuceus. Osciatory otion: If the periodic otion takes pace to and fro aong a straight ine or aong the arc of a circe, the otion is said to be an osciatory otion. Eg: 1) Osciations of a sipe penduu. 2) Vibrations of the prongs of a tuning fork. Sipe Haronic otion:- It is a specia type of periodic otion, in which a partice oves to and fro repeatedy about a ean (i.e., equiibriu) position and the agnitude of force acting on the partice at any instant is directy proportiona to the dispaceent of the partice fro the ean position at that instant. F α - x i.e. F = kx where k- force constant This is caed force aw of SHM. sipe haronic otion is the otion in which the restoring force is proportiona to dispaceent fro the ean position and opposes its increase. Differentia equation of a sipe haronic otion:- A otion in which the restoring force or acceeration is directy proportiona to the dispaceent & acts in a direction opposite to dispaceent. We know that by Newton s II aw, F= a, F = d2 x dt (a) Aso F = - kx (b) On soving the eqn (1), we get where d2 x dt 2 = - kx, d 2 x dt 2 + k x =0 d 2 x dt x = (1) Where = k x= A Cos (t +) (or) x = A Sin (t +) A apitude of SHM is caed anguar frequency. anguar frequency (Unit rad/s) ; (t +) phase of SHM and - phase constant. Characteristics of SHM:- (i).apitude (A):- It is the agnitude of the axiu dispaceent of the partice.it is denoted by A.

2 (ii).phase (t +):- It is the quantity that deterines the state of otion of the partice. It is tie dependent. We have, phase = (t +) ; *At t =0, phase = (phase constant) (iii).anguar frequency ( ) :- (iv).tie period (T):- [ - initia state of the partice] = 2π T rsd/s. or = 2π ν [ But 1 T =ν ] The tie interva after which the osciation repeats itsef is caed tie period. T= 2π where = k Sipe Haronic otion:- It is the sipest for of osciatory otion Motion is SHM When or (i).restoring force α x (ii). At any point this force is directed towards its ean position (iii). If the dispaceent of the partice fro the origin varies with tie as, x (t) = A Cos (t +) or y = A Sin (t +) Sipe Haronic otion and unifor circuar otion:- {Consider a partice which oves aong a circuar path. When the partices oves fro P1 to P2 to P3 etc., then the partices projection aong X axis wi ove fro P1 1 to P2 1 to P3 1 etc., When the partice started oving fro the upper haf of the circe then the projection started oving towards eft and when the partice started oving in the ower haf of the circe then the projection started oving towards right}. Q1. Show that the projection of unifor circuar otion on any diaeter of the circe is S.H.M. Consider a partice P oving round a circe with center O and radius A with unifor anguar speed. At tie t=0, et the partice be at P0 < P0 OB = After tie t= t second, et the partice be at P < P O P0 = t M is the foot of the perpendicuar drawn fro P to the diaeter AB. Fro the right ange triange OMP, we get

3 Cos (t +) = OM OP OM= OP Cos (t +) X = A Cos (t +) (1) Siiary we get {where Y= A Sin (t +) (2) radius of circuar path = Apitude(A) of S.H.M} Eqn.(1) &(2) are siiar to the equation of S.H.M. Therefore it shows that the projection of unifor circuar otion on any diaeter is S.H.M. At t=0 if the partice is at B then = 0, then eqn (1) &(2) reduces to x = A Cos t Expression for veocity(v):- Dispaceent of an SHM is given by & y = A Sin t y= A Sin (t +) Veocity, v = dy dt = A Cos (t +) =A 1 Sin2 (t + ) = A 2 A 2 Sin 2 (t + ) V = A 2 y 2 Specia Cases:- At the ean position y=0, Vax = A At the extree position, y = A, v = 0 Acceeration (a):- a= dv dt = Specia Cases:- d (A Cos (t +)) dt At the ean position y=0, a = 0 At the extree position, y = A, a = 2 A Graphica representation of the variation of dispaceent, veocity and acceeration with tie. = - A 2 sin (t +) = -- 2 y.

4 1. On an average a huan heart is found to beat 72 ties in a inute. Cacuate the period and frequency of the heartbeat. 2.The dispaceent in of a haronic osciator is represented by x=0.25 Cos (6280t - π 3 ) where t is in second. Find (i) Apitude, (2) anguar frequency (3) period and (4) initia phase. Energy of SHM:- An SHM possess both potentia and kinetic energy. Potentia energy is due to the dispaceent against the restoring force and kinetic energy is due its otion. Tota energy of SHM = K.E + P.E = a constant. Kinetic Energy: Ek Let be the ass of the partice executing SHM. Let v be the veocity at any instant. K.E = ½ v 2 = ½ 2 (A 2 x 2 ) we know v = A 2 x 2 Specia cases :- (1). At the ean position x=0, K.E = ½ 2 A 2. (2).At the extree position, x =A, K.E = 0. Potentia Energy: (Ep) - P.E is work required to take the partices against the restoring force. Restoring force F = k x F= - k x Work done to dispace through a sa distance dx is given by, W = F dx. Tota Work done to take the partice fro the ean position 0 to a dispaceent x, x x W= F dx = k x dx = ½ k x 2 = ½ 2 x Specia Cases: (1).At the ean position, x = 0, P.E = 0. (2).Extree position x =A, P.E ax = ½ 2 A 2. Tota Energy = P.E + K.E = ½ 2 (A 2 x 2 ) + ½ 2 x 2. = ½ 2 A 2 or = ½ k A 2 Graphica representation of Tota Energy of SHM

5 3. A partice in SHM crosses the ean position with a veocity 1.6/s. What wi be its veocity when it is haf way towards one of the extreities of otion? 4. The axiu acceeration of a partice perforing SHM is 5/s 2 and its period is 6.28 sec. Find (a) Apitude and (b).veocity when the dispaceent is 4 and (c) acceeration when the veocity is 4 /s. 5. A partice executes SHM of apitude 25 c and tie period 3 sec. what is the iniu tie required for the partice to ove between two points 1.25c, on either side of the ean position? 6. A haronic osciator has a tota energy E. (a).deterine the K.E & P.E when the dispaceent is one haf the apitude. (b).for what vaue of the dispaceent the K.E & P.E are equa. 7. A ass of 1 kg ercury is executing SHM. Its dispaceent is given by x= 0.6 Cos (100 t+ ¼) c. what is the axiu Kinetic energy. 8.A bock whose ass is 1 kg is fastened to a spring of spring constant 50 N/. The bock is pued to a distance x=10c fro its equiibriu position at x=0, on a frictioness surface fro rest at t=0. Cacuate the kinetic energy, potentia energy and tota energy of the bock when it is 5c away fro the ean position. 1. Sipe penduu SOME EXAMPLES OF SHM A sipe penduu is a bob suspended by weightess and inextensibe string. When the bob is taken to one side and reease, it executes SHM. Let be the ass of the bob & be the ength of the penduu. [The distance between the fixed support (O) and the centre of gravity of the bob is known as ength of the penduu]. Let the bob be dispaced fro the equiibriu position A to the position B through a very sa ange (θ). The force due to the weight of the bob acts verticay downwards. g can be resoved in to two coponents (i).g cosθ radiay aong OB and is baanced by the tension T of the string. (ii). But the coponents g sinθ acts towards the equiibriu position. So it is the restoring force. F = - g Sinθ = - g θ (Since θ is very sa Sinθ θ) F= - g x = - g Acceeration of the bob at B is, x But θ = x a= Force ass = - g x = - ( g ) x (1)

6 a α x Since g & are constants. i.e. acceeration is proportiona to the dispaceents. Hence the osciation is sipe haronic otion. We have a = -- 2 y (2) Coparing eqn(1)& (2) we get, 2 = g = g But T = 2π T = 2 π g Thus tie period of a sipe penduu depends on (i).ength of the penduu & (ii).acceeration due to gravity. **A sipe penduu whose period of osciation is two seconds is caed a seconds penduu. (i.e. T = 2 s) Osciations of a spring. Consider a body of ass attached to a spring of sprint constant k. The other end A is fixed to a rigid support. When the body is dispaced through a sa distance x and reeased, it executes SHM. (a).vibration of spring in vertica direction:- Restoring force = - ky When the body is just reeased then acceeration is, a = F = - ky = (- k ) y (1) Since acceeration is directy proportiona to dispaceent. Therefore it is SHM. We have acceeration for SHM a= -- 2 y (2) Coparing eqn(1) & (2), we get 2 = k or = k T= 2π = 2π k ***Spring constant or force constant of a spring (k) is the force required to produce unit eongation in the spring. (F= k y, when y =1 then F =k) Unit of k is N/. Osciation oaded spring cobination:- (i).two springs in parae or (Body connected between two springs(series)):-

7 The ass is suspended by two spring constants k1 and k2. If k is the effective spring constant of the cobination, then k = k1 + k2 T= 2π = 2π k 1 + k 2 If k1 = k2 = k then 2π 2 k (ii).two springs in series The ass is suspended by two spring constants k1 and k2. If k is the effective spring constant of the cobination, then 1 k = 1 k1 + 1 k2 T= 2π = 2π (k1+ k2 ) k1 k2 Daped SHM:- where k = k1 k2 k1+ k2 Daped osciations are periodic osciations whose apitude decreases graduay with tie. It is due to the daped force (friction/viscosity). Daping force (frictiona/viscous force) is proportiona to the veocity of the object. Fd α ϒ v For daped haronic osciator it is seen that: (i). the apitude of osciation decreases exponentiay and where ϒ daping coefficient of the ediu. (ii). the daping force increases the tie period of osciations of the osciator. The free and daped osciations are represented graphicay as shown: Free osciations:- When a body osciates in the absence of externa forces (e.g. friction/ viscosity ). The osciations are said to be free osciations.. It is denoted by ν0 then ν0 = 1 2π k ν = 1 T

Forced osciation:- When an externa periodic force is appied to a daped haronic osciator, the osciator wi vibrate with frequency of appied periodic force.

Resonance:- The phenoenon of producing osciatory otion in a syste by the infuence of an externa periodic force having the sae frequency as that of the natura

Whie crossing a suspension bridge, the sodiers are asked to break steps-when the frequency of arching coincides with the natura frequency of vibration of the

8 Forced osciation:- When an externa periodic force is appied to a daped haronic osciator, the osciator wi vibrate with frequency of appied periodic force. This type of osciation is caed forced osciations. Resonance:- The phenoenon of producing osciatory otion in a syste by the infuence of an externa periodic force having the sae frequency as that of the natura frequency of the syste is caed resonance. For resonance, Frequency of the externa periodic force = natura frequency of the syste e.g: 1).Whie crossing a suspension bridge, the sodiers are asked to break steps-when the frequency of arching coincides with the natura frequency of vibration of the bridge then the bridge osciates with axiu apitude to such an extent that the bridge ay even coapse. This condition is caed Resonance. 2). A vibrating tuning fork when paced near the outh of a particuar ength of air coun produces a oud sound due to resonance. Graphica variation of apitude with driving frequency:-

14 - OSCILLATIONS Page 1

14 - OSCILLATIONS Page 1 14.1 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.