OSCILLATIONS. Syllabus :

Size: px

Start display at page:

Download "OSCILLATIONS. Syllabus :"

Christina Gordon
5 years ago
Views:

1 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 functions. Sipe haronic otion (S.H.M.) and its equation; phase; osciations of a sprin - restorin force and force constant; ener in S.H.M. - inetic and potentia eneries; Sipe penduu - derivation of expression for its tie period; Free, forced and daped osciations, resonance.

2 Einstein Casses, Unit No. 0, 0, Vardhan Rin Road Paza, Vias Puri Extn., Outer Rin Road New Dehi 0 08, Ph. : 96905, 857 PO CONCEPTS C C Sipe Haronic Motion If a partice oves to and fro about a fixed point (equiibriu position) under the appication of a force or torque (caed restorin force or torque) which is direct proportiona to the dispaceent (inear or anuar), directed towards the fixed point, is caed sipe haronic otion. Equation of SHM The necessar and sufficient condition for the otion to be sipe haronic (inear) is that the force shoud be direct proportiona to the dispaceent i.e. F x d x d x F = x or x dt or x 0 with dt The soution of the above differentia equation representin SHM is iven b : x = A sin (t + ) where A is the apitude of the otion, is the anuar (circuar) frequenc = /T (T = tie period) and is phase constant. Practice Probes : d dt. The equation of S.H.M. of a partice is 0, where is a positive constant. The tie period of otion is iven b. A partice is executin S.H.M. of period 4 s. Then the tie taen b it to ove fro the extree position to haf the apitude is / s / s /4 s 4/ s [Answers : () a () b] d C Veocit of SHM is iven b : v Acos( t ) or v dt A x The axiu and iniu veocit of the partice are v ax = A and V in = 0. Practice Probes :. A partice is vibratin in S.H.M. If its veocities are v and v when the dispaceents fro the ean position are and, respective, then its tie period is v v v. A partice is executin S.H.M. Then the raph of veocit as a function of dispaceent is v straiht ine circe eipse hperboa [Answers : () d () c] v v v v d C4 The acceeration in SHM is a. Thus a ax = A and a in = 0 dt Practice Probes :. A partice is executin S.H.M. Then the raph of acceeration as a function of dispaceent is straiht ine circe eipse hperboa [Answers : () a]

3 Einstein Casses, Unit No. 0, 0, Vardhan Rin Road Paza, Vias Puri Extn., Outer Rin Road New Dehi 0 08, Ph. : 96905, 857 PO C5 Ener in SHM (i) Kinetic Ener (K) v (A x ) (ii) Potentia Ener (U) x (iii) Tota ener (E) A = constant Graph for the variation of K, U and E with the position x is iven b : Practice Probes :. A bod executes S.H.M. with an apitude A. Its ener is haf inetic and haf potentia when the dispaceent is A/ A/ A/ A/. When the potentia ener of a partice executin sipe haronic otion is one-fourth of its axiu vaue durin the osciation, the dispaceent of the partice fro the equiibriu position in ters of its apitude a is a/4 a/ a/ a/ [Answers : () c () c] C6 Sipe Penduu The tie period of a sipe penduu of enth L is iven b If enth of the penduu is are, then tie period is iven b : T where R = Radius of the earth L R R so if L, T inutes L. A second penduu is the sipe penduu havin a tie period of s. Practice Probes :. The tie period of a sipe penduu is T. If its enth is increased b %, the new tie period becoes 0.98 T.0 T 0.99 T.0 T

4 Einstein Casses, Unit No. 0, 0, Vardhan Rin Road Paza, Vias Puri Extn., Outer Rin Road New Dehi 0 08, Ph. : 96905, 857 PO 4. The enth of a sipe penduu is increased b 44%. The percentae increase in its tie period wi be 44% % 0% %. A ir is swinin on a swin in the sittin position. How wi the period of swin be affected if she stands up? The period wi now be shorter The period wi now be oner The period wi reain unchaned The period a becoe oner or shorter dependin upon the heiht of the ir. 4. For a sipe penduu the raph between enth and tie period wi be a hperboa paraboa straiht ine none of these [Answers : () d () c () a (4) b] C7 Sprin - Boc Sste The tie period of SHM for a boc of ass connected b a sprin of sprin constant as shown in fiure is iven b T Practice Probes :. The bodies M and N of equa asses are suspended fro two separate assess sprins of constants and, respective. If the two osciate vertica such that their axiu veocities are equa, the ratio of the apitude of M to that of N is / / / /. The vertica extension in a iht sprin b a weiht of, in equiibriu, is 9.8 c. The period of osciation of the sprin, in seconds, wi be A assess sprin, havin force constant, osciates with a frequenc n when a ass is suspended fro it. The sprin is cut into two equa haves and a ass is suspended fro it. The frequenc of osciation wi now be n n n/ n [Answers : () d () a () a] C8 Phsica Penduu The tie period of a phsica penduu is iven b : T I Md I : oent of inertia about the rotationa axis passin throuh point of suspension d : distance of the center of ass fro the point of suspension M : tota ass of the bod

5 Einstein Casses, Unit No. 0, 0, Vardhan Rin Road Paza, Vias Puri Extn., Outer Rin Road New Dehi 0 08, Ph. : 96905, 857 Practice Probes : PO 5. A rod of ass and enth L is suspended fro one of the end point of the rod in vertica pane. The tie period of the rod for sa osciation is iven b L L L L 4 [Answers : () a] C9 C0 Daped Osciation Osciations under the infuence of frictiona force are caed daped osciation. The apitude and hence ener decreases with tie exponentia and eventua the osciator coes to rest. Forced Osciation and Resonance The osciations of a sste under the infuence of an externa periodic force are caed forced osciations. The apitude of these osciations reains constant. If the frequenc of the externa appied force is equa to the natura frequenc of the osciator, resonance is said to occur. If dapin is sa, the apitude of resonant osciations wi becoe ver are. At resonance, the osciator absorbs axiu ener suppied b the externa force.

6 Einstein Casses, Unit No. 0, 0, Vardhan Rin Road Paza, Vias Puri Extn., Outer Rin Road New Dehi 0 08, Ph. : 96905, 857 PO 6 INITIAL STEP EXERCISE. A etaic sphere is fied with water and hun b a on thread. It is ade to osciate. If there is a sa hoe in the botto throuh which water sow fows out, the tie period wi o on increasin ti the sphere is ept o on decreasin ti the sphere is ept reain unchaned throuhout first increase, then it wi decrease ti the sphere is ept and the period wi not be the sae as when the sphere was fu of water.. A sipe penduu of enth suspended fro the ceiin of a train is osciatin when the train is at rest. If the train starts ovin with a constant acceeration, the tie period of the penduu wi be a a a. A wea daped haronic osciator of frequenc n is driven b an externa periodic force of frequenc n. When the stead state is reached, the frequenc of the osciator wi be n n (n + n )/ n n 4. The period of a partice in S.H.M. is 8 s. At t = 0 it is at the ean position. The ratio of the distances traveed b it in the first and the second is (C) (D) 5. A bod is on a rouh horizonta surface which is ovin horizonta in S.H.M. of frequenc Hz. The coefficient of static friction between the bod and the surface is 0.5. The axiu vaue of the apitude for which the bod wi not sip aon the surface is approxiate 9 c 6 c 4.5 c c 6. A pan with a set of weiht is attached to a iht sprin. The period of vertica osciations is 0.5 s. When soe additiona weihts are put in the pan, the period of osciations increases b 0. s. The extension caused b the additiona weihts is. c.7 c.8 c 5.5 c 7. A penduu coc, which eeps correct tie at sea eve, oses 5 s per da when taen to the top of a ountain. If the radius of the earth is 6400, the heiht of the ountain is The iniu phase difference between the two sipe haronic osciations sin t cost and = sin t + cos t is /6 /6 / 7/ 9. A sipe penduu is suspended fro the roof of a troe that oves free down a pane of incination. The tie period of osciation of the penduu is sin cos tan 0. A bod executes S.H.M. of period s under the infuence of one force, and S.H.M. of period 4 s under the infuence of a second force. When both the forces act siutaneous in the sae direction, the period of osciation wi be 7 s 5 s s.4 s. The tie period of a sipe penduu easured inside a stationar ift is T. If the ift starts acceeratin upwards with an acceeration /, the tie wi be T T T T

7 Einstein Casses, Unit No. 0, 0, Vardhan Rin Road Paza, Vias Puri Extn., Outer Rin Road New Dehi 0 08, Ph. : 96905, 857 PO 7. A second s penduu is paced in a space aborator orbitin around the earth at a heiht R fro the earth s surface where R is the radius of the earth. The tie period of the penduu wi be zero / s 4 s infinite. The percentae chane in radius of the earth is % without chane in its ass. The percentae chane in tie period of the sipe penduu is ½ % % %. % 4. Let the coefficient of inear expansion of the assess strin of sipe penduu is. If the teperature is increased b t then the fractiona chane in tie period of the sipe penduu is t t /4 t / t 5. A sprin of force constant is divided in : :. The force constant of the onest sprin is 6 6. The frequenc of vertica osciations of the three sprin-ass sste, shown in the fiure is FINAL STEP EXERCISE (OBJECTIVE). A sipe penduu has a tie period T. The penduu is copete iersed in a non-viscous iquid whose densit is /0th of that of the ateria of the bob. The tie period of the penduu iersed in the iquid is 9 T T T. A inear haronic osciator of force constant 0 6 N/ and apitude 0.0 has a tota echanica ener of 60 Joue. Its T 0 axiu potentia ener is 00 J axiu inetic ener is 00 J axiu potentia ener is 60 J both and are correct. When a bod is suspended fro two iht sprins separate, the periods of vertica osciations are T and T. When the sae bod is suspended fro the two sprins connected in series, the period wi be T + T T T (T T ) / T T 4. In the above probe if the bod is suspended fro the two sprins connected in parae, the tie period wi be T T T T TT T T TT T T

8 Einstein Casses, Unit No. 0, 0, Vardhan Rin Road Paza, Vias Puri Extn., Outer Rin Road New Dehi 0 08, Ph. : 96905, 857 PO 8 5. The dispaceent of a partice executin periodic otion is iven b (M )x / 4cos t sin(000t) This expression a be considered to be a resut of the superposition of two three four five independent haronic otions. 6. A unifor cinder of ass M and cross-sectiona area A is suspended fro a fixed point b a iht sprin of force constant. The cinder is partia subered in a iquid of densit. If it is iven a sa downward push and reeased, it wi osciate with tie period M A M M A M A 7. Two partices execute S.H.M. of the sae apitude and frequenc aon the sae straiht ine. The pass one another when oin in opposite directions each tie their dispaceent is haf their apitude. The phase difference between the is A person nora weihin 60 stands on a patfor which osciates up and down sipe haronica with a frequenc Hz and an apitude 5 c. If a achine on the patfor ives the person s weiht, then ( = 0 /s, = 0), the axiu readin of the achine wi be 08 the axiu readin of the achine wi be 90 the iniu readin of the achine wi be both and are correct 9. A ass M is suspended fro a assess sprin. An additiona ass stretches the sprin further b a distance x. The cobined ass wi osciate on the sprin with tie period /(M ) x (M ) / x ( / ) /(M ) x 0. Eectron in an oscioscope are defected b two utua perpendicuar osciatin eectric fieds such that at an tie the dispaceents due to the are iven b x = A cos t, = A cos t. Then 6 the path of the eectron is a straiht ine havin the equation x = a circe havin the equation x + = A an eipse havin the equation x x A / 4 an eipse havin the equation x x + = A /4. Two inear sipe haronic otions of equa apitudes and frequencies and are ipressed on a partice aon x and axes respective. If the initia phase difference between the is /, the resutant path foowed b the partice is iven b x A x x A x x A 4x x A 8x. A piece of wood has diensions a, b and c. Its reative densit is d. It is foatin in water such that the side a is vertica. If it is pushed down a itte and then reeased, the tie period of osciation wi be abc ad bc d bcd

9 Einstein Casses, Unit No. 0, 0, Vardhan Rin Road Paza, Vias Puri Extn., Outer Rin Road New Dehi 0 08, Ph. : 96905, 857. A U tube of unifor bore of cross-sectiona area a is set up vertica with open ends up. A iquid of ass and densit d is poured into it. The iquid coun wi osciate with a period PO 9 ANSWERS (INITIAL STEP EXERCISE) ad a d ad 4. A bod of ass fas fro a heiht h onto the pan of a sprin baance. The asses of the pan and sprin are neiibe. The force constant of the sprin is. The bod stics to the pan and osciates sipe haronica. The apitude of osciation is / ( / ) (h / ) ( / )( (h / )) ( / )( (h / ) ). d. d. b 4. d 5. d 6. b 7. a 8. c 9. b 0. d. b. d. b 4. d 5. c 6. b ANSWERS (FINAL STEP EXERCISE). c. d. d 4. b 5. b 6. a 7. c 8. d 9. a 0. c. c. c. d 4. b

10 PO 0 AIEEE ANALYSIS [00]. If a sprin has tie period T, and is cut into n equa parts, then the tie period of each part wi be 4. A chid swinin on a swin in sittin position, stands up, then the tie period of the swin wi Tn T/n nt T increase decrease. In a sape haronic osciator, at the ean position inetic ener is iniu, potentia ener is axiu both inetic and potentia eneries are axiu inetic ener is axiu, potentia ener is iniu both inetic and potentia eneries are iniu. A sprin of force constant 800 N/ has an extension of 5 c. The wor done in extendin it fro 5 c to 5 c is reain sae increase if the chid is on and decreases if the chid is short 5. A bod executes sipe haronic otion. The potentia ener (P.E.), the inetic (K.E.) and tota ener (T.E.) are easured as a function of dispaceent x. Which of the foowin stateents is true? P.E. is axiu when x = 0 K.E. is axiu when x = 0 T.E. is zero when x = 0 K.E. is axiu when x is axiu 6 J 8 J J 4 J AIEEE ANALYSIS [00] 6. A ass M is suspended fro a sprin of neiibe ass. The sprin is pued a itte and then reeased so that the ass executes SHM of tie period T. If the ass is increased b, the tie period becoes 5T/. Then the ratio of /M is 6/9 5/ /5 5/9 7. Two partices A and B of equa asses are suspended fro two assess sprins of sprin constant and, respective. If the axiu veocities, durin osciation, are equa, the ratio of apitudes of A and B is 8. The enth of a sipe penduu executin sipe haronic otion is increased b %. The percentae increase in the tie period of the penduu of increased enth is 4% 0% % % 9. The dispaceent of a partice varies accordin to the reation x = 4(cos t + sin t). The apitude of the partice is Einstein Casses, Unit No. 0, 0, Vardhan Rin Road Paza, Vias Puri Extn., Outer Rin Road New Dehi 0 08, Ph. : 96905, 857

11 < [004] Einstein Casses, Unit No. 0, 0, Vardhan Rin Road Paza, Vias Puri Extn., Outer Rin Road New Dehi 0 08, Ph. : 96905, 857 PO AIEEE ANALYSIS [004/005] 0. The bob of a sipe penduu executes sipe haronic otion in water with a period t, whie the period of osciation of the bob is t 0 in air. Neectin frictiona force of water and iven that the densit of the bob is (4/) 000 /. What reationship between t and t 0 is true? t = t 0 t = t 0 / t = t 0 t = 4t 0 [004]. A partice at the end of a sprin executes sipe haronic otion with a period t, whie the correspondin period for another sprin is t. If the period of osciation with the two sprins in series is T, then T = t + t T = t + t T = t + t T = t + t [004]. The tota ener of a partice, executin sipe haronic otion is x x independent of x x ½ where x is the dispaceent fro ean position. [004]. A partice of ass is attached to a sprin (of sprin constant ) and has a neutra anuar frequenc 0. An externa force F(t) proportiona to cos t ( 0 ) is appied to the osciator. The dispaceent of the osciator wi be proportiona to 0 ( 0 ) ( 0 0 ) [004] 4. In forced osciation of a partice the apitude is axiu for a frequenc of the force, whie the ener is axiu for a frequenc of the force, then = > < when dapin is sa and > when dapin is are 5. The function sin (t) represents a sipe haronic otion with a period / a sipe haronic otion with a period / a periodic, but not sipe haronic, otion with a period /. a periodic, but not sipe haronic, otion with a period /. [005] 6. Two sipe haronic otions are represented b the equations = 0. sin 00 t and = 0. cos t. The phase difference of the veocit of partice with respect to the veocit of partice is 6 6 [005] 7. If a sipe haronic otion is represented b d x x 0 dt, its tie period is [005] 8. The bob of a sipe penduu is a spherica hoow ba fied with water. A pued hoe near the botto of the osciatin bob ets sudden unpued. Durin observation, ti water is coin out, the tie period of osciation woud reain unchaned increase towards a saturation vaue first increase and then decrease to the oriina vaue first decrease and then increase to the oriina vaue [005]

12 Einstein Casses, Unit No. 0, 0, Vardhan Rin Road Paza, Vias Puri Extn., Outer Rin Road New Dehi 0 08, Ph. : 96905, 857 PO AIEEE ANALYSIS [006] 9. Startin fro the oriin a bod osciates sipe haronica with a period of s. After what tie wi its inetic ener be 75% of the tota ener? s s 6 s Hz 4 0. The axiu veocit of a partice, executin sipe haronic otion with an apitude 7, is 4.4 s. The period of osciation is 0. s 00 s 0.0 s 0 s. A coin is paced on a horizonta patfor which underoes vertica sipe haronic otion of anuar frequenc. The apitude of osciation is radua increased. The coin wi eave contact with the patfor for the first tie for an apitude of / at the hihest position of the patfor at the ean position of the patfor for an apitude of /. AIEEE ANALYSIS [007]. Two sprins, of force constant and, are connected to a ass as shown. The frequenc of osciation of the ass is f. If both and are ade four ties their oriina vaues, the frequenc of osciation becoes f f/4 f/ 4f. A partice of ass executes sipe haronic otion with apitude a and frequenc v. The averae inetic ener durin its otion fro the position of equiibriu to the end is a v a v 4 a v 4 a v 4. The dispaceent of an object attached to a sprin and executin sipe haronic otion is iven b x = 0 cos t etre. The tie at which the axiu speed first occurs is 0.5 s 0.75 s 0.5 s 0.5 s 5. A point ass osciates aon the x-axis accordin to the aw x = x 0 cos (t /4). If the acceeration of the partice is written as a = A cos (t + ), then A = x 0, = /4 A = x 0, = /4 A = x 0, = /4 A = x 0, = /4 ANSWERS AIEEE ANALYSIS. b. c. b 4. b 5. b 6. a 7. a 8. b 9. a 0. c. b. c. b 4. a 5. b 6. c 7. d 8. c 9. c 0. c. d. a. c 4. c 5. a TEST YOURSELF. A sipe penduu is osciatin in a ift. Let the tie period of the osciation is T. Consider the foowin cases : Case. Lift is ovin with constant veocut Tie Period. Lift is ovin with constant acceeration (downward) T 4 Which of the above tie period is iniu : T T T T 4 (upward or downward) T. Lift is ovin with constant acceeration (upward) T

13 Einstein Casses, Unit No. 0, 0, Vardhan Rin Road Paza, Vias Puri Extn., Outer Rin Road New Dehi 0 08, Ph. : 96905, 857 PO. If a hoe is bored aon a diaeter of the earth and a stone is dropped into the hoe, it wi reach the centre of the earth and stop there reach the other side of the earth and stop there execute S.H.M. about the centre of the earth execute osciator, but not sipe haronic, otion about the centre of the earth.. Two asses and are suspended toether b a assess sprin of constant. When the asses are in equiibriu, is ent reoved. Then the anuar frequenc of osciation of and the apit of osciation are respective. /, /,, /( ) /( ), 4. The ratio of the apitudes of the sipe haronic osciations iven b = A sin t and = (A/) sin t + (A/) cos t is / 5. A partice is subjected to two sipe haronic otions aon x and axes : x = a sin t and = a sin t. The resutant trajector is a sine curve circe eipse straiht ine 6. When a partice osciates sipe haronica, its potentia ener varies periodica. If the frequenc of osciation of the partice is n, the frequenc of potentia ener variation is n/ n n 4n 7. A boc of ass is paced on a frictioness horizonta tabe. Sprins of force constants and are attached on either side of it. The other ends of the sprins are fixed as shown in the fiure. If the boc is dispaced a itte horizonta and eft to osciate, the anuar frequenc of osciation wi be. c. c. b 4. d 5. d / ( ) / ( ) ( ) 6. c 7. a 8. b 9. b 0. d / / 8. Sprins of constants,, 4, 8,..., are connected in series. A ass is attached to one end and the sste is aowed to osciate. The tie period is Two sipe penduus havin enths and 6 are both iven sa dispaceents in the sae directions at the sae instant. The wi aain be in phase at the ean position after the shorter penduu has copeted n osciations where n is / A fat horizonta board oves up and down in S.H.M. of apitude A. Then the saest perissibe vaue of tie period, such that an object on the board a not ose contact with the board, is A A A A ANSWERS

Simple Harmonic Motion (SHM)

Simple Harmonic Motion (SHM) Phsics Sipe Haronic Motion (SHM) www.testprepart.co abe of Content. Periodic otion.. Osciator or Vibrator otion. 3. Haronic and Non-haronic osciation. 4. Soe iportant definitions. 5. Sipe haronic otion.