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1 Oscillations Equations 0. Out of the followin functions representin otion of a particle which represents SHM I) y = sinωt cosωt 3 II) y = sin ωt III) IV) 3 y = 5cos 3ωt 4 y = + ωt+ ω t a) Only IV does not represent SHM b) I and III c) I and II d) Only I. he otion which is not siple haronic is a) Vertical oscillations of a sprin b) Motion of siple pendulu c) Motion of a planet around the sun d) Oscillation of liquid colun in a U-tube Which one of the followin equations of otion represents siple haronic otion? a) Acceleration = kx+ kx b) Acceleration = -k (x + a) 0 c) Acceleration = k (x + a) d) Acceleration = kx he function sin ( t) ω represents a) A periodic, but not siple haronic otion with a period / ω b) A periodic, but not siple haronic otion with a period / ω c) A siple haronic otion with a period / ω

2 d) A siple haronic otion with a period / ω A particle executes siple haronic oscillation with an aplitude a. he period of oscillation is. he iniu tie taken by the particle to travel half of the 006 aplitude fro the equilibriu position is a) 4 b) 8 c) 6. he otion of a particle varies with tie accordin to the relation 005 y = a(sinωt+ cos ωt) a) he otion is oscillatory but not SHM b) he otion is SHM with aplitude a c) he otion is SHM with aplitude a d) he otion is SHM with aplitude a d) 7. Which of the followin functions represents a siple haronic oscillation? a) sinωt cosωt b) sin ω t c)sinωt+ sin ωt d) sinωt sin ωt 8. he iniu phases difference between two siple haronic oscillations y = sinωt+ cosωt; y = sinωt+ cosωtis a) 7 b) c) d) he displaceent of a particle fro its ean position (in etre) is iven by y = 0.sin(0 t+.5 )cos(0t+.5 ). he otion of the particle is a) Periodic but not SHM b) Non periodic

3 c) Siple haronic otion with period 0.s d) Siple haronic otion with periodic 0.s 0. he displaceent tie raph of a particle executin SHM is as shown in the fiure 008 he correspondin force-tie raph of the particle is a) b) c) d) Velocity, Acceleration and Enery. wo siple haronic otions of anular frequency 00 and 000 rad s have the sae displaceent aplitude. he ratio of their axiu accelerations is a) : 0 b) :0 c) 3 :0 d) 4 :0. A point perfors siple haronic oscillation of period and the equation of otion is iven by x = asin( ωt+ / 6). After the elapses of what fraction of the tie period the velocity of the point will be equal to half of its axiu velocity a) 8 b) 6 c) 3 d)

4 3. wo points are located at a displaceent of 0 and 5 fro the source of oscillation. he period of oscillation is 0.05s and the velocity of the wave is 300s. What is the phase difference between the oscillations of two points? a) 3 b) 3 c) d) 6 4. A particle is executin SHM. hen, the raph of velocity as a function of 007 displaceent is a/an a) Straiht line b) Circle c) Ellipse d) Hyperbola 5. he particle executin siple haronic otion has a kinetic enery K cos ω t. he 0 axiu values of the potential enery and the total enery are respectively K0 a) 0 and K 0 b) and K 0 c) K 0 and K 0 d) K0 and K 0 6. Which one of the followin stateents is true for the speed v and the acceleration a, of a particle executin siple haronic otion? a) When v is axiu, a is axiu b) Value of a is zero, whatever ay be the value of v c) When v is zero, a is zero d) When v is axiu, a is zero

5 A particle executes SHM; its tie period is 6s. If it passes throuh the centre of oscillation then its velocity is s at ties s. hen aplitude will be a) 7. b) 4 c c) 6 c d) A particle executin SHM has aplitude 0.0 and frequency 60 Hz. he axiu acceleration of particle is 004 a) 60 s b) 80 s c) 0 s d) 44 s 9. he anitude of acceleration of particle executin SHM at the position of axiu displaceent is a) Zero b) Miniu c) Maxiu d) None of these 0. If for a particle executin SHM, the equation of SHM is iven as y= acosωt. hen 003 which of the followin raph represents the variation in its potential enery? a) II, IV b) I, III c) III, IV d) I, II. A particle of ass oscillates with siple haronic otion between points x and x, the equilibriu position bein O. Its potential enery is plotted. It will be as iven below in the raph

6 0 a) b) c) d) ie Period and Frequency. A particle of ass is located in a one diensional potential field where potential enery is iven by V(x) = A ( cos px) where A and p are constants. he period of sall oscillations of the particle is 00 a) b) c) d) Ap Ap A 3. A body is executin SHM when its displaceent lare the ean position are 4c and 5c it has velocity 0s and 8s respectively. Its periodic tie t is a) sec 3 b) sec c) 3 sec d) sec 4. One-fourth lenth of a sprin of force constant k is cut away. he force constant of the reainin sprin will be a) 3 4 k b) 4 k c) k d) 4k 3 5. he equation of a daped siple haronic otion is anular frequency of oscillation is d x dx 0 Ap + b + kx=. hen the dt dt

7 a) ω k b = 4 / b) ω k b = 4 / c) ω k b = 4 / d) ω k b = Assertion (A): he periodic tie of a hard sprin is less as copared to that of a soft sprin. Reason (R): he periodic tie depends upon the sprin constant, and sprin constant is lare for hard sprin. a) Both assertion and reason are true and reason is the correct explanation of assertion. b) Both assertion and reason are true but reason is not the correct explanation of assertion. c) Assertion is true but reason is false. d) Both assertion and reason are false. 7. A body executes siple haronic otion under the action of force F with a tie period 4 5 s. If the force is chaned to F it executes siple haronic otion with tie period 3 5 s. If both forces F and F act siultaneously in the sae direction on the body, its tie period will be a) s b) s c) s d) s 8. A siple pendulu perfors siple haronic otion about x = 0 with an aplitude a, and tie period. he speed of the pendulu at a 3 a) a b) c) 3 a a x = will be a 3 d) 9. A siple pendulu of lenth l has a axiu anular displaceentθ. he axiu kinetic enery of the bob is a) l( cos θ ) b) 0.5 l c) l d) l

8 A particle of ass is executin oscillation about the oriin on the x-axis. Its 3 potential enery is U( x) = k[ x], where k is a positive constant. If the aplitude of oscillation is a, then its tie period is a) Proportional to a b) Independent of a c) Proportional to a d) Proportional to 3. wo pendulus have tie periods and 5/4. hey start SHM at the sae tie fro the ean position. What will be the phase difference between the after the bier pendulu copleted one oscillation a) b) 0 90 c) a 3/ 0 60 d) 3. he axiu displaceent of the particle executin SHM is c and the axiu acceleration is (.57) c s. Its tie period is a) 0.5 s b) 4.0 s c).57 s d) 3.4 s A rectanular block of ass and area of cross-section A floats in a liquid of density ρ. If it is iven a sall vertical displaceent fro equilibriu it underoes oscillation with a tie period. hen a) ρ b) c) d) A ρ 34. A particle executes siple haronic otion with a frequency f. hen frequency with which the potential enery oscillates is a) f b) f/ c) f d) Zero 30 0

9 Siple Pendulu wo siple pendulu first of bob ass M and lenth L, second of bob ass M and lenth L. M = M and L = L If the vibrational eneries of both are sae. hen which is correct? a) Aplitude of B reater than A b) Aplitude of B saller than A c) Aplitude will be sae d) None of the above 36. he ass and diaeter of a planet are twice those of earth. he period of oscillation of pendulu on this planet will be (if it a second s pendulu on earth) a) / s b) s c) s d) /s 37. Assertion (A): he percentae chane in tie period is.5%, if the lenth of siple pendulu increases by 3%. Reason (R): ie period is directly proportional to lenth of pendulu. a) Both assertion and reason are true and reason is the correct explanation of assertion. b) Both assertion and reason are true but reason is not the correct explanation of assertion. c) Assertion is true but reason is false. d) Both assertion and reason are false. 38. A clock S is based on oscillation of a sprin and a clock p is based on pendulu otion. Both clock run at the sae rate on earth and on a planet havin the sae density as earth but twice the radius. a) S will run faster than P. b) P will run faster than S. c) Both will run at the sae rate as on the earth. d) Both will run at the sae rate which will be different fro that on the earth.

10 39. he tie period of a siple pendulu of lenth L as easured in an elevator descendin with acceleration 3 is 3L 3L 3L a) b) c) d) 40. A pendulu has tie period in air when it is ade to oscillation in water, it 008 acquired a tie period =. hen density of the pendulu bob is equal to (density of water = ) a) b) c) d) None of these 4. A coin is placed on a horizontal platfor which underoes vertical siple haronic otion of anular frequencyω. he aplitude of oscillation is radually increased. he coin will leave contact with the platfor for the first tie a) At the ean position of the platfor b) For an aplitude of ω c) For an aplitude of ω L 3 d) At the hihest position of the platfor 4. A heavy sall-sized sphere is suspended by a strin of lenth l. he sphere rotates uniforly in a horizontal circle with the strin akin an anle θ with the vertical. hen, the tie-period of this conical pendulu is l a) t = b) l sinθ t = c) l cosθ t = d) t = l cosθ 43. he lenth of the second s pendulu is decreased by 0.3c when it is shifted to Chennai fro London. If the acceleration due to ravity at London is 98c s, the acceleration due to ravity at Chennai is (assue = 0 ) a) 98c s b) 978c s c) 984c s d) 975c s

11 he tie period of a siple pendulu in a stationary train is. he tie period of a ass attached to a sprin is also. he train accelerates at the rate 5s.If the new tie periods of the pendulu and sprin be p and s respectively, then a) p s = b) p > s c) p < s d) Cannot be predicted 45. Assertion (A): Water in a U-tube executes SHM, the tie period for ercury filled up to sae heiht in the U-tube be reater than that in case of water. Reason (R): he aplitude of an oscillatin pendulu oes on increasin. a) Both assertion and reason are true and reason is the correct explanation of assertion. b) Both assertion and reason are true but reason is not the correct explanation of assertion. c) Assertion is true but reason is false. d) Both assertion and reason are false. 46. ie period of a siple pendulu is. If its lenth increases by %, the new tie 005 period becoes a) 0.98 b).0 c) 0.99 d) he aplitude of an oscillatin siple pendulu is 0c and its period is 4s. Its speed after s when its passes throuh its equilibriu position is a) Zero b).0s c) 0.3s d) 0.4s 48. A siple second pendulu is ounted in a rocket. Its tie period will decrease when the rocket is a) Movin up with unifor velocity b) Movin up with unifor acceleration c) Movin down with unifor acceleration d) Movin around the earth in eostationary orbit

49. If the lenth of a pendulu is ade 9 ties and ass of the bob is ade 4 ties, then the value of ties period becoes a) 3 b) 3/ c) 4 d) 004 50.

12 49. If the lenth of a pendulu is ade 9 ties and ass of the bob is ade 4 ties, then the value of ties period becoes a) 3 b) 3/ c) 4 d) he period of oscillation of a siple pendulu is in a stationary lift. If the lift oves upwards with acceleration of 8, the period will a) Reain the sae b) Decreases by / c) Increase by /3 d) None of these 5. wo sprin are connected to a block of ass M placed on a frictionless surface as shown below. If both the sprins have a sprin constant k, the frequency of oscillation of block is a) k M b) k M c) k M he tie period of a ass suspended fro a sprin is. If the sprin is cut into four equal parts and the sae ass is suspended fro the of the parts, then the new tie period will be d) M k a) b) c) 4 d) 53. Pendulu after soe tie becoes slow in otion and finally stops due to a) Air Friction b) Earth s Gravity c) Mass of Pendulu d) None of these

13 54. wo sprins of force constants k and k are connected to a ass as shown below. he frequency of oscillation of the ass is a) k b) k c) 3k 55. Assertion (A): he aplitude of an oscillatin pendulu decreases radually with tie. Reason (R): he frequency of the pendulu decreases with tie. a) Both assertion and reason are true and reason is the correct explanation of assertion. b) Both assertion and reason are true but reason is not the correct explanation of assertion. c) Assertion is true but reason is false. d) Both assertion and reason are false. Key ) c ) c 3) b 4) b 5) c 6) c 7) a 8) b 9) c 0) c ) b ) d 3) b 4) c 5) d 6) d 7) a 8) d 9) c 0) b ) c ) b 3) b 4) b 5) a 6) a 7) a 8) d 9) a 30) a 3) b 3) b 33) b 34) c 35) b 36) b 37) c 38) b 39) a 40) b 4) b 4) c 43) b 44) c 45) d 46) d 47) a 48) a 49) a 50) c 5) b 5) a 53) a 54) c 55) c d) k

14 . d y dt y y = sinωt cosωt And 3 y = 5cos 3ωt 4 Condition and equation SHM. Concept. 3. Concept 4. Here y = sin ωt dy = ω sinωtcosωt = ωsin ωt dt d y = ω cos dt For SHM, t = ω 5. y = asinωt a y = a = asinωt d y dt ωt y Hints Equations are satisfyin this y t t = + ω + ω is not periodic and y 3 = sin ωt is periodic but not Or sinω t = = sin 6 Or ω t = ort = 6 6ω

15 Or t = ω = 6. y = a(sinωt+ cos ωt) Or Or Or y = a sinωt+ cosωt sin y a cos = sinωt+ cosωt 4 4 y = a sin ωt+ 4 his is the equation of siple haronic otion with aplitude a 7. Let y = sinωt cosωt dy = ω cosωt+ ωsinωt dt d y = sin t cos dt Or Or ω ω ω ωt a = ω (sinωt cos ωt) a= ω y Or a y 9. Given y = 0.sin(0 t+.5 )cos(0t+.5 ) y = 0.sin (0 t+.5 ) y = 0.sin (0 t+ 3 ) his equation represents siple haronic otion of anular frequency 0. ie period = = = = 0.s ω Concept

16 Velocity, Acceleration and Enery. aax = ω A ( aax ) ω Or = ( aax ) ω ( aax ) (00) Or = = :0 = ( a ) (000) 0 ax. x = asin ωt+ 6 dx v= = aωcos ωt+ dt 6 Or Or aω = aωcos ωt+ 6 t = = = 6ω 6 hus, at velocity of the point will be equal to half of its axiu velocity 3. Path different Δ x = 5 0 = 5 ie period, = 0.05s Frequency v = 0Hz = 0.05 = Velocity, v = 300s v 300 Wavelenth λ = = = 5 v 0 Hence, phase difference Δ φ = Δ x = 5 = λ v = ω ( a x ) v a ω x a v + ω x = ω a + = Which is the equation of an ellipse

17 5. Concept 6. Concept 7. v= aω cosωt = a..cos. 6 6 Or 6 a = = 7.c 8. Maxiu acceleration of particle = aω = a( f) 9. Concept = 4a f = (60) = 44 s 0. he potential enery is axiu at extree position (where y = ± a) and zero at ean position so, raph I is correct. Also, fro y= acosωt y =± aat tie t = 0 Hence, raph III is also correct. U = kx At equilibriu position (x = 0), potential enery is iniu. At extree position x and x, its potential eneries are U = kx andu = kx. V (x) = A (-cos px) dv F = = Ap sin px dx For sall oscillations, we have ie Period and Frequency F Ap x

18 Hence, the acceleration would be iven by F Ap a = = x F Also, a = = ω x But, ω = Ap Or = = ω Ap v = ω ( a y ) 0 = ω ( a 4 ) And 8 = ω ( a 5 ) So, 0 8 = ω (5 4 ) = (3 ω) Or ω = ie, t = ω t = = sec k l 4 k = k 3 5. Displaceent of daped oscillator is iven by frequency of daped oscillator b = ω 0 k b = 4 6. Concept 9. Heiht of bob at axiu anular displaceent bt/ x xe ω t = sin( + φ) where ω = anular h= l lcos θ = l( l cos θ )

19 Also, PE = KE h = l( cos θ ) du U = k x F = dx = 3k x.. (i) Also, \ x = asinωt And d x + ω x = 0 dt Acceleration, d x F = a= dt d x a = = ω x dt = ω x (ii) Fro eqns (i) and (ii) ω = = = ω 3. Concept 3kx 3. Maxiu acceleration = 3kx = 3 ka ( sin ωt) a = Aω.57 = 33. Up thrust (upwards) = Axρ a = Axρ Aρ Or a = x = ω x 4 4 = = 4 his is the equation of siple haronic otion. ie period of oscillation s

20 = ω = Aρ A 35. Frequency, Or n l n = n l L = = n l L n = n n > n Enery, E = ω a = n a And a n a n = a n l Given, n > nand = a > a. So, aplitude of B saller than A GM 36. Gravity = R M R = = M R earth e p e planet p e p Siple Pendulu Also, = e p p e = p

21 p = s l 37. = l 38. Δ = Δl l Δ = 3=.5% 4 = GρR Or R 3 For pendulu block, will increase on the planet so tie period will decreases. But for sprin clock, it will not chane. Hence, P will run faster than S 39. he effective acceleration in a lift descendin with acceleration 3 is eff = = 3 3 ie period of siple pendulu L = = eff L /3 = 40. he effective acceleration of a bob in air and water are iven as l = and = σ ρ = = l = σ ρ = ρ [ σ = ] 3L Puttin = = ρ = ρ

22 4. Concept 43. L 4 = = L 4 = = Since, lenth is decreased is less than L L = Or ( L L ) = Or = = 98 3 = 978cs 44. ie period of siple pendulu placed in a train acceleratin at the rate of iven by = k as It is independent of as well as a. hence, when the train acceleration, the tie period of the siple pendulu decreases and that of sprin reains unchaned. Hence, p < and s < i.e, p < 45. Concept 46. l / s Δ Δl = l Δ = = (%) = % = 00 = is =.0

23 47. A = 0c, = 0., = 4s, t = s y = Asinωt dy = v= Aω cos( ωt) = A cos t = cos = cos = 0 dt l 48. = l Since, tie period of second pendulu decreases, so, it iplies that effective value of is increasin. hus, it eans that rocket is acceleration upwards. l 49. = =, l = l l = 9l l = = = l 9 3 = 3 = = + 8 = 9 5. = 9 = = 3 3 = + = k k k k eq k k eq = Frequency of oscillation 5. = (i) k Now we know that, k k f = f M = M

24 Sprin constant lenth Or k. (ii) x k = 4k So, new tie period of sae ass suspended fro one of the parts, = =. = 4k k 53. Concept 54. Let F and F be the restorin forces produced then F kx and F kx otal restorin force is F = F+ F = kx kx = (3 k) x Hence, frequency k n = 55. Concept

OSCILLATIONS

OSCILLATIONS OSCIAIONS Important Points:. Simple Harmonic Motion: a) he acceleration is directly proportional to the displacement of the body from the fixed point and it is always directed towards the fixed point in