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(C)g , 22 3 g.(D)2g , 23g

.Sol: amax = .2Aor g = .2Aor g = .2 2 21 2 A . Aor g = .2 12 . 3or g = 2.2.g2. . and if occurs at extreme at upper extreme position.. A= sin .t + 3 cos .t

or 2 = sin .t0 + 3 cos .t0or 2 = 2sin(.t0 + .)Buttan 3 31. . . . 60º3.. . .or 0 2 2sin t3. .. . .. . . . . or 0 t3 2

. .

. . .or 0 t2 3 6. . .. . . .or 0t 26 6 g. .. .. Hence option (A) is correct.

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SIMPLE HARMONIC MOTIONwww.physicsashok.in 81THINKING PROBLEMS1. Which ofthe following functions are (a) aperiodic (b) periodic but not simple harmonic (c) simple harmonic : (i)sin 2.t (ii) 1 + cos 2.t (iii) a sin .t + b cos .t (iv) sin .t + sin 2.t + cos 2

.t (v) sin3 .t (v) log (1 + .t) (vii)exp (.t) ?2. Can everyoscillatorymotion be treated as simple harmonicmotionin the limit ofsmall amplitude ?3. What would happen to themotionof an oscillating systemif the sign of the force termin equation F =  kx ischanged ?4. (a) Can amotion be periodic but not oscillatory ?(b) Can amotion be oscillatory but not simple harmonic ?If your answer is yes, give an example and if not, explainwhy ?5. Can a bodyhave accelerationwithout having velocity ?6. Determine whether or not the following quantitie scan be in the same directio

nfor a simple harmonicmotion :(a) displacement and velocity, (b) velocityand acceleration, (c) displacement and acceleration.7. (a) Canwe ever construct a simple pendulumstrictlyaccording to its definition ?(b) Is themotion of a simple pendulumlinear SHMor angular SHM?8. Agirl is swinging in a sitting position.Howwill the period of swing be affected if :(a)The girl stands upwhile swinging ?(b)Another girl of samemass comes and sits next to her ?9. Ahollowmetal sphere is filledwithwater and a small hole ismade at its bottom. It is hanging bya long threadand ismade to oscillate.Howwill the period ofoscillation change ifwater is allow

ed to flowthrough the holetill the sphere is empty ?10. The resultant of two simple harmonicmotions at right angles and of the samefrequency is always a circularmotion.True or false?Explain.SOLUTION OF THINKING PROBLEMS1. (a) Functions (vi) log (1 + .t) and (vii) exp (.t) increase (or decrease) continuouslywith time and can neverrepeat themselfso are aperiodic.(b) Function (ii) (1+ cos 2.t), (iv) sin .t + sin 2.t +cos 2.t and (v) sin3 .t are periodic [i.e., f(t + T) = f(t)] withperiodicity (./.), (2./.) and (2./.) respectively but not simple harmonic as for

 these functions (d2y/dt2) isnot .  y.(c) Functions (i) sin 2.t and (iii) a sin .t + b cos .t, i.e., (a2 + b2)1/2 sin[.t + tan1 (b/a)] are simple harmonic[with time period (./.) and (2./.) respectively] as for these (d2y/dt2) .. y.2. No, not at all. In the limit of small amplitudes onlythose oscillatorymotions can be treated as simple harmonicfor which the restoring force (or torque) becomes linear. For example, the oscillatorymotion of a simple orspring pendulumormotion of atoms inamolecule becomes simple harmonic in the limit ofsmall amplitude asrestoring force (or torque) becomes linear while in case of oscillatorymotion of a ballbetween two inclined

planes or two perfectly elasticwalls, themotion does not become simple harmonicevenfor vanishingysmallamplitude.

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3. If the sign is changed in the force equation, accelerationwill not be opposite to displacement and hence theparticlewillnot oscillate, butwillaccelerate inthedirectionofdisplaement. So themotionwillbecome acceleratedtranslatory. However, equations ofmotion cannot be applied to analyse themotionas acceleration (= .2y) isnot constant.

Mathematical analysis shows that in this situation both velocity and displacement will increase exponentiallywithtime.

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SIMPLE HARMONIC MOTIONwww.physicsashok.in 824. Yes;Uniformcircularmotion, (b)Yes;When a ball is thrown froma height on a perfectlyelastic plane surfacethemotion is oscillatory but not simple harmonic as the restoring force F =mg =constt. and not F .  x]

5. Yes; In SHMat extreme position velocityis zero but acceleration ismaximum= .2A.6. (a)Yes;whenthe particle ismoving fromequilibriumposition to extreme position, (b)Yes;whenthe particle ismoving fromextreme to equilibriumposition, (c)No; as inSHMdisplacement is alwaysopposite to acceleration.]7. (a) No, (b)Angular SHM.8. (a) decreases, (b) unchanged9. Time periodwill first increase, reaches amaximumand thenwill decrease.10. False. The generalmotion is elliptical.Acircle is a particular case of ellipsewhen itsmajor axis is equal to itsminor axis. In general, themotion is described by

2 222 2d y 2xycos sina b ab. . . . .When a = b and . = ./2,x2 + y2 = a2ASSERTION-REASON TYPEAstatement of Statement-1 is given and a Corresponding statement of Statement-2is given just belowit ofthe statements,mark the correct answer as  

(A) If both Statement-1 and Statement-2 are true and Statement-2 is the correctexplanation ofStatement-1.(B) If both Statement-1 and Statement-2 are true and Statement-2 isNOT correct explanation ofStatement-1.(C) If Statement-1 is true but Statement-2 is false.(D) If both Statement-1 and Statement-2 are false.(E) IfStatement-1 is false but Statement-2 is true.1. Statement-1 : The force acting on a particle moving along xaxis is F = kx + v0t, where k is a constant.Statement-2 :To an observermoving along x-axiswith constant velocity v0, it represents SHM.2. Statement-1 : Aparticle executing simple harmonicmotioncomes to rest at the e

xtreme positions.Statement-2 :The resultant force on the particle is zero at these positions.3. Statement-1 : Soldiers are asked to break stepswhile crossing the bridges.Statement-2 :The frequencyofmarchingmaybe equalto the naturalfrequencyof bridgeandlead to resonancewhich can break the bridge.4. Statement-1 : The functionY= cos2 .t + sin .t does not represent a simple harmonicmotion.Statement-2 : Sumof two harmonic functionmaynot be harmonicmotion.5. Statement-1 : Ablock ofmassmis attached to a spring inside a trolley ofmass 2mas shown in figure, all surfaces are smooth. Spring is stretched andreleased. Both trolleyand block oscillate simple harmonicallywith same timeperiod and same amplitude.

m2mStatement-2 : Inabsence ofexternalforce,centreofmass ofsystemofparticles

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does not accelerate.

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SIMPLE HARMONIC MOTIONwww.physicsashok.in 83MATCH THE COLUMN1. Asimple harmonic oscillator consists ofa block attached to a springwith k= 200N/m.Theblock slides ona frictionless horizontal surface,with equilibriumpoint x = 0.Agraph of the blocks velocity v as a

function of time t is shown. Correctlymatch the required informationin the column Iwith the values given in the column II. (use .2= 10)220 0.10 0.20t(s)V(m/s)Column-I Column-II(A) The blocksmass in kg (P) 0.20(B) The blocks displacement at t = 0 inmetres (Q) 200(C) The blocks acceleration at t = 0.10 s inm/s2 (R) 0.20(D) The blocksmaximumkinetic energyin Joule (S) 4.0

2. Matchthe followingColumn-I Column-II(A) Linear combination of two SHMs (P) T = 2.Rg (R is radius of the earth)(B) y =Asin .1t +Asin(.2t + .) (Q) SHMfor equal frequencies and amplitude(C) Time periodof a pendulumofinfinite length. (R) Superpositionmay not be a SHMalways.(D) Maximumvalue of time period ofan oscillating (S) amplitude will be 2 A for .1 = .2 andpendulum. phase difference of ./2.3. Matchthe followingColumn-I Column-II

(A) Aconstant force acting along the line of (P) The time periodSHMaffects(B) Aconstant torque acting along the arc of (Q) The frequencyangular SHMaffects.(C) Aparticle falling on the block executing SHM (R) themeanpositionwhen the later crosses themean position affects(D) Aparticle executing SHMwhen kept on a (S) The amplitudeuniformly accelerated car affects.

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SIMPLE HARMONIC MOTIONwww.physicsashok.in 84LEVEL  11. Two particlesP andQdescribe SHMof same amplitude a, same frequencyf along the same straight line.Themaximumdistance between the two particles is a 2 . The phase difference between

the particle is :(A) zero (B) ./2 (C) ./6 (D) ./32. Arod of length l is inmotion such that its endsA and B aremoving along x-axis andy-axis respectively. it is given thatd 2dt.. rad/s always. P is a fixed point on the rod.LetMbe the projection ofPon x-axis. For the time interval inwhich . changes from 0 to ./2, choose the correct statement,

yxPMlAB(A) The acceleration ofMis always directed towards right(B)Mexecutes SHM(C)Mmoveswith constant speed (D)Mmoveswith constant acceleration3. The coefficient of frictionbetween block ofmassmand 2mism= 2 tan .. There isnofriction between blockofmass 2mand inclined plane. Themaximumamplitude of two

block systemfor which there is no relativemotion between both the blocks.km2m(A) g sin .km (B)mg sink.(C)3mg sin

k.(D) None of these4. Graph shows the x(t) curves for three experiments involving a particular spring-block systemoscillating inSHM. The kinetic energy of the systemismaximumat t = 4 sec. for the situation :124sec.t(in sec)30x

(A) 1 (B) 2 (C) 3 (D) Same in all5. Aparticle ofmassm= 2 kg executes SHMin xy-plane between pointsAand Bunderaction of force x y

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F . F i . F j..Minimumtime taken by particle to movefromAto B is 1 sec.At t = 0 the particle is at x =2 and y=2.Then Fx as functionof time t isA(2, 2)y

xB(2, 2)(A) 4.2 sin .t (B) 4.2 cos .t (C) 4.2 cos .t (D) None of these6. The speed v of a particlemoving along a straight line,when it is at a distance (x) froma fixed point of the lineis given byv2 = 108  9x2(allquantities are in cgs units) :(A) themotionis uniformlyaccelerated along the straight line(B) themagnitude of the acceleration at a distance 3cmfromthe point is 27 cm/sec2(C) themotionis simple harmonic about the given fixed point.

(D) themaximumdisplacement fromthe fixed point is 4 cm.

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SIMPLE HARMONIC MOTIONwww.physicsashok.in 857. The time period of an ideal simple pendulumis given by : T = 2. l / gThe time period of actual simple pendulum(T´) is slightlydifferent due to smalldamping (friction). Then(A) T´ > T (B) T´ < T (C) T = T´ (D) None

8. Asmallbob of a simple pendulumcontainswater. Its periodic time isT.During oscillation, the temperature ofsurrounding is lowered so that water gets freezed. The newtime period is T´. Then:(A) T > T´ (B) T < T´ (C) T = T´ (D) None9. Two simple pendulaAand B are shown in the figure. The phase difference betweenA and B is :(A) . (B) ./2l l(C) 0 (D) ./4 A B10. Abodyexecutes simple harmonicmotionunder the action of a force F1witha time

period (4/5) second. If theforce is changed to F2 it executes SHM with a time period (3/5) second. If boththe forces F1 and F2 actsimultaneously in the same direction on the body, its time period in second is : (A) 12/25 (B) 24/25 (C) 35/24 (D) 25/1211. What should be the displacement of a simple pendulumwhose amplitude is A, at which potential energy is1/4th of the total energy ?(A) A/ 2 (B)A/2 (C)A/4 (D) A/ 2 212. The potential energyUof a particle is given byU= {20 + (x  4)2} J.Totalmechanical energy of the particleis 36 J. Select the correct alternative(s)

(A) The particle oscillates about point x= 4m(B) The amplitude of the particle is 4m(C) The kinetic energy of the particle at x = 2mis 12 J(D) Themotion of the particle is periodic but not simple harmonic13. Arod ofmassMand lengthL is hinged at its centre ofmass so that it can rotate in a vertical plane. Two springs each of stiffness k are connected at its ends,asshown in the figure. The time period of SHMisL, MkHinge

k(A)2 M6k. (B)2 M3k. (C)2 MLk. (D)M6k

.14. Aparticlemoves along the x-axis according to the equation x =Asin2 .t(A) The particle oscillates about the origin (B) The particle oscillates about t

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he point x =A(C) The particle oscillateswith a period T = ./. (D) The particle oscillateswith amplitudeA/215. In the arrangement shown in figure the pulleys are smooth and massless. Thespring k1 and k2 aremassless. The time period of oscillation of themassmismk k1 (A) 2

1 22 mk k.. (B)1 22 2mk k..(C) 1 21 2

m(k k )22k k.. (D) 1 21 2m(k k )2k k..

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SIMPLE HARMONIC MOTIONwww.physicsashok.in 8616. The equationofSHMofa particle oscillating alongthe x-axis is givenbyx=3 sint6. .. . . . . . . cm.The acceleration

of the particle at t = 1 s is(A) 1.5 .2 cms2 (B) 2.6 .2 cms2 (C) 2.6 .2 cms2 (D) 1.5 .2 cms217. For the systemshown in the figure, initiallythe spring is compressed byadistance a fromits natural lengthandwhenreleased, itmoves to a distanceb fromits equilibriumposition. The dicrease in amplitude for one half -cycle (a to +b) iskµ ma b(A)µmgk (B)

2µmgk (C)µmg2k (D) none of these18. Themotion of a particle is given x =Asin .t +B cos .t. Themotion of the particle is(A) not simple harmonic (B) simple harmonicwith amplitudeA+B(C) simple harmonicwith amplitude (A+B)/2 (D) simple harmonicwith amplitude A2 . B219. Two massesm1 and m2 are suspended together by amassless spring of force constant K.When the massesare in equilibriumis removedwithout disturbing the system. The amplitude of oscillation is

(A)m2g/K (B)m1g/K (C) (m1+m2)g/K (D) (m1m2)g/K20. Two bodiesMandNof equalmasses are suspended fromtwo seperatemassless springs of spring constantsk1 and k2 respectively. If the two bodies oscillate vertically such that theirmaximumvelocities are equal, theratio of the amplitude of vibration ofMto that onNis :(A) k1/k2 (B) 1 2 k / k (C) k2/k1 (D) 2 1 k / k21. Aparticle executing SHMwhilemoving fromone extremityis found at distance x1, x2, x3 fromthe centre at theends of three successive seconds, the period is(A) . (B) ./2. (C) 2./. (D) 1/.where ..= cos1(x1 + x3)/2x2

22. One end of a spring of force constant k is fixed to a verticalwall and the other to a body ofmassmresting ona smooth horizontal surface. There is anotherwall at a distance x0 fromthebody. The spring is then compressed by 2x0 and released. The time taken mx0to strike thewall is(A)1 k6 m. (B)mk (C)2 m

3 k.(D)

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k3 m.23. In the above problemthe velocitywithwhich the body strikes the other wall is (A) 0x m

k (B) 03k xm (C) 0k xm (D) 02x mk24. Aparticlemoves intheXYplane according to the equation: r . .2i . 4j..sin .t. Themotion ofthe particle is:(A) parabolic (B) circular (C) straight line (D) None of these

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SIMPLE HARMONIC MOTIONwww.physicsashok.in 8725. The speed v of a particlemoving along a straight line, when it is at a distance x froma fixed point on the line isgiven byV= (108  9x2)1/2.All quantities are in SI units.(A) Themotionis uniformlyaccelerated along the straight line.

(B) Themagnitude of acceleration at a distance 3mfromthe fixed point is 27m/s2.(C) Themagnitude of acceleration at a distance 3mfromthe fixed point is 9m/s2.(D) Themaximumdisplacement of the particle fromthe fixed point is 4m.26. The potential energyUof a particle is given byU= (2.5X2 + 100) Joule. If themass of the particle is 0.2 kg,then:(A) themotion of the particle is SHM. (B) themean position isX= 0.(C) angular frequency of the oscillation is 5 rad/s (D) the time period of oscillation is 1.26 sec.27. In SHM:(A) displacement and velocitymaybe in the same direction.(B) displacement and velocity can never be in the same direction.

(C) velocity and accelerationmay be in the same direction.(D) displacement and acceleration can never be in the same direction.28. Aparticlemoves in theXYplane according to the equation r . .3i . j.cos5 t.. Themotionof theparticle is:(A)Along a straight line (B) along an ellipse (C) periodic (D) along a parabola29. The potential energyof a particle ofmass 2 kg,moving alongthe xaxis is givenbyU(x) = 16(x2  2x) J,wherex is inmetres. Its speed at x = 1 mis 2 ms1 :(A) Themotion of the particle is uniformlyaccelerated(B) Themotion of the particle is oscillatory fromx = 0.5mto x= 1.5m.(C) Themotion of the particle is simple harmonic(D) The period of oscillation of the particle is ./2 s.

30. If a SHMis given by y= (sin .t + cos .t)m,which of the following statementsis/are true ?(A) The amplitude is 1m (B) The amplitude is 2 m.(C) Particle starts itsmotion fromy= 1m. (D) Particle starts itsmotion fromy= 0m.31. Three simple harmonic motions in the same direction having the same amplitude a and same period aresuperposed. If each differs in phase fromthe next by45º, then :(A) the resultant amplitude is (1 + 2 ) a.(B) the phase of the resultantmotion relative to the first is 90º.(C) the energy associatedwith the resultingmotion is (3 + 2 2 ) times the energy associatedwith any single

motion.(D) the resultingmotion is not simple harmonic.32. Alinear harmonic oscillator offorce constant 2 × 103N/mand amplitude 0.01mhasa totalmechanical energyof 160 J. Its :(A)maximumpotentialenergyis 100 J. (B)maximumkinetic energy is 100 J.(C)maximumpotential energy is 160 J. (D)minimumpotentialenergyis zero.

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SIMPLE HARMONIC MOTIONwww.physicsashok.in 8833. Auniformcylinder of lengthLandmassMhaving cross-sectional areaAis suspended,with its lengthvertical,froma fixed point by amassless spring, such that it is half-submerged in a liquid of density . at equilibrium

position.When the cylinder is given a small downward push and released it starts oscillating verticallywithsmall amplitude. If the force constant of the spring is k, the frequencyof oscillation of the cylinder is :(A)1/ 2 1 k A g2 M. . . .. .. .. (B)1/ 2 1 k A g2 M. . . .

. .. .. (C)1 k gL2 1/ 22 M. . . .. .. .. (D)1/ 2 1 k A g2 A g. . . .. .. . ..34. A particle of mass m is executing oscillations about the origin on the x-axis. Its potential energy isV(x) = k|x|3where k is a positive constant. If the amplitude of oscillations isa, then its time period Tis

(A) proportional to 1/ a (B) independent of a [JEE, 98](C) proportional to a (D) proportional to a3/235. Aparticle free to move along the x-axis has potential energy given byU(x) = k[1  exp(x2)] for . < x < +., where k is a positive constant of appropriatedimensions. The(A) at point awayfromthe origin, the particle is in unstable equilibrium.(B) for any finite nonzero value of x, there is a force directed awayfromthe origin.(C) ifits totalmechanical energyis k/2, it has itsminimumkinetic energyat the origin.(D) for smalldisplacements fromx= 0, themotion is simple harmonic. [JEE, 99]36. Three simple harmonic motions in the same direction having the same amplitud

e a and same period aresuperposed. If each differs in phase fromthe next by45º, then [JEE, 99](A) the resultant amplitude is (1. 2) a(B) the phase of the resultantmotion relative to the first is 90º(C) theenergyassociatedwiththeresultingmotionis (3. 2 2) timesthe energyassociatedwithanysinglemotion.(D) the resultingmotion is not simple harmonic.37. The period of oscillation of simple pendulumof lengthL suspended fromthe roof of a vehiclewhichmoveswithout friction down an inclined plane of inclination . is given by [JEE, 2000] (A)2 L

gcos.. (B)

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2 Lgsin.. (C)2 Lg. (D)

2 Lg tan..38. Aparticle executes simple harmonicmotionbetween x= Aand x= +A.The time takenfor it to go from0 toA/2 is T1 and to go fromA/2 toAisT2. Then [JEE(Scr), 01](A) T1 < T2 (B) T1 > T2 (C) T1 = T2 (D) T1 = 2T239. Aparticle is executing SHMaccording to y = a cos .t. Thenwhich of the graphs represents variations ofpotential energy: [JEE(Scr), 03]I II

iP.E.III IVP.E.x(A) (I)&(III) (B) (II)&(IV) (C) (I)&(IV) (D) (II)&(III)

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SIMPLE HARMONIC MOTIONwww.physicsashok.in 8940. Ablock Pofmassmis placed on a frictionless horizontal surface.Another blockQ of same mass is kept on P and connected to the wallwith the help of a spring ofspring constant k as shownin the figure. µS is the coefficient of friction

QPk µSbetween P andQ. The blocksmove together performing SHMof amplitudeA.Themaximumvalue of the friction force between P andQis [JEE, 04](A) kA (B) kA/2 (C) zero (D)µSmg41. Asimple pendulumhas time period T1.When the period of suspensionmoves vertically up according to theequation y= kt2where k = 1m/s2 and t is time then the time period of the pendulumis T2 then (T1/T2)2 is[JEE(Scr), 05](A) 5/6 (B) 11/10 (C) 6/5 (D) 5/4

42. Function x =Asin2.t + B cos2.t + C sin .t cos .t represents SHM [JEE, 06](A) for any value ofA, B and C (expect C = 0) (B) ifA= B; C = 2B, amplitude = | B 2 |(C) ifA= B; C = 0 (D) ifA= B; C = 2B, amplitude = | B |43. Astudent performs an experiment for determination of g = (4.2l/T2)l = 1mandhe commits anerror of .l.Forthe takes the time on n oscillationswiththe stopwatch of least count .T and he commits a humanerror of 0.1sec. For which of the following data, themeasurement of gwillbemost accurate ? [JEE, 06]. l .T n Amplitude of oscillation(A) 5mm 0.2 sec 10 5mm(B) 5mm 0.2 sec 20 5mm

(C) 5mm 0.1 sec 20 1mm(D) 1mm 0.1 sec 50 1mm44. Thextgraphofaparticleundergoing simpleharmonicmotionis shownbelow. the acceleration of the particle at t = 4/3 s is110 4 12t(s)x(cm)8(A) 3 / 32 .2cm/s2 (B) .2/32 cm/s2(C) .2/32 cm/s2 (D)  3 / 32 .2cm/s2 [JEE, 09]

45. Auniformrod of lengthLandmassMis pivoted at the centre. Its two ends areattached to two springs of equal spring constants k. The springs are fixed to rigidsupports as shown in the figure, and the rod is free to oscillate in the horizontalplane. The rod is gently pushed through a small angle . in one direction andreleased. the frequencyof oscillation is [JEE, 09](A)1 2k2. M (B)1 k2. M (C)1 6k

2. M (D)1 24k2. M

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46. ThemassMshownin the figure oscillates in simple harmonicmotionwith amplitudeA.The amplitude of thepoint Pis [JEE, 09]P M(A) k1 k2 12k A

k (B) 21k Ak(C) 11 2k Ak . k (D) 21 2k Ak . k

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[Take g = 10m/s2]Answer the following questions based on above information :4. At what point will the block leave the platform?(A) 2.5 cmfrommean positionwhen acceleration is acting down and velocity is in upward direction.(B) 2.5 cmfrommean positionwhen platformismoving up.(C) 2.5 cmabovemean positionwhen platformismoving down.

(D) 2.5 cmbelowthemean position.5. Mark the correct statement(s).(A) Normal contact force between the platformand block is constant.(B)As platformapproachesmean position frombottom, the normal contact force between the block andplatformincreases.(C)As platformmoves up awayfrommean position, the normalcontact force between the block and platformdecreases.(D) Both (B) and (C) are correct.6. At what point, the block returns to the platform?(A) 1.3 cmabove equilibriumposition (B) 1.3 cmbelowequilibriumposition

(C) 4.3 cmabove equilibriumposition (D) 4.3 cmbelowequilibriumposition

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SIMPLE HARMONIC MOTIONwww.physicsashok.in 91LEVEL  21. Aringofmassmcanfreelyslide ona smoothverticalrod.Thering is symmetricallyattached with two springs, as shown in the fig., each of stiffness k. Each spring

makes an angle . with the horizontal. If the ring is slightly displaced vertically,determine its time period.k kmrod2. Ablock B ofmassm= 0.5 kg is attachedwith upper end of a vertical spring of forceconstant K= 1000Nm1 as shown in fig.Another identical block A falls froma heighth = 49.5 cmon the block B and gets stuckwith it. The combined body starts to performverticaloscillations.

hABCalculate amplitude of these oscillations. (g= 10ms2)3. One end of an ideal spring is fixed to awall at originOand axis of spring isparallel to x-axis.Ablock ofmassm= 1 kg is attached to free end of the spring and it is performing SHM. Equation of position of the block inco-ordinate systemshown in Figure is x = 10 + 3 sin(10t), where t is in second and x in cm.Another block ofmassM=3 kg,moving towards the originwith velocity30 cm/sec collides with the block performing SHM at t = 0 and getsstuck to it. Calculate

1 kg3 kgx(i) newamplitude of oscillations, O(ii) newequation for position of the combined bodyand (iii) loss of energyduring collision.Neglect friction.4. Two identical blocks Aand B ofmassm= 3 kg are attached with ends of an idealspring of force constantK = 2000 Nm1 and rest over a smooth horizontal floor.Another identical block C moving with velocityv0 = 0.6ms1 as shown in fig. 100 strikes the blockA and gets stuck to it. Calculate for subsequent motion

m V K 0(i) velocityofcentre ofmass of the system, m m(ii) frequencyof oscillations of the system,(iii) oscillation energyof the system, and (iv)maximumcompressionof the spring.5. In the arrangement shown in fig, pulleys are small and light and springs areideal.K1, K2, K3 and K4 are force constants of the springs.Calculate period ofsmall vertical oscillations of block ofmassm.K2 K4K1 K3m6. Fig., shows a particle ofmassm= 100 gm, attachedwith four identical springs,eachoflengthl =10 cm. InitialtensionineachspringisF0 =25 newton.Neglecting

gravity, calculateperiodofsmalloscillationsoftheparticle alonga line perpendicularto the plane of the figure. p

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A mBCD7. In the arrangement shown in Fig., bodyB is a solid cylinder of radius R = 10cmwithmassM= 4 kg. It can rotatewithout friction about a fixed horizontal axisO

Ablock A ofmass m = 2 kg suspended by an inextensible thread is wrappedaround the cylinder.Ahorizontal light spring of force constant K= 100 Nm1fixed at one end keeps the systemin static equilibrium. CalculateARO(i) initial elongationinthe spring and B(ii) period of small vertical oscillations of the block. (g = 10ms2)

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SIMPLE HARMONIC MOTIONwww.physicsashok.in 928. Arigid rod ofmassmwith a ballofmassMattached to the free end is restrainedto oscillate in a verticalplane as shown in the figure. Find the naturalfrequencyofoscillation.

M3/4LL/4km9. Find the period of small oscillations in a vertical plane performed by a ball ofmassmfixed at themidle of ahorizontally stretched string of length l. The tension of the string is assumedto be constant and equal toF >> mg.10. Aparticlemoves along the x-axis according to the lawx = a cos .t. Find the distance that the particle covers

during the time interval t =0 to t.11. Find the amplitude, frequencyand epoch ofthe simple harmonicmotionrepresented byx=3 sin .t +4 cos .t.12. Ahelical spring elongates 10 cmwhen subjected to a tension of 5N. Find themasswhich should be attachedto the bottomof the spring so that when pulled down and released themasswillvibrate twice per second. Findalso itsmaximumvelocitywhen the amplitude of vibration is 1 cm.13. Find the frequency of small oscillations of the arrangement illustrated in figure. Theradius of the pulley isR, itsmoment of inertia relative to the axis of rotationis I, themass of the body ism, and the spring stiffness is k. The mass of the thread and

thespring are negligible, the thread does not slip over the pulleyand there is no friction inthe axis of the pulley.mR14. Aplankwith a body ofmassmon it executes simple harmonicmotionof cyclicfrequency .= 11 rad s1 between the levels 1 and 2 separated by a distance 2aas shown infig. Find :(a) the force that the body exerts on the plank as it moves fromlevel 1 to 2 whena = 4 cm;

a maO2(b) theminimumvalue of awhenthe bodystarts falling behind the plank; 1(c) the amplitude of oscillation at whichthe body jumps up to a height h= 50 cmrelative to level 1.15. (a)Ablock ofmassmis tied to one end of a stringwhich passes over a smoothfixed pulleyAand under a lightsmoothmovable pulleyB. Theother end of the string is attached to the lower end ofa spring ofspring constantk2. Find the period of smalloscillations ofmassmabout its equilibriumposition.m

ABk2

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k1Figure (a)mABk2k1

Figure (b)(b)Ablock ofmassmis attached to one end ofa light inextensible string passing over a smooth light pulleyBand under another smooth light pulleyAas shownin the figure.The other end of a string is fixed to a ceiling.Aand B are held by springs of spring constants k1 and k2. Find angular frequencyof small oscillation of thesystem.

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SIMPLE HARMONIC MOTIONwww.physicsashok.in 9316. Find the angular frequencyofmotion of blockmfor smallmotion of rodBDwhenweneglect the inertial effects of rod BD. Spring constant are k1 and k2.Neglect frictionforces.

BDCk b 1k2m17. Find the angular frequency for small oscillation of block ofmassmin the arrangement shown in the figure.Neglect themass of the rod.l/2 l/2k2k1

mk318. ArodAB ofmassMis attached as shown belowto a spring of constant k.Asmall block ofmassmis placed on the rod at its free endA.(i) if endAismoved down througha smalldistance d and released, determine theperiod of vibration. bmkBAl(ii) determine the largest allowable value of d if the blockmis to remain at all 

times in contactwith the rod.19. A rod ofmass mand length l is hinged at its upper end and carries a block of mass M at its lower end. Two springs having the spring constants k1 and k2respectively are attached to the rod at the distances b and c fromthe hinge asshown in the figure. Find the angular frequencyofvibrationfor smalloscillationof the system. Mlbmk1

k2C20. Auniformbroad of lengthL andweightWis balanced on a fixed semi-circularcylinder of radiusR as shown in the figure. If the plank is tilted slightlyfromitsequilibriumposition, determine the period ofits oscillations.lr21. Aplankwith a body ofmassmplaced on it start moving straight up according tothe lawy = a(1  cos .t),where y is the displacement fromthe initial position, .= 11 rad/s. Find :(a) The time dependence of the force that the body exerts on the plank.(b) Theminimumamplitude of oscillation ofthe plank at which the bodystarts falli

ng behind the plank.22. Aparticle ofmassmfree to move in the xy plane is subjected to a forcewhose components are Fx =kx and

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Fy = ky, where k is a constant. The particle is released when t = 0 at the point(2, 3). Prove that thesubsequentmotion is simple harmonic along the straight line 2y  3x = 0.23. In the shown arrangement, both the springs are in their natural lengths. The coefficient of friction betweenm2 andm1 isµ. There is no friction betweenm1and the surface. If the blocks are displaced slightly, they together perform

simple harmonicmotion.Obtaion.mm12 k1k2(a) Frequencyof such oscillations.(b) The conditionif the frictional force on blockm2 is to act in the direction of its displacement frommean position.(c) If the condition obtained in (b) ismet, what canbemaximumamplitude of theiroscillations ?

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SIMPLE HARMONIC MOTIONwww.physicsashok.in 94LEVEL  31. Statewhether true or falseTwo simple harmonicmotions are represented by the equationsx1 = 5sin [2.t + ./4] and x2 = 5 2 (sin 2.t + cos2.t) their amplitudes are in th

e ratio 1 : 2 [REE,96]2. Ablock is kept on a horizontal table. the table is undergoing simple harmonicmotion of frequency 3Hz in ahorizontalplane.The coefficient of static frictionbetweenblock and the table surface is 0.72. Find themaximumamplitude of the table at which the block does not slip on the surface. [REE,96] 3. Abob ofmassMis attached to the lower end of a vertical string of lengthL andcross sectional areaA.TheYoungsmodulus of thematerial of the string isY. If the bob executes SHMin the verticaldirection, find thefrequencyofthese oscillations. [REE, 2000]

4. Adiatomicmolecule has atoms ofmassesm1 andm2. The potentialenergyof themolecule for the interatomicseparation r is given byV(r) = A+B(r  r0)2,where r0 is the equilibriumseparation,andAandBare positiveconstants. The atoms are compressed towards eachother fromtheir equilibriumpositions and released.Whatis the vibrational frequencyof themolecule ? [REE, 01]5. Two massesm1 andm2 connected bya light spring ofnatural length l0 is compressed completelyand tied byastring. This systemwhilemoving with a velocity v0 along +ve xaxis pass through the origin at t = 0.At thisposition the string snaps. Position ofmassm1 at time t is given bythe equation.x1(t) = v0t A(1  cos .t)

Calculate : [JEE, 03](a) Position of the particlem2 as a function of time.(b) l0 in terms ofA.6. Asmallbodyattached to one end of a verticallyhanging spring is performing SHMabout itsmean position with angular frequency . and amplitude a. If at a height y* fromthe meanposition the bodygets detached fromthe spring, calculate the value of y* so that the heightmy0H attained by the mass ismaximum. The body does not interact with the spring dur

ing itssubsequent motion after detachment. (a.2 > g). [JEE, 05]

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SIMPLE HARMONIC MOTIONwww.physicsashok.in 95Answer KeyASSERTION-REASON TYPEQ. 1 2 3 4 5Ans. A C A B D

MATCH THE COLUMN1. [(AR), (BP), (CQ), (DS)]2. [(AQR), (BQRS), (CP), (DP)]3. [(ARS), (BRS), (CPQS), (DRS)]LEVEL  1Q. 1 2 3 4 5 6 7 8 9 10Ans. B B C A B BC A B A AQ. 11 12 13 14 15 16 17 18 19 20Ans. C ABC A CD D D B D B DQ. 21 22 23 24 25 26 27 28 29 30Ans. C C B C B ABC ACD AC BCD BQ. 31 32 33 34 35 36 37 38 39 40

Ans. AC BC B A D AC A A A BQ. 41 42 43 44 45 46Ans. C ABD D D C DPASSAGEQ. 1 2 3 4 5 6Ans. C B A A C DLEVEL  21.T 2 m2k. . 2. 5 cm 3. (i) 3 cm; (ii) x = 10 + 3sin(5t + .)cmor x = 10  3sin(5t) cm; (iii) 0.135 Joule4. (i) 0.2m/sec.; (ii)

5 10 Hz.; (iii) 0.09 Joule; (iv) 3 10 mm. 5.1 2 3 4T 4 m 1 1 1 1k k k k. .. . . . . . . . .6. 0.02 . sec. 7. (i) 20 cm; (ii) 0.4 . sec. 8.f 1 3k2 27M 7m.

. . 9.T mF. . l

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SIMPLE HARMONIC MOTIONwww.physicsashok.in 9610. S = a[n + 1  cos (.t  n2.

)] when n is even; S = [n + cos (.t  n2.)] when n is odd.11. A= 5, f = ./2. and epoch = 53º 12. m= 310 g, vmax = 12.5 cm/sec. 13. 2f 1 k2 m 1/R.. .14. (a)a 2 N mg 1 cos t

g. . .. . . . . . . ; (b) min 2a . g. ; (c) a = 0.2 m.15. (a) 1 21 2m(k k )T 2k k.. . ; (b) 1 21 2

1 k k2 m(k k ). .. 16. . .21 22 21 2k k Cm k C k (a b). .. .

17. 2 312 31 4k k km k 4k. .. . . . . . . .18. (i)22T 2 (m M/ 3)kb.

. . l; (ii)2

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max 2g (m M/ 3)dkb.. l19.

2 21 22(k b k c ) (M/m/ 2)g(M m/ 3). .. ..ll 20.T 2 L 3 gr4

. . 21. (a) N = m(gta.2 cos.t); (b) 8 km.23. (a) 1 21 21 k k2 m m.. . ; (b) 1 12 2k mk m. ; (c) 1 2 21 2 2 1µ(m m )m g

m k m k..LEVEL  31. True 2. 2cm 3.1 YA2. ML 4. 1 21 21 m m 2f 2B(m m ). ..

5. (a) 12 02mx v t A(1 cos t)m. . . . ; (b) 102m1 Am. .

. . . . . .l 6. 2y mg g a

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k. . ..

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OPTICS

DHANALAKSHMI NAGARNEAR ANNAMAIAH CIRCLE,

TIRUPATI.PH NO. 9440025125

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OPTICSwww.physicsashok.in 1Review of Concepts(a) Due to reflection, none of frequency, wavelength and speed of light change.(B) Law of reflection :(i) Incident ray, reflected ray and normal on inident point are coplanar.

(ii) The angle of incidence is equal to angle of reflection.Some important points : In case of plane mirror :(i) For real object, image is virtual.(ii) For virtual object, image is real.(iii) Image size = Object size.(iv) The converging point of incident beam behaves as object.(v) If incident beam on optical instrument (mirror, lens etc) is converging in nature, object isvirtual.(vi) If incident beam on the optical instrument is diverging in nature, the object is real.(vii) The converging point of reflected or refracted beam from an optical instru

ment behaves aimage.(viii) If reflected beam or refracted beam from an optical instrument is converging in nature,image is real.RealImagePVirtualObjectPnn

(ix) If reflected beam or refracted beam froman optical instrument is diverging in nature,PP RealObjectVirtualObjectnn ...

.image is virtual.(x) For solving the problem, the reference frame is chosen in which optical instrument (mirror,lens, etc.) is in rest.(xi) The formation of image and size of image is independent of size of mirror.(xii) Visual region and intensity of imagedepend on size of mirror.

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OPTICSwww.physicsashok.in 2(xiii) If the plane mirror is rotated through an angle . , the reflected ray and image is rotatedthrough an angle 2 . in the same sense.(xiv) If mirror is cut into a number of pieces, then the focal length does not c

hange.(xv) The minimum height of mirror required to see the full image of a man of height h is h/2.ImageRestObject(xvi)vImageRestObject(xvii)

v sin.v sin.v cos. v cos..ImageRestObject(xviii)vvm 2 vm - v Object ImageIn rest(xix)vm 2 vm

ImageObject(xx)vvm 2 vm + v(C) Number of images formed by combination of two plane mirrors : The images formed bycombination of two plane mirrors are lying on a circle whose centre is at the meeting points of mirrors.Also, object is lying on that circle.Here, ..

n . 360where . = angle between mirrors.(i) If .360.is even number, the number of image is n1.(ii) If .360.is odd number and object is placed on bisector of angle between mirrors, thennumber of images is n1.(iii) If .360.is odd and object is not situated on bisector of angle between mirrors, then the 

number of images is euqal to n.(D) Law of reflecteion in vector form :Let e1 = unit vecotr along incident ray.

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Let e 2 = unit vector along reflected raynn 1 .2 .n = unit vector along normal on point of incidenceThen, e 2 . e1 . 2 .e1.n .n 

(e) Spherical mirrors :(i) It is easy to solve the problems in geometrical optics by the help of co-ordinate sign convention.

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OPTICSwww.physicsashok.in 3x xy yx x

y yx xy yx xy yx xy y(ii) The mirror formula is .

. ..1u1 1Also, R = 2 .These formulae are only aplicable for paraxial rays.(iii) All distances are measured from optical centre. It means optical centre is taken as origin.(iv) The sign convention are only applicable in given values.(v) The transverse magnification isobject size uimage size ..

. . .1. If object and image both are real, . is negative.2. If object and image both are virtual, . is negative3. If object is real but image is virtual; . is positive.4. If object is virtual but image is real, . is positive.DFdSun..5. Image of star; moon or distant object is formed

at focus of mirror.If y = the ddistance of sun or moon from earth.D = diameter of moon or suns disc.. = focal length of the mirrord = diameter of the image. = the angle subtended by sun or moons discThen tan . = . = .. dyDHere, . is in radian.Laws of Refraction1. (a)The incident ray, the refracted ray and normal on incidence point are copl

anar.(B) .1 sin.1 . .2 sin.2 ...cons tant ..1

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.2

.2

.1(C) Snells law in vector form:.1.22 .

1 .n Let, e1 = unit vector along incident raye 2 = unit vecotr along refracted.

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OPTICSwww.physicsashok.in 4n = unit vector along normal on incidence point.Then .1 .e1 . n . . .2 .e 2 . n ..Some important points :(i) The value of absolute retractive index . is always greater or equal to one.

(ii) The value of refractive index depends uponmaterial of medium, colour of light and temperatureof medium.(iii) When temperature increases, refractive index decreases.(iv) Optical path is defined as product of geometrical path and refractive index.i.e., optical path = .x(v) For a given time, optical path remains constant.i.e., .1 x1 . .2 x2 . . cons tant. dtdxdt

dx 221.1 . .. .1 c1 . .2 c2.2112c. c.

.i.e., c. . 1(vi) The frequency of light does not depend upon medium.. c1 . ..1 , c2 . . .2.121221

cc... .... .. . 12. (a) When observer is in rarer medium and object is in denser medium:Then apparent depth. . real depth(B) When object is in air and observer is in

AirObserverDenser medium

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(.)P ObjectApparentdepthPReal

depthdenser medium:real position. . apparent position(C) The shift of object due to slab is . .... ....x . t 1. 1(i) This formula is ony applicable when observer is inrarer medium.

(ii) The object shiftness does not depend upon theP P QObjectshiftness= xtposition of object.(iii) Object shiftness takes place in the direction ofincidence ray.(D) The equivalent rerfractive index of a combination of a number of slabs for normal incidenceisi

iitt.... .Here, .ti = t1 + t2 + .......t1 .1.2 t2..

..

.

.

.

.2211iti t t

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OPTICSwww.physicsashok.in 5(e) The apparent depth due to a number of media isiti.

. .itrd.(f) The lateral shifting due to a slab is d = t sec r sin (i  r).3. (a) Cricital angle : When a ray passes from denser medium ( .2 )to rarer medium ( .1), then for 90° angle of refraction, thecRarer.1

Denser.290°corresponding angle of incidence is critical angle.Mathematically,2sinC 1...(B) (i) When angle of incidence is lesser thancritical angle, refraction takes place. thecorresponding deviation is

sin sini i11 2 . . .... ...... . .for i = CRarer medium(. ) 1

irc i ii < C i = C Denser medium(. ) 2(ii) When angle of incidence is greater thancritical angle, total internal reflection takesplace. the corresponding deviation is. . . . 2i when i > C4. The . . i graph is(i) Critical angle depends upon colour of light, material of medium,ci

../2and temperature of medium.

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(ii) Critical angle does not depend upon angle of incidenct.PRISM(a) Deviation produced by prism is ..i.i..A.(B) r + r = A(C) For grazing incidence, i = 90°(D) For grazing emergence, i = 90°(e) For not transmitting the ray from prism, . > cosec

2AAB Cn n r r i i (f) For limiting angle of prism, i = i = 90°, the limitingangle of prism = 2C where C is critical angle. If angleof prism exceeds the limiting values, then the raysare totally reflected.(g) . . i graph for prism:

(h) For minimum deviation,(i) i = i and r = r (ii)2sin A2A sin m ...... . .. .i..m

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(a) Lens formula is only applicable for thin lens.(B) r = 2 . formula is not applicable for lens.(C) object size um image size.. .(D) Magnification formula is only applicable when object is perpendicualr to opt

ical axis.(e) Lens formula and the magnification formula is only applicable when medium on both sides oflenses are same.(f) f(+ve)(i)f(-ve)(ii)f(-ve)(iii)f(+ve)(iv)

f(-ve)(v)f(+ve)(vi)(g) This lens formula is applicable for converging as well diverging lens. .1 .1 .Thin lens makers formula : 2. .... .... . .... ..

.

.

. . .

.

. 1 1 22 1r1r1 1

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OPTICSwww.physicsashok.in 72. (a) Thin lens formula is only applicable for paraxial ray.(B) This formula is only applicable when medium on both sides of lens are same.(C) Intensity is proportional to square of aperture.(D) When lens is placed in a medium whose refractive index is greater than that

of lens. i.e.,.1 . .2 . Then converging lens behaves as diverging lens and vice versa.(e) When medium on both sides of lens are not same. Then both focal lengths arenot same toeach other.(f) If a lens is cut along the diameter, focal length does not change.(g) If lens is cut by a vertical, it converts into two lenses of different+ff1 f2focal lengths.i.e.,

1 21 1 1.....(h) If a lens is made of a number of layers of different refractiveindex (shown in figure).Then number of images of an object formed by the lens is+ + ++ + +.1.

2.3.4.5.6equal to number of different media.(i) The minimum distance between real object and image in is 4 . .(j) The equivalent focal elngth of co-axial combination of two d

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1 21 1F1(l) (i) Power of thin lens,FP . 1

(ii) Power of mirror isFP . . 1(m) If a lens is silvered at one surface, then the system behaves as an equivalent mirror, whosepower P = 2PL + PmHere, PL = Power of lens = . .... .... . ...

. ...

.

. . .1 1 22 1r1r1Pm = Power of silvered surfaceFm. . 1

Here,2F r2m . , where r2 = radius of silvered surface.FP . . 1Here, F = focal length of equivalent mirror.

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.

. . .

. . . .

. .(A) (B) (C) (D) (E)

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OPTICSwww.physicsashok.in 911. Assertion (A) : A convex lens of glass .. .1.5. behave as diverging lens when immersed in carbondisulphide of higher refractive index .. .1.65..Reason (R) : A diverging lens is thinner in the middle and thicker at the edges.

 (A) (B) (C) (D) (E)12. Assertion (A) : A biconvex lens of focal length 10 cm is split into two equal parts by a plane parallel toits principal axis. The focal length of each part will be 20 cm.Reason (R) : The focal length depends on how many parts the convex lens has been split.(A) (B) (C) (D) (E)13. Assertion (A) : Radius of curvature of a convex mirror is 20 cm. If a real object is placed at 10 cmfrom pole of the mirror, image is formed at infinity.Reason (R) : When object is placed at focus, its image is formed at infinity.

(A) (B) (C) (D) (E)14. Assertion (A) : For a prism of refracting angle 60° and refractive index 2 , minimum deviation is30°.Reason (R) : At minimum deviation, 1 2 302r . r . A . .(A) (B) (C) (D) (E)15. Assertion (A) : Image formed by concave lens is not always virtual.Reason (R) : Image formed by a lens if the image is formed in the direction of ray of light with(A) (B) (C) (D) (E)16. Assertion (A) : Minimum deviation for a given prism does not depend on the r

efractive index . ofthe prism.Reason (R) : Deviation by a prism is given by . . 1 2 . . i . i . A and does not have the term . .(A) (B) (C) (D) (E)Level # 1.Objective Type QuestionMultiple Choice Question with ONE correct answer :1. Two plane mirrors M1 and M2 are inclined to each other at 70°. A ray incident on the mirror M1 at an angle. falls on M2 and is then reflected parallel to M1 for(A) . = 45° (B) . = 50° (C) . = 55° (D) . = 60°

2. A light ray is incident on a horizontal plane mirror at an angle of 45°.At what angle should a second plane mirror be placed in order that thereflected ray finally be reflected horizontally from the second mirror,as shown in figure.(A) . = 30° (B) . = 24°(C) . = 22.5° (D) . = 67.5°3. A plane mirror is placed in y-z plane facing towards negative x-axis. The mirror is moving parallel to yaxiswith a speed of 5 cm/s. A point object P is moving infront of the mirror with avelocity (3 cm/s)i+ (4cm/s) j  + (5 cm/s)k. Find the velocity of image with respect to mirror

(A) (3 cm/s)i+ (4 cm/s) j  + (5 cm/s)k(B) (3 cm/s)i

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+ (4 cm/s) j  + (5 cm/s)k (C) (3 cm/s)i (4 cm/s) j   (5 cm/s)k(D) none of the above.

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OPTICSwww.physicsashok.in 104. The size of the face of a dancer is 24 cm x 16 cm. Find the minimum size of a plane mirror required to seethe face of dancer completely by(i) one eyed dancer. (ii) two eyed dancer.

(Distance between the eyes is 4 cm.)(A) (i) 12 x 8 cm2 (ii) 12 x 6 cm2(B) (i) 8 x 10 cm2 (ii) 12 x 2 cm2(C) (i) 10 x 12 cm2 (ii) 9 x 8 cm2(D) (i) 12 x 2 cm2 (ii) 6 x 13 cm25. A bullet of mass m2 is fired from a gun of mass m1 with horizontal velocity v. A plane mirror is fixed at gunfacing towards bullet. The velocity of the image of bullet formed by the plane mirror with respect to bulletis(A) . ...

. ...

.12m1 m (B) . . .... ... .11 2m

m m(C). . ..12 1m2 m m(D) none of these6. In the given figure, the angle of reflection is(A) 30° (B) 60°(C) 45° (D) none of these.7. Two plane mirrors A and B are aligned parallel to each other, as shown in

figure. A light ray is incident at an angle of 30° at a point just inside oneend of A. The plane of incidence coincides with the plane of figure. The 0.2 mAB30°2 3 mmaximum number of times the ray undergoes reflections (excluding thefirst one) before it emerges out is(A) 28 (B) 30 (C) 32 (D) 348. A point source of light B is placed at a distance L in front of the centre of amirror of width d hung vertically on a wall as shown. A man walks in front ofthe mirror along a line parallel to the mirror at a distance 2L from it as shown

.dB

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L2 LThe greatest distance over which he can see the image of the light sourcein the mirror is(A) ½ d (B) d (C) 2d (D) 3d9. A plane mirror having a mass m is tied to the free end of a masslessspring of spring constant k. The other end of the spring is attached to

a wall. The spring with the mirror held vertically to the floor can slidealong it smoothly. When the spring is at its natural length, the mirrorWallis found to be moving at a speed of v cm/s. The separation between kthe images of a man standing before the mirror, when the mirror isin its extreme positions(A)kv m (B) km2v

(C)k2v m (D) k4v m10. Two spherical mirrors M1 and M2, one convex and other concave having same radiusof curvature R are arranged coaxially at a distance 2R (consider their pole separationto be 2R). A bead of radius a is placed at the pole of the convex mirror a shown. TheM1 M2ratio of the size of the first three images of the bead is(A) 1 : 2 : 3 (B) 1 : 3

: 121(C) 41: 111: 131(D) 3 : 11 : 4111. An object is placed in front of a convex mirror at a distance of 50 cm. A plane mirror is introduced

covering the lower half of the convex mirror. If the distance between the object and the plane mirror is 30cm, there is no parallax between the images formed by the two mirrors. The radius of curvature of theconvex mirror (in cm) is(A) 60 (B) 50 (C) 30 (D) 25

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OPTICSwww.physicsashok.in 1112. A rectangular glass slab ABCD of refractive index n1, is immersed in water of refractive index n2 (n1 > n2).A ray of light is incident at the surface AB of the slab as shown. The maximum value of the angle of

incidence .max , such that the ray comes out only from the other surface CD is given by(A) .... ..... .. .... ...

. ...

. ..

. . .11 221 1ncos sin nnsin n (B) .... .

..

.. .

. .

.

.

. .

.

.

. ..

.

. ..

. . .

2111nsin n cos sin 1(C) . .... ... .21 1n

sin n (D) . .... ..

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. .11 2nsin n13. Two thin slabs of refractive indices .1 and .

2 are placed parallel toeach other in the x-z plane. If the direction of propagation of a rayin the two media are along the vectors jb ia r 1 . . and jd ir 2 . c .xy.2.1then we have

(A) .1a . .2b (B) 2 222 21c dca ba.....

(C) .1 (a2 + b2) = .2 (c2 + d2) (D) none of these14. A man stands on a glass slab of height . and inside an elevator acceleratedupwards with a. Thebottom of the slab appears to have shifted with respect to the man by a distance (if the R. I. of the glassis .g)(A) less then . .g (B) greater than . .g (C) equal to . .g (D) cant be said.15. A ray of light travels from a medium of refractive index . into air. If theangle of incidence at the plane

surface of separation is . and the corresponding angle of deviation is D, the variation D with . is showncorrectly by the figure.C C CDD D DD1 D1 D1 D2 D2D2(0, 0) . ./2 (0, 0) . (0, 0) . ./2 (0, 0) . ./2(A) (B) (C) (D)16. An observer can see through a pin hole at the top end of a thin rod of heighth placed as shown in the figure. Beaker height is 3h and its radius is h.

When the beaker is filled with a liquid up to a height 2h, he can see theh2 h

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3 hEyelower end of the rod. Then refractive index of the liquid is(A) 23(B)2

3 (C) 25(D) 25.(Assume that the distance between rod and the wall is negligible).17. A glass sphere of radius 5 x 102 m has a small bubble 2 x 102 m from its centre. Bubble is viewed alongthe diameter of the sphere, from the side on which it lies. If refractive indexof glass is 1.5 then how farfrom the surface will the bubble appear?(A) 2.1 cm (B) 2.5 cm (C) 1.5 cm (D) 2.0 cm

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OPTICSwww.physicsashok.in 1218. A ray of light travelling in a transparent medium falls on a surface separating the medium from air at anangle of incident 45°. The ray undergoes total internal reflection. If . is the refractive index of the medium

with respect to air, select the possible value(s) of . from the following :(A) 1.3 (B) 1.4 (C) 1.5 (D) 1.719. A tank contains a transparent liquid of refractive index n the bottom of whichis made of a mirror as shown. An object O lies at a height d above the mirror.A person P vertically above the object sees O and its image in the mirror anddOPfinds the apparent separation to be(A) 2nd (B) n 12d

. (C)n2d(D) .1 n.nd .20. A fish looks up at the surface of a perfectly smooth lake. The surface appears dark except a circulararea directly above it. The plane angle . that this illuminated region subtendsis(A) 48.6° (B) 24.3° (C) 97.2° (D) 12.15°21. A ray of light enters an anisotropic medium from vacuum at grazing incidence. If . is the angle made by

the reflected ray inside the medium with the interface and n( . ) is the refractive index of the mediumthen,(A) n( . ) sin . = 1 (B) n( . ) cos . = 1 (C) 1sinn( ) ...(D) 1cosn( ) ..

.22. The slab of a material of refractive index 2 shown in figure hasa curved surface. APB of radius of curvature 10 cm and a planesurface CD. On the left of APB is air and on the right of CD iswater with refractive indices as given in figure. An object O isplaced at a distance of 15 cm from pole P as shown. Thedistance of the final image of O from P, as viewed from the left is(A) 20 cm (B) 30 cm (C) 40 cm (D) 50 cm23. An object is placed at a distance of 12 cm from a convex lens on its principal axis and a virtual image ofcertain size is formed. On moving the object 8 cm away from the lens, a real image of the same size asthat of virtual image is formed. The focal length of the lens in cm is

(A) 15 (B) 16 (C) 17 (D) 1824. A spherical surface of radius of curvature R separates air (refractive index 1.0) from glass (refractive

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index 1.5). The centre of curvature is in the glass. A point object P placed inair is found to have a realimage Q in the glass. The line PQ cuts the surface at the point O and PO = OQ. The distance PO is equalto(A) 5 R (B) 3 R (C) 2 R (D) 1.5 R25. A lens of focal length . is placed in between an object and screen fixed at

a distance D. The lens formstwo real images of object on the screen for two of its different positions, a distance x apart. The two realimages have magnifications m1 and m2 respectively (m1 > m2).(A)m1 m2x.. . (B) m1 m2 = 1 (C) 4DD2 . x2. . (D) all the above

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OPTICSwww.physicsashok.in 1326. A liquid of refractive index 1.33 is placed between two identicalplano-convex lenses, with refractive index 1.50. Two possibleP Qarrangement P and Q are shown. The system is

(A) divergent in P, convergent in Q.(B) convergent in P, divergent in Q.(C) convergent in both(D) divergent in both.27. A lens of refractive index . is put in a liquid of refractive index .. . Ifthe focal length of the lens in air is. , its focal length in liquid will be(A). ... . .. ... . .1(B)

. ... .. .1.

. .. . .(C). .. ... . .... . .1(D) .. . ...... .28. A convergent lens is placed inside a cell filled with a liquid. The lens has a focal length +20 cm when inair and its material has a refractive index 1.50. If the liquid has a refractive index 1.60, the focal length of

the system(A) 160 cm (B)  24 cm (D) 80 cm (D) + 80 cm29. A double convex lens, made of glass of refractive index 1.5, has focal length 6 cm. The radius of curvatureof one surface is double than that of other surface. The small radius of curvature has value(A) 4.5 cm (B) 6 cm (C) 4 cm (D) 9 cm30. If the distance between a projector and screen is increased by 1%, then illumination on the screendecreases by(A) 1 % (B) 2 % (C) 3 % (D) 4 %31. A lens forms a sharp image of a real object on a screen. On inserting a para

llel slide between the lensand the screen with its thickness along the principal axis of the lens it is found necessary to shift thescreen parallel to itself d away from the lens for getting image sharply focused on it. If the refractiveindex of the glass relative to air is ., the thickness of slab is(A) .d(B) .d (C) 1d. ..(D) . .

.. .1 d32. A thin convex lens in used to form a real image of a bright point object. Th

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eapeture of the lens is small. A graph, shown is obtained by plotting a suitableOXY-1.

parameter Y against another suitable parameter x.If . = the focal length of the lensu = object distancev = image distanceand Real Positive Convention is used then(A) (uV) . x; (u + V) . y (B) (u + V) . x; (uV) . y(C) u . x;vu. y (D) u1. x;

v1. y

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OPTICSwww.physicsashok.in 1433. Which of the following best represents object distance u vs image distance v graph for a convex lens.(A)y

.(B)y.(C)y.(D)y.34. Three thin prisms are combined as shown in figure. The refractive indices of the crown glass for red,

yellow and violet rays are .r, .y and .v respectively and those for the flint glass are .r, .y and .vrespectively. The ratio A/A for which there is no net angular dispersion.(A). .12 1yy. .. .

(B)y yy y2 ... .. . .(C). .. . y yy y1

1. . .. . ...(D)y y2 y . y... . .. .35. A point object is placed at distance of 0.3 m from a convex lens of focal le

ngth0.2 m cut into two equal halves, each of which is displaced by 0.0005 m, as shown

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in figure. If C1 and C2 be their optical centres then,(A) an image is formed at a distance of 0.6 m from C1 or C2 along principal axis.(B) two images are formed, one at a distance of 0.6 m and other at a distance OC1C2of 1.2 m from C1 or C2 along principal axis.

(C) an image is formed at a distance of 0.12 m from C1 or C2 along principal axis.(D) two images are formed at a distance of 0.6 m from C1 or C2 along principal axisat a separation of 0.003 m.36. A glass prism of refractive index 1.5 is immersed in water (refractive index 4/3). A light beam incidentnormally on the face AB is totally reflected to reach on the face BC if (1983)A B.(A)sin 8

9. . (B)2 sin 83 9. . (C)sin 23. .37. A ray of light from a denser medium strike a rarer medium at an angle of incidence i (see Figure). Thereflected and refracted rays make an angle of 90° with each other. The angles of reflection and refractionare r and r The critical angle is

i rr'(A) sin.1 .tan r . (B) sin.1 .tani. (C) sin.1 .tan r.. (D) tan.1 .sin i.

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OPTICSwww.physicsashok.in 1538. Two coherent monochromatic light beams of intensities . and 4 . are superposed. The maximum andminimum possible intensities in the resulting beam are(A) 5 . and . (B) 5 . and 3 . (C) 9 . and . (D) 9 . and 3 .

39. An isosceles prism of angle 120° has a refractive index 1.44. The parallel monochromatic rays enterthe prism parallel to each other in air as shown. The rays emerge from the opposite faces120°(A) are parallel to each other (B) are diverging(C) make an angle 2 [sin1 (0.72)  30°] with each other(D) make an angle 2 sin1 (0.72) with each other40. A diminished image of an object is to be obtained on a screen 1.0 m from it. This can be achieved byappropriately placing(A) a concave mirror of suitable focal length (B) a convex mirror of suitable fo

cal length(C) a convex lens of focal length less than 0.25 m (D) a concave lens of suitable focal length41. A concave lens of glass, refractive index 1.5 has both surfaces of same radius of curvature R. Onimmersion in a medium of refractive index 1.75, it will behave as a(A) convergent lens of focal length 3.5 R (B) convergent lens of focal length 3.0 R(C) divergent lens of focal length 3.5 R (D) divergent lens of focal length 3.0R42. A hollow double concave lens is made of very thin transparent material. It can be filled with air or eitherof two liquids L1 and L2 having refractive indices 1 . and .2 respectively ..2 .

 .1 .1. . the lens willdiverge a parallel beam of light if it is filled with(A) air and placed in air (B) air and immersed in L1(C) L1 and immersed in L2 (D) L2 and immersed in L143. A diverging beam of light from a point source Is having divergence angle . , falls symmetrically on aglass slab as shown. The angles of incidence of the two extreme rays are equal.If the thickness of theglass slab is t and the refractive index n, then the divergence angle of the emergent beam is(A) zero (B) . (C)sin 1 1

n. . .. .. . (D)2sin 1 1n. . .. .. .44. A ray of light passes through four transparent media with refractive indices 1 . , 2 . , 3 . and 4 . asshown in the figure. the surfaces of all media are parallel. If the emergent ray CD is parallel to the

incident ray AB, we must have(A) 1 2 . . . (B) 2 3 . . . (C) 3 4 . . . (D) 4 1 . . .

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OPTICSwww.physicsashok.in 1645. A given ray of light suffers minimum deviation in an equilateral prism p, Additional prism Q and R ofidentical shape and of the same material as P are now added as shown in the figure. The ray will now

sufferP Q R(A) greater deviation (B) no deviation(C) same deviation as before (D) total internal reflection46. Which one of the following spherical lenses does not exhibit dispersion? The radii of curvature of thesurfaces of the lenses are as given in the diagrams.(A) R1 R2 (B) R (C) R R (D) R47. Two plane mirrors A and B are aligned parallel to each other, as shown in the figure A light ray isincident at an angle 30° at a point just inside one end of A. The plane of incidence coincides with the

plane of the figure. The maximum number of times the ray undergoes reflections (including the firstone) before it emerges out is(A) 28 (B) 30 (C) 32 (D) 3448. The size of the image of an object, which is at infinity, as formed by a convex lens of focal length 30 cmis 2 cm. If a concave lens of focal length 2 0 cm is placed between the convex lens and the image at adistance of 26 cm from the convex lens, calculate the new size of the image.(A) 1.25 cm (B) 2.5 cm (C) 1.05 cm (D) 2 cm49. A ray of light is incident at the glass-water interface at an angle i, it emerges finally parallel to thesurface of water, then the value of g . would be

(A) .4 3.sini (B) 1 sin i (C) 4 3 (D) 150. A beam of white light is incident on glass air interface from glass to air such that green light just sufferstotal internal reflection. The colors of the light which will come out to air are(A) Violet, Indigo, Blue (B) All colors except green(C) Yellow, Orange, Red (D) White light

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OPTICSwww.physicsashok.in 1751. An equilateral prism is placed on a horizontal surface. A ray pQ is incident onto it. For minimum deviationPQ

RS(A) PQ is horizontal (B) QR is horizontal(C) RS is horizontal (D) Any one will be horizontal52. A source emits sound of frequency 600 Hz inside water. The frequency heard in air will be equal to(velocity of sound in water = 1500 m/s, velocity of sound in air = 300 m/s)(A) 3000 Hz (B) 120 Hz (C) 600 Hz (D) 6000 Hz53. A point object is placed at the centre of a glass sphere of radius 6 cm andrefractive index 1.5. Thedistance of virtual image from the surface is(A) 6 cm (B) 4 cm (C) 12 cm (D) 9 cm

54. A convex lens is in contact with concave lens. the magnitude of the ratio of their focal length is 2 3 .Their equivalent focal length is 30 cm. What are their individual focal lengths? (A) 15, 10 (B) 10, 15 (C) 75, 50 (D) 75, 5055. A container is filled with water .. .1.33. upto a heightof 33.25 cm. A concave mirror is placed 15 cm abovethe water level and the image of an object placed atthe bottom is formed 25 cm below the water level.Focal length of the mirror is(A) 15 cm(B) 20 cm(C) 18, 31 cm

(D) 10 cmMultiple Choice Question with ONE or MORE THAN ONE correct answer:56. A convex lens of focal length 40 cm is in contact with a concave lens of focal length 25 cm. The powerof the combination is(A) 1.5 dioptres (B) 6.5 dioptres (C) +6.5 diopres (D) +6.67 dioptres57. A converging lens is used to form an image on a screen. When the upper halfof the lens is covered byan opaque screen(A) half the image will disappear(B) complete image will be formed(C) intensity of the image will increase

(D) intensity of the image will decrease.58. A short linear object of length b lies along the axis of a concave mirror of focal length f at a distance ufrom the pole of the mirror. The size of the image is approximately equal to(A)1 2 b u ff. . .. .. .(B)1 2 b fu f

. .. . . . .(C)

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b u ff. . .. .. .(D)2 b f

u f. .. . . . .59. A beam of light consisting of red, green and blue colours is incident on a right angled prism, figure. Therefractive indices of the material of the prism for the above red, green and blue wavelengths are 1.39,1.44 and 1.47 respectively. The prism will

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OPTICSwww.physicsashok.in 1845°(A) separate part of the red colour from the green and blue colours(B) separate part of the blue colour from the red and green colours(C) separate all the three colours from one another.

(D) not separate even partially any colour from the other two colours.60. A thin prism P1 with angle 4° and made from glass of refractive index 1.54 iscombined with anotherthin prism P2 made from glass of refractive index 1.72 to produce dispersion without deviation. Theangle of the prism P2 is(A) 5.33° (B) 4° (C) 3° (D) 2.6°61. Two thin convex lenses of focal lengths f1 and f2 are separated by a horizontal distance d (whered . f1 . d . f2 ) and their centres are displaced by a vertical separation . asshown in Figure.Taking the origin of coordinates O, at the center of the first lens the x and y

coordinates of the focalpoint of this lens system, for a parallel beam of rays coming from the left, are given by:(A)1 21 2x f f , yf f. . .. (B). . 1 21 2 1 2,

f f dx yf f d f f. .. .. . .(C). . . . 1 2 1 11 2 1 2,f f d f d f dx y

f f d f f d. . . .. .. . . . (D). . 1 2 11 2, 0f f d f dx yf f d. .. .. .

62. Which of the following form(s) a virtual and erect image for all positions of the object?(A) Convex lens (B) Concave lens (C) Convex mirror (D) Concave mirror.

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63. A ray of light travelling in a transparent medium falls on a surface separating the medium from air at anangle of incidence of 45°. The ray just undergoes total internal reflection. If nis the refractive index ofthe medium with respect to air, select the possible value(s) of n from the following:(A) 1.3 (B) 1.4 (C) 1.5 (D) 1.6

64. A concave mirror is placed on a horizontal table, with its axis directed vertically upwards. Let O be thepole of the mirror and C its centre of curvature. A point object is placed at C. It has a real image, alsolocated at C. If the mirror is now filled with water, the image will be.(A) real, and will remain at C(B) real, and located at a point between C and .(C) virtual, and located at a point between C and O(D) real, and located at a point between C and O

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OPTICSwww.physicsashok.in 19Fill in the blanks:1. A light wave of frequency 5 x 1014 Hz enters a medium of refractive index 1.5, In the medium thevelocity of the light wave is .................. and its wavelength is .........

....... (2 Marks)2. A convex lens A of focal length 20 cm and a concave lens B of focal length 5cm are kept along thesame axis with a distance d between them. If a parallel beam of light falling on A leaves B as a parallelbeam, then d is equal to .......... cm.3. A monochromatic beam of light of wavelength 6000Å in vacuum enters a medium ofrefractive index1.5. In the medium its wavelength is ..., its frequency is ............. (1985)4. In youngs double-slit experiment, the two slits act as coherent sources of equal amplitude A and ofwavelength . . In another experiment with the same set-up the two slits are source

s of equal amplitudeA and wavelength . , but are incoherent. The ratio of the intensity of light at themidpoint of thescreen in the first case to that in the second case is ................. (1986)5. A thin lens of refractive index 1.5 has 7a focal length of 15 cm in air. when the lens is placed in amedium of refractive index43 , its focal length will become ........... cm. (1987)6. A point source emits sound equally in all directions in a non-absorbing medium. Two points P and Qare at a distance of 9 meters and 25 meters respectively from the source. The ratio of amplitudes of

the waves at P and Q is .................... (1989)7. A slab of a material of refractive index 2 shown in Figure, has a curved surface APB of radius ofcurvature 10 cm and a plane surface CD. On the left of APB is air and on the right of CD is water withrefractive indices as given in the figure.ABCDP C On1=1.0 n2=2.0

20 cm15 cm343n =An object O is placed at a distance of 15 cm from the pole P as shown. The distance of the final imageof O from P, as viewed from the left is ............... (1991)8. A thin rod of length 3fis placed along the optic axis of a concave mirror of focal length f such that its

image which is real elongated, just touches the rod. The magnification is ............. (1991)9. A ray of light undergoes deviation of 30° when incident on an equilateral prism

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 of refractive index 2 .The angle made by the ray inside the prism with the base of the prism is ............. (1992)10. A light of wavelength 6000Å in air, enters a medium with refractive index 1.5. Inside the medium itsfrequency is ........ Hz and its wavelength is ............... Å. (1997)

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OPTICS11.Two thin lenses, when in contact, produce a combination of power +10 diopters. When they are 0.25 mapart, the power reduces to +6 diopters. The focal length of the lenses are ...... m and .... m.

(1997)

12.A ray of light is incident normally on one of the faces of a prism of apex angle 30° and refractive index2 . The angle of deviation of the ray is .......... degrees.(1997)True / False :

13. Theintensityof light at adistancerfrom theaxisof alongcylindrical sourceisinverselyproportionalto r. (1981)

14. A convex lens of focal length 1 meter and a concave lens of focal length 0.25 meter are kept 0.75meter apart. A parallel beam of light first passes through the convex lens, then through the concavelens and moves to a focus 0.5 m away from the concave lens. (1983)15. A beam of white light passing through a hollow prism give no spectrum. (1983)16. A parallel beam of white light fall on a combination of a concave and a convex lens, both of the samematerial. Their focal lengths are 15 cm and 30 cm respectively for the mean wavelength in white light.On the other side of the lens system, one sees coloured patterns with violet colour at the outer edge.

(1988)Table Match17. Match List I and List II and select the correct answer using the codes given below the lists:The arrangement shows different lenses made of substance of refractive index 1.5 and kept in air. R1= 30 cm, R2 = 60 cm. Match the focal lengthsTable I Table III.R1 R2 A. 120 cmII.R1 R2 B. +40 cm

III. R1 R2 C.40 cmR1

R2

IV.D.+120 cm(A) I-A, II-B, III-D, IV-C(B) I-C, II-A, III-B, IV-D(C) I-D, II-C, III-A, IV-B(D) I-B, II-D, III-C, IV-A

www.physicsashok.in 20

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OPTICSwww.physicsashok.in 2118.Table I Table III. An object is placed at focus before A. Magnification is  .a convex mirror

II. An object is placed at the centre of B. Magnification is +0.5curvature before a concave mirrorIII. An object is placed at focus before C. Magnification is +1a concave mirror.IV. An object is placed at centre of curvature D. Magnification is  1before a convex mirror.(A) I-B, II-D, III-A, IV-E (B) I-A, II-D, III-C, IV-B(C) I-C, II-B, III-A, IV-E (D) I-B, II-E, III-D, IV-C19. Match the followings:Table I Table IIA. Magnification m = +1 (i) Convex mirrorB. Magnification

23m . . (ii) Plane mirrorC. Magnification32m . . (iii) Concave mirror(A) A . (ii) B . (iii) C . (i) (B) A . (i) B . (ii) C . (iii)(C) A . (ii) B . (i) C . (iii) (D) A . (iii) B . (ii) C . (i)20. For a concave mirror of focal length 20 cm, match the followings:Table I Table IIObjective distance Nature of imageA. 10 cm (i) Magnified, inverted and real

B. 30 cm (ii) Equal size, inverted and realC. 40 cm (iii) Smaller, inverted and realD. 50 cm (iv) Magnified, erect and virtual(A) A . II, B . I, C . III, D . IV (B) A . IV, B . I, C . II, D . III(C) A . I, B . IV, C . III, D . II (D) A . IV, B . I, C . III, D . II.PASSAGE TYPE QUESTIONSTHE NEXT QUESTIONS REFER TO THE FOLLOWING PASSAGEThe ciliary muscles of eye control the curvature of the lens in the eye and hence can alter the effectivefocal length of the system. When the muscles are fully relaxed, the focal length is maximum. When themuscles are strained the curvature of lens increases (that means radius of curva

ture decreases) andfocal length decreases. For a clear vision, the image must be on retina. The image distance is thereforefixed for clear vision and it equals the distance of retina from eye-lens. It is about 2.5 cm for a grownupperson.

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OPTICSwww.physicsashok.in 22A person can theoretically have clear vision of objects situated at any large distance from the eye. Thesmallest distance at which a person can ciliary muscles are most strained in this position. For an

average grown-up person, minimum distance of object should be around 25 cm.A person suffering for eye defects uses spectacles (eye glass). The function oflens of spectacles isto form the image of the objects within the range in which person can see clearly. The image of thespectacle lens becomes object for eye-lens and whose image is formed on retina.The number ofspectacle lens used for the remedy of eye defect is decided by the power of thelens required and thenumber of spectacle-lens equal to the numerical value of the power of lens withsign. For example,power of lens required is +3 D (converging lens of focal length

1003 cm), then number of lens will be+3.For all the calculations required you can use the lens formula and lens makers formula. Assume thatthe eye lens is equiconvex lens. Neglect the distance between eye lens and the spectacle lens.1. Minimum focal length of eye-lens of a normal person is(A) 25 cm (B) 2.5 cm (C)259cm (D)25

11cm2. Maximum focal-length of eye lens of normal person is(A) 25 cm (B) 2.5 cm (C)259cm (D)2511cm3. A near-sighted man can clearly see object only upto a distance of 100 cm andnot beyond this. The

number of the spectacles lens necessary for the remedy of this defect will be(A) + 1 (B)  1 (C) + 3 (D)  34. A far-sighted man cannot see object only upto a distance of 100 cm from his eyes. The number of thespectacles lens that will make his range of clear vision equal to an average grown up person is(A) + 1 (B)  1 (C) + 3 (D)  35. A person who can see objects clearly from distance 10 cm to . , then we can say that the person is(A) normal sighted person (B) near-sighted person(C) far-sighted person (D) a person with exceptional eyes having no eye defect.THE NEXT QUESTIONS REFER TO THE FOLLOWING PASSAGESpherical aberration in spherical mirrors is a defect which is due to dependence

 of focal length ' f 'on angle of incidence '. ' as shown in figure. is given bysec

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2f . R . R .C FPrincipal axis Pole (P)f..

where R is radius of curvature of mirror and . is the angle of incidence. The rays which are closed toprincipal axis are called paraxial rays and the rays far away from principal axis are called marginalrays. As a result of above dependence different rays are brought to focus at different points and theimage of a point object is not a point.

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OPTICSwww.physicsashok.in 236. If p f and m f represent the focal length of paraxial andmarginal rays respectively, then correct relationshipis:(A) f p . fm (B) p m f . f (C) p m f . f (D) None

7. If angle of incidence is 60°, then focal length of this rays is:(A) R (B) 2R(C) 2R (D) 08. The total deviation suffered by the ray falling on mirror at an angle of incidence equal to 60° is:(A) 180° (B) 90° (C) Cant be determined (D) None9. For paraxial rays, focal length approximately is:(A) R (B) 2R(C) 2R (D) None10. Which of the following statements are correct regarding spherical aberration

:(A) It can be completely eliminated(B) it cant be completely eliminated but is cant be minimised by allowing either paraxial or marginalrays to hit the mirror(C) It is reduced by taking large aperture mirrors(D) NoneTHE NEXT QUESTIONS REFER TO THE FOLLOWING PASSAGERainbow is formed during rainy season due to refraction and total internal reflection of rays falling onsuspended water droplets. When rays of the sun fall on rain drops, the rain drops disperse the light anddeviate the different colours by refraction and total internal reflection to the

 eye of the observer. Aperson observing the drops will see different colours of the spectrum at different angles. The rainbowwhich results from single total internal reflection is called primary rainbow and secondary rainbow isformed due to two total internal reflections suffered by rays falling on water drops.ABCDV

RRVRedRedvioletvioletRays fromsunSecondaryrainbowPrimaryrainbow

Figure shows formation of rainbow due to four drops A, B, C and D. The light surffers only one totallinternal reflection in drops C and D forming primary rainbow. Secondary rainbow

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 is formed by drops Aand B where light suffers two total linternal reflections.11. Rainbow is an arc of:(A) Circle (B) Ellipse(C) Parabola (D) Cant be determined12. The visibility of the rainbow is due to:(A) All rays (B) Rays undergoing maximum deviation

(C) Rays undergoing minimum deviation (D) None

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OPTICSwww.physicsashok.in 2413. In primary rainbow, the colour of outer edge is:(A) Blue (B) Violet (C) Red (D) None14. In secondary rainbow, the colour of inner edge is:(A) Red (B) Violet (C) Indigo (D) None

15. The necessary condition for the observer to see rainbow is:(A) Sun, observers eye and the centre of the rainbow arc lie on the same line(B) Sun, observers eye and the centre of the rainbow arc lie on the different line(C) From any position provided sun is at the back of the observer (D) NoneTHE NEXT QUESTIONS REFER TO THE FOLLOWING PASSAGEThe laws governing the behavior of the rays namely rectilinear propagation, laws of reflection andrefraction can be summarised in one fundamental also known as Fermats principle.According to thisprinciple a ray of light travels from one point to another such that the time taken is at a stationary value

(maximum or minimum). If c is the velocity of light in a vacuum, the velocity in a medium of refractiveindex n iscn , hence time taken to travel a distance l isnlc . If the light passes through a number ofmedia, the total time taken is1 nlc. . . ... . and1 n dl

c . . If refractive index varies continuously. Now, .nlis the total optical path, so that Fermats principle states that the path of a ray is such that the opticalpath in at a stationary value. This principle is obviously in agreement with the fact that the ray arestraight lines in a homogenous isotropic medium. It is found that it also agrees with the classical laws ofreflection and refraction.16. If refractive index of a slab varies as . .1. x2 where x is measured from one end, then optical pathlength of a slab of thickness 1 m is:(A)

43m (B)34m (C) 1 m (D) None17. The optical path length followed by ray from point A to B given that laws of reflection are obeyed asshown in figure is:A BP(A) Maximum (B) Minimum (C) Constant (D) None18. The optical path length followed by ray from point A to B given that laws of

 reflection are obeyed asshown in figure isA B

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(A) Maximum (B) Minimum (C) Constant (D) None

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OPTICS OPTICS19. The optical path length followed by ray from point A to B given that laws of refraction are obeyed asshown in figure isAB

(A) Maximum (B) Minimum (C) Constant (D) None20. The optical path length followed by ray from point A to B given that laws of refraction are obeyed asshown in figure isA and B are fociiof ellipseA B(A) Maximum (B) Minimum (C) Constant (D) NoneTHE NEXT QUESTIONS REFER TO THE FOLLOWING PASSAGEOne hard and stormy night you find yourself lost in the forest when you come upon a small hut.Entering it you see a crooked old woman in the corner hunched over a crystal bal

l. You are about tomake a hasty exit when you hear the howl of wolves outside. Taking another lookat the gypsy youdecide to take your chances with the wolves, but the door is jammed shut. Resigned to bad situationyour approach her slowly, wondering just what is the focal length of that niftycrystal ball.21. If the crystal ball is 20 cm in diameter with R.I. = 1.5, the gypsy lady is1.2 m from the central of ball,where is the image of the gypsy in focus as you walk towards her?(A) 6.9 cm from the crystal ball (B) 7.9 cm from the crystal ball(C) 8.9 cm from the crystal ball (D) None22. The image of old lady is:

(A) real, inverted an enlarged (B) erect, virtual and small(C) erect, virtual and magnified (D) real, inverted and diminished23. The old lady moves the crystal ball closer to her wrinkled old face. At some point you can no longer getan image of her. At what object distance will there be no change of the gypsy formed?(A) 10 cm (B) 5 cm (C) 15 cm (D) NoneTHE NEXT QUESTIONS REFER TO THE FOLLOWING PASSAGE

The table below contains some physical properties of common optical materials. The refractive indexof a material is a measure of the amount by which light is bent upon entering th

e material. Thetransmittance range is the range of wavelengths over which the material is transparent.

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OPTICSwww.physicsashok.in 26Physical Properties of Optical MaterialsMaterial Refractiveindex for lightof 0.589 .m

Transmittancerange (.m)Useful rangefor prisms(.m)ChemicalresistanceLithiumfluoride1.39 0.12-6 2.7-5.5 PoorCalciumfluoride

1.43 0.12-12 5-9.4 GoodSodiumchloride1.54 0.3-17 816 PoorQuartz 1.54 0.20-3.3 0.20-2.7 ExcellentPotassiumbromide1.56 0.3-29 1528 PoorFlint glass* 1.66 0.352.2 0.35-2 ExcellentCesium iodide 1.79 0.37.0 15-55 Poor*Flint glass is lead oxide doped quartz.24. According to the table, which material(s) will transmit light at 25 .m  (A) Potassium bromide only (B) Potassium bromide and cesium iodide

(C) Lithium fluoride and cesium iodide (D) Lithium fluoride and flint glass25. A scientist hypothesizes that any material with poor chemical resistance would have a transmittancerange wider than 10.m. The properties of which of the following materials contradicts this hypothesis (A) Lithium fluoride (B) Flint glass(C) Cesium iodide (D) Quartz26. When light travels from one medium to another, total internal reflection can occur if the first mediumhas a higher refractive index than the second. Total internal reflection could occur if light were travellingfrom 

(A) Lithium fluoride of flint glass(B) potassium bromide to cesium iodide(C) quartz to potassium bromide(D) flint glass to calcium fluoride27. Based on the information in the table, how is the transmittance range related to the useful prismrange (A) The transmittance range is always narrower than the useful prism range(B) The transmittance range is narrower than or equal tot he useful prism range(C) The tranmittance range increases as the useful prism range decreases(D) The tranmittance range is wider than and includes within it the useful prism range28. The addition of lead oxide to pure quartz has the effect of 

(A) decreasing the transmittance range and the refractive index(B) decreasing the transmittance range and increasing the refractive index(C) increasing the transmittance range and the useful prism range

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(D) increasing the transmittance range and decreasing the useful prism range.THE NEXT QUESTIONS REFER TO THE FOLLOWING PASSAGEA periscope viewing system is to be used to observe the behavior of primates ina large environmentallycontrolled room on the upper floor of a large research facility. The periscope,like those used on

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OPTICS OPTICSsubmarines, is essentially a large, folded-path, low power telescope (using prisms to fold the light path).A sketch of the preliminary design appears below. Like all Newtonian telescopes, it uses a relatively longfocal length objective lens to form a real image in front of the eyepiece lens (

of shorter focal length). Theobserver looks through the eyepiece lens to see the final image, in the same manner that one would usea magnifying glass.

The distance between the lenses is approximately equal to the sum of their focal lengths. The eyepiece,

(A) The focal length of the eyepiece lens is too short.in this design, can be moved forward or back in order to focus on the primates as they move closer to

or further away from the objective lens.29. The total tube length of the three sections is to be 4 m. The objective lens available has a focal lengthof 3 m. What should the focal length of the eyepiece lens be?(A) 0.75 m (B) 1 m (C) 1.33 m (D) 7 m30. A visitor seeing the sketch points out an important flaw that will require a design change. what is theflaw?(B) The images of the primates will be inverted(C) The objective lens should be a diverging lens.(D) The prisms cannot be used in this way.31. A visitor seeing the sketch points out an important flaw that will require a design change. What is the

flaw?(A) The focal length of the eyepiece lens is too short.(B) The images of the primates will be inverted(C) The objective lens should be a diverging lens.(D) The prisms cannot be used in this way.32. What will be the approximate magnification of this periscope?(A) 0.67x (B) 1x (C) 3x (D) 300x33. The prisms (454590° prisms) turn the light path through 90° by total internal reflection from theinside hypotenuse faces of the prisms when the incident angle is 45° as in the sketch.(C)

Can one use crown glass with an index of refraction of 1.52 for the prism?(A) yes, because the critical angle for crown glass is 47°(B) yes, because the critical angle for crown glass is 41°.No, because the critical angle for crown glass is exact