fiziks Institute for NET/JRF, GATE, IIT JAM, M.Sc. Entrance, JEST, TIFR and GRE in Physics CLASSICAL MECHANICS SOLUTIONS NET/JRF (JUNE-2011)

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1 CLASSICAL MECHANICS SOLUTIONS NET/JRF (JUNE-) b, where a and b are ositive onstants. The angular frequen of sall osillations about the iniu of the otential is 8 b 8 a () 8 a / b 8 b / a Q. A artile of unit ass oves in a otentialv a Ans: b V b 4 b 4 a a b 3. a k V Sine, and k where is stable equilibriu oint. Solution: V a V 6b 6b Hene k a a 8a at 4 b a Thus, 8a. 4 b. a Q. The aeleration due to gravit g on the surfae of Earth is aroiatel.6 ties that on the surfae of Mars. Given that the radius of Mars is about one half the radius of Earth, the ratio of the esae veloit on Earth to that on Mars is aroiatel..3 ().3 5. Ans: () Solution: Esae veloit = gr Esae veloit of Earth Esae veloit of Mass gr e e e.3 where gr R R and ge.6. g Q3. The Hailtonian of a sste with n degrees of freedo is given b H q q ;,... t,... n n;, with an eliit deendene on the tie t. Whih of the following is orret? Different hase trajetories annot interset eah other. H alwas reresents the total energ of the sste and is a onstant of the otion. () The equations q H /, H / q are not valid sine H has eliit tie i i i deendene. An initial volue eleent in hase sae reains unhanged in agnitude under tie evolution. H.No. 4 D, Ground Floor, Jia Sarai, Near IIT, Hauz Khas, New Delhi 6 Phone: / i

2 Ans: Q4. The Lagrangian of a artile of harge e and ass in alied eletri and agneti fields is given b L v ea v e, where A and are the vetor and salar otentials orresonding to the agneti and eletri fields, resetivel. Whih of the following stateents is orret? The anoniall onjugate oentu of the artile is given b v e The Hailtonian of the artile is given b H A e () L reains unhanged under a gauge transforation of the otentials Under a gauge transforation of the otentials, L hanges b the total tie derivative Ans: L Solution: V t Q5. Consider the dea roess in the rest frae of the. The asses of the, and are M, M and zero resetivel. A. The energ of is M M M Ans. : Solution: Fro onservation of energ M M M M E E. () M M MM E M and 4 E sine oentu of and is sae. Ans: 4 4 M, M E E E E M M M M E and E E M E M M M E E E B. The veloit of is M M M M M M M M H.No. 4 D, Ground Floor, Jia Sarai, Near IIT, Hauz Khas, New Delhi 6 Phone: / () M M. M M

3 Solution: Veloit of E v M M M M v M M 4M M M M v v 4M M M M M M M M 4M M 4 4 M M v. M M Q6. The Hailtonian of a artile of unit ass oving in the -lane is given to be: H in suitable units. The initial values are given to be,,, and, b the artiles in the -lane and the. During the otion, the urves traed out lane are both straight lines a straight line and a herbola resetivel () a herbola and an ellise, resetivel both herbolas Ans: Solution: H Solving Hailtonion equation of otion H H and. H and H. After solving these four differential equation and eliinating tie t and using boundar ondition one will get and H.No. 4 D, Ground Floor, Jia Sarai, Near IIT, Hauz Khas, New Delhi 6 Phone: /

4 NET/JRF (DEC-) Q7. A double endulu onsists of two oint asses attahed b strings of length l as shown in the figure: The kineti energ of the endulu is Ans: l l os () l os l os Solution: Let o-ordinate, and,. K. E. l sin, l os l os, l sin, los l os l sin l sin los l os, l sin l sin Put the value of,,, in K.E equation, one will get T l os. Q8. A onstant fore F is alied to a relativisti artile of rest ass. If the artile starts fro rest at t =, its seed after a tie t is Ft Ft / Ft Ft / tanh () e F t Ans: Solution: d F dt Ft. At t, so, l l Ft u u Ft F t Ft u Ft Ft. H.No. 4 D, Ground Floor, Jia Sarai, Near IIT, Hauz Khas, New Delhi 6 Phone: /

5 Q9. The otential of a diatoi oleule as a funtion of the distane r between the atos is a b given bv r 6 r r. The value of the otential at equilibriu searation between the atos is: 4a / b a / b () a / b a / 4b Ans: a b V Solution: V r, for equilibriu 6 r r r a b r r b 6 7 a 6 r r 6 b b b 6a r 6 r r 6a a b 6 a b V r a a a. a b b b 4b 4b a a Q. Two artiles of idential ass ove in irular orbits under a entral otential Ans: 6 V r kr. Let l and l be the angular oenta and r, r be the radii of the orbits l r resetivel. If, the value of is: l r / () / Solution: V l eff kr, where l is angular oentu. r V eff l kr 3 Condition for irular orbit r r l 4 r l r. l r r l r l Thus sine. l r r l r l Q. A artile of ass oves inside a bowl. If the surfae of the bowl is given b the equation z a, where a is a onstant, the Lagrangian of the artile is r r gar a r () r r r sin gar a r H.No. 4 D, Ground Floor, Jia Sarai, Near IIT, Hauz Khas, New Delhi 6 Phone: / r r r r gar 5

6 Ans: fiziks Solution: L z gz, where z a It has lindrial setr. Thus r os rsin, rsin ros. ros, rsin and z arr., z ar. So, L a r r r gar. Q. A lanet of ass oves in the inverse square entral fore field of the Sun of ass M. If the sei-ajor and sei-inor aes of the orbit are a and b, resetivel, the total energ of the lanet is: GM GM a b a b GM a b () GM a b a a b Ans: Solution: Assue Sun is at the entre of ellitial orbit. GM GM Conservation of energ v v a b Conservation of oentu L va vb a v v b GM GM v v a b b a ba v GM b ab a b a v v GM b ab b v GM a b a GM b GM E v GM a a b a a GM b GM b b a GM a b a a b a b a v b s a v H.No. 4 D, Ground Floor, Jia Sarai, Near IIT, Hauz Khas, New Delhi 6 Phone: /

7 Q3. An annulus of ass M ade of a aterial of unifor densit has inner and outer radii a and b resetivel. Its rinile oent of inertia along the ais of setr erendiular to the lane of the annulus is: Ans: 4 4 b a M M b a b a () M b a M b a Q4. The trajetor on the z z - lane (hase-sae trajetor) of a ball bouning erfetl elastiall off a hard surfae at z = is given b aroiatel b (neglet frition): P Z P Z z z P Z P Z () z z Ans: Solution: H Pz gz and Pz E gz. H.No. 4 D, Ground Floor, Jia Sarai, Near IIT, Hauz Khas, New Delhi 6 Phone: /

8 NET/JRF (JUNE-) Q5. The bob of a sile endulu, whih undergoes sall osillations, is iersed in water. Whih of the following figures best reresents the hase sae diagra for the endulu? () Ans: Solution: When sile endulu osillates in water it is daed osillation so alitude ontinuousl derease and finall it stos. 9 Q6. Two events searated b a (satial) distane 9, are siultaneous in one inertial Ans: frae. The tie interval between these two events in a frae oving with a onstant seed.8 (where the seed of light 8 3 / s) is 6s 4s () s s Solution: t Put ' ' 9 ' ' 9 and t t. Then t t ' v v ' t v. 8 t t 4se v ' ' ' ' ' ' ' t t v v t t. v v v v H.No. 4 D, Ground Floor, Jia Sarai, Near IIT, Hauz Khas, New Delhi 6 Phone: /

9 Q7. If the Lagrangian of a artile oving in one diensions is given b L V Hailtonian is V V Ans: L Solution: Sine H L and () V V. H V H V H V. Q8. A horizontal irular latfor utes with a onstant angular veloit direted vertiall uwards. A erson seated at the entre shoots a bullet of ass horizontall with seed v. The aeleration of the bullet, in the referene frae of the shooter, is v to his right v to his left the () v to his right v to his left Ans: Solution: Veloit of bullet = fore v F. vˆ j, Angular veloit= kˆ. There will be oriolis F viˆ a viˆ. r, has the value r rˆ ˆ () 3 Q9. The Poisson braket Ans: Solution: r iˆ j ˆzkˆ / z r z r, =, /, iˆ ˆj kˆ, r r r r r r z z z z z r rˆˆ r r r r z H.No. 4 D, Ground Floor, Jia Sarai, Near IIT, Hauz Khas, New Delhi 6 Phone: /

10 Q. Consider the otion of a lassial artile in a one diensional double-well Ans: V. If the artile is dislaed infinitesiall fro the iniu 4 on the -ais (and frition is negleted), then the artile will eeute sile haroni otion in the right well with an angular otential frequen the artile will eeute sile haroni otion in the right well with an angular frequen () the artile will swith between the right and left wells the artile will aroah the botto of the right well and settle there 4 Solution: V V 3 V 4 V. At,,. so V is aiu. Thus it is unstable oint V 4 V and it is stable equilibriu oint with. Q. What is roer tie interval between the ourrene of two events if in one inertial frae events are searated b and our 6.5s aart? 6.5 s 6. s () 5.75s 5. s Ans: Solution: Proer tie interval r 7.5 t t se. 3 H.No. 4 D, Ground Floor, Jia Sarai, Near IIT, Hauz Khas, New Delhi 6 Phone: /

11 NET/JRF (DEC-) Q. A solid linder of height H, radius R and densit ρ, floats vertiall on the surfae of a Ans: liquid of densit. The linder will be set into osillator otion when a sall instantaneous downward fore is alied. The frequen of osillation is g H g H Solution: Fro Newton s law of otion a g Agh where A is area of ross setion, AH. () g gh AHa AHg Agh a g H H Q3. Three artiles of equal ass () are onneted b two idential assless srings of stiffness onstant (K) as shown in the figure K K If, and 3 denote the horizontal dislaeent of the asses fro their resetive equilibriu ositions the otential energ of the sste is Ans: K 3 H g H K 3 3 () 3, () K K Solution: V K K 3 V K K3 3V K 3 3 Q4. A lanet of ass oves in the gravitational field of the Sun (ass M ). If the seiajor and sei-inor aes of the orbit are a and b resetivel, the angular oentu of the lanet is 3 3 GM a b GM a b () GM ab a b GM ab a b H.No. 4 D, Ground Floor, Jia Sarai, Near IIT, Hauz Khas, New Delhi 6 Phone: /

12 Ans: Solution: Assue Sun is at the entre of ellitial orbit. GM GM Conservation of energ v v a b Conservation of oentu L va vb v a v b v b s a v GM GM v v a b a b a v v GM b ab b a ba v GM b ab b v GM a b a b v GM a b a b GMab L va GM a a ba ba L GM ab a b Q5. The Hailtonian of a sile endulu onsisting of a ass attahed to a assless Ans: string of length l is H gl os dl is: dt () l g g sin sin l l g l dl dt os L t. If L denotes the Lagrangian, the value of l os H. l Solution: L, H where gl os L iqi H H H, P l i, L gl os l. H.No. 4 D, Ground Floor, Jia Sarai, Near IIT, Hauz Khas, New Delhi 6 Phone: /

13 Hene we have to alulatel, H whih is onl defined into hase sae i.e. and. Then L gl os l L H L H g L and l t L, H sin dl dt g sin l Q6. Two bodies of equal ass are onneted b a assless rigid rod of length l ling in the -lane with the entre of the rod at the origin. If this sste is rotating about the z-ais with a frequen ω, its angular oentu is l / 4 l / () l l Ans: Solution: Sine rod is assless i.e. M. Moent of inertia of the sste I r r, and r l l l l I I. Angular oentu, J I and J. 4 4 Q7. Whih of the following set of hase-sae trajetories is not ossible for a artile obeing Hailton s equations of otion? P P r l () P P Ans: Solution: Phase urve does not ut eah other H.No. 4 D, Ground Floor, Jia Sarai, Near IIT, Hauz Khas, New Delhi 6 Phone: /

14 Q8. The uon has ass 5 MeV / and ean life tie. s in its rest frae. The ean Ans: distane traversed b a uon of energ 35MeV before deaing is aroiatel, Solution: Sine 5 3 k. () k MeV E 35MeV and. E E v 5 35 v 35 5 v v. 94. Now, t t v, t 6..s t t 6.6 s 8 9 Now the distane traversed b uon is vt k. NET/JRF (JUNE-3) Q9. The area of a dis in its rest frae S is equal to (in soe units). The dis will aear Ans: distorted to an observero oving with a seedu with reset to S along the lane of the dis. The area of the dis easured in the rest frae of the observer O is ( is the seed of light in vauu) u / u / u () Solution: Area of dis fro S frae is i.e. a or a a Area of dis fro S frae is u where b a. u u a b a a u u H.No. 4 D, Ground Floor, Jia Sarai, Near IIT, Hauz Khas, New Delhi 6 Phone: /

15 Q3. A lanet of ass and an angular oentu L oves in a irular orbit in a otential, V r k / r, where k is a onstant. If it is slightl erturbed radiall, the angular frequen of radial osillations is Ans: Solution: 3 k / L V eff L r k r k / L 3 (). For irular orbit L r r r 3 k / V eff Veff L k 3 3 3k / L L k L. Thus r r 3 k r d V k dr eff r rr 3L r k, k r rr 3L L k k L k 3k k L L r k L dv dr rr k. 3 L Q3. The nuber of degrees of freedo of a rigid bod in d sae-diensions is d 6 () d / Ans: () Q3. A sste is governed b the Hailtonian H where a and b are onstants and a b d d!, are oenta onjugate to and resetivel. For what values of a and b will the quantities 3 be onserved? and a 3, b a 3, b () a, b 3 a, b 3 Ans: Solution: Poisson braket 3, H and, H ( b3) (3 bb ) and ( a ) ( aa ) a, b 3 H.No. 4 D, Ground Floor, Jia Sarai, Near IIT, Hauz Khas, New Delhi 6 Phone: /

16 Q33. The Lagrangian of a artile of ass oving in one diension is given b L b whereb is a ositive onstant. The oordinate of the artile t at tie t is given b: (in following and are onstants) b t t t bt bt bt bt () os sin osh sinh Ans: d L L d Solution: Equation of otion b b b dt dt d dt b d b t dt bt t NET/JRF (DEC-3) Q34. Let A, B and C be funtions of hase sae variables (oordinates and oenta of a ehanial sste). If, reresents the Poisson braket, the value of A, B, C A, B, C is given b B, C, A () A, C, B C, A, Ans: Solution: We know that Jaobi identit equation A, B, C B, C, A C, A, B Now A, B, C A, B, C B, C, A C, A, Q35. A artile oves in a otentialv Ans: oentu is/are onstant(s) of otion? z B None L, L and L z () onl L and B. Whih oonent(s) of the angular L onl L z H.No. 4 D, Ground Floor, Jia Sarai, Near IIT, Hauz Khas, New Delhi 6 Phone: /

17 z Solution: A artile oves in a otential V r V r,, r sin os r sin sin os r V r,, r sin os Now is li-o-ordinate i.e L z is onstant of otion. Q36. The Hailtonian of a relativisti artile of rest ass and oentu is given Ans: b H Lagrangian is () Solution: L L H V, in units in whih the seed of light. The orresonding V L V V L V V H Now L H H / V Put value L V Q37. A endulu onsists of a ring of ass M and radius R susended b a assless rigid rod of lengthl attahed to its ri. When the endulu osillates in the lane of the ring, the tie eriod of osillation is l R g l g / 4 R () Ans: () R Rl l g R l / 4 Solution: The oent of inertia about ivotal oint is given b g R Rl l H.No. 4 D, Ground Floor, Jia Sarai, Near IIT, Hauz Khas, New Delhi 6 Phone: /

18 I I. Md MR M( l R) If ring is dislaed b angle then otential energ is Mg( l R)os The Lagrangian is given b L I V( ) = ( ( ) ) MR M l R Mg ( l R )os d L L dt ( MR M ( l R) ) Mg( l R)sin For sall osillation sin ( MR M ( l R) ) Mg( l R) R Rl l Tie eriod is given b. g R l Q38. Consider a artile of ass attahed to two idential srings eah of length l and sring onstant k (see the figure). The equilibriu onfiguration is the one where the srings are unstrethed. There are no other eternal fores on the sste. If the artile is given a sall dislaeent along the -ais, whih of the following desribes the equation of otion for sall osillations? o o 3 k k k () k l l Ans: Solution: The lagrangian of sste is given b L V ( ) The otential energ is given b o k k V( ) l l l l V( ) k l l For sall osillation one an aroiate otential b Talor eansion o H.No. 4 D, Ground Floor, Jia Sarai, Near IIT, Hauz Khas, New Delhi 6 Phone: /

19 ( ) kl V l 4 V( ) kl 4 l 8 l kl V( ) 4 l 4 V( ) k. 4l So Lagrangian of sste is given b 4 4l L k 3 d L L k The Lagranges equation of otion. dt l NET/JRF (JUNE-4) Q39. The tie eriod of a sile endulu under the influene of the aeleration due to gravit g is T. The bob is subjeted to an additional aeleration of agnitude 3 g in the horizontal diretion. Assuing sall osillations, the ean osition and tie eriod of osillation, resetivel, of the bob will be o to the vertial and 3 T o 3 to the vertial and T / () o 6 to the vertial and / T o to the vertial and T / 3 Ans: () Solution: T l g T g 3g g T l g 4g g l T g T T g g 3g T os g, Tsin 3g tan 3 o 6 H.No. 4 D, Ground Floor, Jia Sarai, Near IIT, Hauz Khas, New Delhi 6 Phone: /

20 Q4. A artile of ass and oordinate q has the Lagrangian is a onstant. The Hailtonian for the sste is given b L q qq, where q q () q q Ans: Solution: H q L where L q qq L q qq q q q H q L H q L H q L q q q q q q q q q q q q q Q4. The oordinates and oenta, i,,3 Ans. : braket relations i j ij H i i of a artile satisf the anonial Poisson,. If C 3 3 and C are onstants of C3 C, C, then otion, and if 3 3 C, C C and C, C C 3 3, and, (), and,, and, C C C C C C 3 3 C C C C C C 3 3 C C C C C C 3 3 Solution: C, C, 3 3 C3 3 3 H.No. 4 D, Ground Floor, Jia Sarai, Near IIT, Hauz Khas, New Delhi 6 Phone: /

21 C, C 3 C C C C C C C C C C C C C, C C C, C 3 C C C C C C C C C C C C , C C C Q4. The reentl-disovered Higgs boson at the LHC eerient has a dea ode into a hoton and a Z boson. If the rest asses of the Higgs and Z boson are 5 GeV/ and 9 GeV/ resetivel, and the deaing Higgs artile is at rest, the energ of the hoton will aroiatel be 35 3 GeV 35GeV () 3GeV 5GeV Ans. : () Solution: HB PH ZB Fro onservation of oentu PP P P P P Now E E E H P Z B H B E E M PH ZB HB P P and E H E Z B P M Z B 4 4 EZ EP EZ EP MZ P P B H B H B M M EZ E B P H M M M 4 ZB ZB HB HB zb EP M H H B M HB E P H E E M ZB PH HB M M EP H 3.GeV 4 5 H z Q43. A anonial transforation relates the old oordinates q, to the new ones P B M H B B Q, b the relations Q q and P / q. The orresonding tie indeendent generating funtion is P/ q qp H.No. 4 D, Ground Floor, Jia Sarai, Near IIT, Hauz Khas, New Delhi 6 Phone: / () q / P qp

22 Ans. : Solution: Qq ; P /q F q F P q q F q P f P ( ) F Q q P F q P f q ( ) oaring both side f( q) f( P) F q P NET/JRF (DEC-4) Q44. The equation of otion of a sste desribed b the tie-deendent Lagrangian is t L e V dv dv d d dv dv () d d Ans: t Solution: L e V L t e and d L L dt L V e d V V e e e e e dt t t t t t V t V e 4 Q45. A artile of ass is oving in the otential V t a b where a, b are 4 ositive onstants. The frequen of sall osillations about a oint of stable equilibriu is a / a / () 3 a / 6 a / Ans: V a b 4 4 Solution: H.No. 4 D, Ground Floor, Jia Sarai, Near IIT, Hauz Khas, New Delhi 6 Phone: /

23 V 3 a b ab a, b V a 3 b V At, a (Negative so it is unstable oint) V a a3b a a b b (Positive so it is stable oint) V a Q46. The radius of Earth is aroiatel 64 k. The height h at whih the aeleration due to Earth s gravit differs fro g at the Earth s surfae b aroiatel % is 64 k 48 k () 3 k 6 k Ans: () g h g h g h Solution: h3 k.. g R g R g R Q47. Aording to the seial theor of relativit, the seed v of a free artile of ass and total energ E is: Ans: () () v E E v E v E v E Solution: 4 v v E v v E E E H.No. 4 D, Ground Floor, Jia Sarai, Near IIT, Hauz Khas, New Delhi 6 Phone: /

24 Q48. The Hailtonian of a lassial artile oving in one diension is H 4 q where is a ositive onstant and and q are its oentu and osition resetivel. Given that its total energ E E the available volue of hase sae deends on E as 3 / 4 E E () E is indeendent of E Ans: Solution: H q Phase area 4 dq E A dq E A E E A E / /4 3/4 4 E E / /4 Vq E q /4 / E E H q, q. As a Q49. A ehanial sste is desribed b the Hailtonian result of the anonial transforation generated b Fq Q the new oordinate Q and oentu P beoes Q,, the Hailtonian in q Q P Q Q P P () P Q P 4 Q P Ans: Solution: H q Q q, F F q, Q F q Q q (i) F Q P P q (ii) q P H.No. 4 D, Ground Floor, Jia Sarai, Near IIT, Hauz Khas, New Delhi 6 Phone: /

25 Fro equation (i) and (ii) QP q P 4 Q P 4 H q Q P P P Q5. The robe Mangalaan was sent reentl to elore the lanet Mars. The inter-lanetar art of the trajetor is aroiatel a half-ellise with the Earth (at the tie of launh), Sun and Mars (at the tie the robe reahes the destination) foring the ajor ais. Assuing that the orbits of Earth and Mars are aroiatel irular with radii R E and R M, resetivel, the veloit (with reset to the Sun) of the robe during its voage when it is at a distane r R E R M r fro the Sun, negleting the effet of Earth and Mars, is RE RM GM r R R r E M GM RE RM r R R E Earth M r Sun R E R M Mars () R rr E GM M GM r Ans: Solution: Total energ E K /a where a ajor ais and a RE RM. v GM GM r RE RM RE RM r v GM r R R E M H.No. 4 D, Ground Floor, Jia Sarai, Near IIT, Hauz Khas, New Delhi 6 Phone: /

26 NET/JRF (JUNE-5) Q5. A artile oves in two diensions on the ellise 4 8. At a artiular instant it is at the oint,, and the -oonent of its veloit is 6 (in suitable units). Then the -oonent of its veloit is 3 () 4 Ans. d d Solution: dt dt v 8 v 68 v 3 Q5. Consider three inertial fraes of referene A, B and C. the frae B oves with a v veloit with reset to A, and C oves with a veloit with reset to B in the sae diretion. The veloit of C as easured in A is 3 7 Ans. Solution: v, u u v 4 uv A S () 7 u Q53. If the Lagrangian of a dnaial sste in two diensions is L, then its Hailtonian is H H () H H Ans. () L Solution: L (i) B S C u C / C / 3 7 H.No. 4 D, Ground Floor, Jia Sarai, Near IIT, Hauz Khas, New Delhi 6 Phone: /

27 L or (ii) ut in equation (i) H L ut value of and H Q54. A artile of ass oves in the one diensional otential 3 4 V where 3 4,. One of the equilibriu oints is. The angular frequen of sall osillations about the other equilibriu oint is 3 () 4 Ans. V V 3 4 Solution: V Sring onstant k ve k Q55. A artile of unit ass oves in the -lane in suh a wa that t t t t derived fro the otential and. We an onlude that it is in a onservative fore-field whih an be Ans. Solution: and and and () that is ossible for L V H.No. 4 D, Ground Floor, Jia Sarai, Near IIT, Hauz Khas, New Delhi 6 Phone: /

28 Q56. A artile oves in one diension in the otential V k t, where k t is a tie Ans. deendent araeter. Then d V, the rate of hange of the eetation value V of the dt otential energ is dk k dk dt dt k () dk dt Solution: H k t d V k V V H k t k t V H dt t t,,, V k t dt t d k k k t t Q57. Let q and be the anonial oordinate and oentu of a dnaial sste. Whih Ans. of the following transforations is anonial?. Q q and P. Q q and P q neither nor both and () onl onl q P Solution: For A: Q, P Q P Q P Q, P.. (Not anonial) q q For B : Q q, P q Q, (anonial) H.No. 4 D, Ground Floor, Jia Sarai, Near IIT, Hauz Khas, New Delhi 6 Phone: /

29 Q58. Whih of the following figures is a sheati reresentation of the hase sae trajetories (i.e., ontours of onstant energ) of a artile oving in a one-diensional V 4 otential 4 () Ans. Solution: V 4 4 V, V V ve for (unstable oint) = + ve for (stable oint) E E E H.No. 4 D, Ground Floor, Jia Sarai, Near IIT, Hauz Khas, New Delhi 6 Phone: /

30 NET/JRF (DEC-5) Q59. Two asses eah, are laed at the oints, a, a and a, a asses, eah, are laed at the oints a, inertia of the sste are Ans.:,4a 4 a,8a () Solution: I i i zi ii z i i I a a a a I 6a a and aa, 4 a,4a and two. The rinial oents of 8 a,8a Siilarl, I 6a and I zz a I I, I I z z z z I I i ii i a a a a Moent of inertia tensor 6a a I a 6a a I I a Eigen value of atries is rinial oent of inertia, whih is given b a I, 4 a I, 8 a 3 I z So, I 4a and I 8a Q6. The Lagrangian of a sste is given b 5 L q q k q q qq 4 where and k are ositive onstants. The frequenies of its noral odes are k, 3k k 3 73 () 5 k, k k, 6k Ans.: H.No. 4 D, Ground Floor, Jia Sarai, Near IIT, Hauz Khas, New Delhi 6 Phone: /

31 5 Solution: L q q k q q qq 4 4 k L q q q 4q qq qq 4 T, 4 V k k 4 k 4k The seular equation V 4 k 4k 4 k k k k k, 4 k k 4 k 4 4k 4 3k 7 k 4 3k 6 k k k 3k k 3k, Q6. Consider a artile of ass oving with a seed v. If T R denotes the relativisti kineti energ and T N its non-relativisti aroiation, then the value of T R TN for T v., is R () Ans.: None of the otions is orret. Solution: TN v, TR v ( v.) Now, v. T R T v N TN T R T T R N T R.75 T R v v. H.No. 4 D, Ground Floor, Jia Sarai, Near IIT, Hauz Khas, New Delhi 6 Phone: /

32 Q6. A anonial transforation, q P, Q is erfored on the Hailtonian H q via the generating funtion, F q ot Q of the following grahs shows sheatiall the deendene of Q t Qt. If Qt on t? Q, whih () Q t Qt Ans.: Solution: H q, F q otq F F F, P, K H q Q t F q ot Q..(i) q F P ose Q q Q P P ose Q q..(ii) Fro (i) and (ii) PosQ F F ; t t K H t t K H q ut the value of and q K P using equation of otion Q and P H.No. 4 D, Ground Floor, Jia Sarai, Near IIT, Hauz Khas, New Delhi 6 Phone: /

33 K Q P P P onstant K K Q Q Q onstant P onstant P P Q Qt fro boundar ondition Q t Therefore, otion is orret. Q63. The Lagrangian of a artile oving in a lane s given in Cartesian oordinates as Ans.: Solution: L In olar oordinates the eression for the anonial oentu r (onjugate to the radial oordinate r ) is r sin r os r os r sin () r os r sin r sin r os L ros, rsin ros rsin, r sin ros Lr sin os r sin os rr os rr sin L Pr r sin os r os sin r P r sin r os r H.No. 4 D, Ground Floor, Jia Sarai, Near IIT, Hauz Khas, New Delhi 6 Phone: /

34 Q64. Let, t and, t NET/JRF (JUNE-6) be the oordinate sstes used b the observers O and O, resetivel. Observer O oves with a veloit v along their oon ositive - ais. If t and t are the linear obinations of the oordinates, the Lorentz transforation relating O and O takes the for and and () and and Ans: Solution: t v t vt v v v v t v v v v t v v v t v v v v t t vt t v v v v v v t v v t Q65. A ball of ass, initiall at rest, is droed fro a height of 5 eters. If the oeffiient of restitution is.9, the seed of the ball just before it hits the floor the seond tie is aroiatel (take g 9.8 / s ) 9.8 / s 9. / s ()8.9 / s 7. / s Ans: () Solution: veloit just before hitting first tie is H.No. 4 D, Ground Floor, Jia Sarai, Near IIT, Hauz Khas, New Delhi 6 Phone: /

35 v gh / s After hitting veloit will be ev v 8.9 / s veloit hitting before seond tie will be sae as v Q66. The Hailtonian of a sste with generalized oordinate and oentu q, is H q. A solution of the Hailtonian equation of otion is (in the following A and B are onstants) A Be q e B At At, At A At () Ae, q e B Ans: Solution: H q A Ae, q e B At At A Ae, q e B A t A t Fro Hailton s equation H d q (i) q dt H dq dt fro equations (i) and (ii) d dq q q q (ii) Integrating both sides, ln ln q ln A q A (iii) fro equation (i) d q A dt d Adt lnb ln At Be B At Putting this value of in equation (iii) gives Hene, the orret otion is q A e B At H.No. 4 D, Ground Floor, Jia Sarai, Near IIT, Hauz Khas, New Delhi 6 Phone: /

36 Q67. A anonial transforation q, Q, P, F qp qpon the Hailtonian H q, is ade through the generating funtion q q 4 where and are onstants. The equations of otion for QP, are 4 P Q and P Q P () Q and Ans.: Solution: F qp, qp P P Q Q 4P Q and Q P P Q and P Q This is F te generating funtion so F q & F P Q & & qp Q q q Q Q P H Q, P H P 4 QP P Q Q Q 4 4 4P Q Q and H Q P P Q Q68. The Lagrangian of a sste oving in three diensions is L k k 3 3 The indeendent onstants of otion is/are energ alone onl energ, one oonent of the linear oentu and one oonent of the angular oentu () onl energ, one oonent of the linear oentu onl energ, one oonent of the angular oentu Ans. : H.No. 4 D, Ground Floor, Jia Sarai, Near IIT, Hauz Khas, New Delhi 6 Phone: /

37 Solution: The otion is in 3D. So don t get onfine with, 3 the are atuall, z, Langrangian is then L L L L z k k z, when,, z So, not an oonent at Linear oentu is onserve. Now transfor the Lagrangian to Hailtonian P 4 4 P Pz H k k z H so energ is onserved t Now let us assue L Pz zp dl dt L, H L t L, H Pz zp, H, H Pz Pz, H z, H P z P, H z L, H, P P, kz z, P z P, kz P 4 4 P z z Pz Pz P k. z P z k. z 4 4 k z k z z k z k z dl dt. Siilarl dl dt and dl z dt Q69. For a artile of energ E and oentu (in a frae F ), the raidit is defined E 3 as ln. In a frae F oving with veloit v,, with reset to E 3 F, the raidit will be ln ln () ln ln Ans: H.No. 4 D, Ground Floor, Jia Sarai, Near IIT, Hauz Khas, New Delhi 6 Phone: /

38 E 3 Solution: ln E 3 E 3 Then ln E 3 E Where 3 3 v E E v 3 Put the value of 3 and E one will get E 3 E ln 3 ln ln v E 3 E 3 ln v E 3 E 3 E 3 ln ln E 3 NET/JRF (DEC-6) Q7. A ball of ass is droed fro a tall building with zero initial veloit. In addition to gravit, the ball eerienes a daing fore of the for, where is its instantaneous veloit and is a onstant. Given the values kg, kg/ s and g Ans. : Solution: / s the distane travelled (in etres) in tie t in seonds, is t e t t e t t () 5t e 5t d dt d g dt Putting the values of, and g and silifing we obtain d d dt dt The general solution of this equation is t t e t Using the initial onditions, H.No. 4 D, Ground Floor, Jia Sarai, Near IIT, Hauz Khas, New Delhi 6 Phone: /

39 We obtain, and Hene the required solution is t t e t Q7. A relativisti artile oves with a onstant veloit v with reset to the laborator frae. In tie, easured in the rest frae of the artile, the distane that it travels in the laborator frae is v Ans. : v () v v Solution: Fro Partile tinitial t tfinal t v v vt vt, v / v / v / v / vt t vt t v v / v / Q7. A artile in two diensions is in a otential Ans. : V,. Whih of the following (aart fro the total energ of the artile) is also a onstant of otion? () Solution: V, H d dt, H t L v, H,,, H.No. 4 D, Ground Floor, Jia Sarai, Near IIT, Hauz Khas, New Delhi 6 Phone: /

40 Q73. The dnais of a artile governed b the Lagrangian L k kt desribes an undaed sile haroni osillator a daed haroni osillator with a tie varing daing fator () an undaed haroni osillator with a tie deendent frequen a free artile Ans. : Solution: L k kt L L kt, k kt d L L kt kkkt dt So otion is equivalent to free artile Q74. The araboli oordinates, are related to the Cartesian oordinates, b and. The Lagrangian of a two-diensional sile haroni osillator of ass and angular frequen is 4 () 4 Ans. : Solution: For two diensional Haroni osillation L H.No. 4 D, Ground Floor, Jia Sarai, Near IIT, Hauz Khas, New Delhi 6 Phone: /

41 ,, L 4 L Q75. After a erfetl elasti ollision of two idential balls, one of whih was initiall at rest, the veloities of both the balls are non zero. The angle between the final, veloities (in the lab frae) is () Ans. : Solution: Angle between two artile Conservation of oentu u v os v os (i) v sin v sin (ii) onservation of kineti energ u v v (iii) Fro (i) and (ii) u v v vv os os sin sin u v v vv os (iv) u v v (v) v v v v vv os u v v H.No. 4 D, Ground Floor, Jia Sarai, Near IIT, Hauz Khas, New Delhi 6 Phone: /

42 os Q76. Consider irular orbits in a entral fore otential V r n k, where k and r n. If the tie eriod of a irular orbit of radius R is T and that of radius R is T T, then T n Ans. : () Solution: V eff J k n r r, 3 n () r r r Veff J nk 3 n n n J r r 4 r nk r 3 n r n r n/ T n r T T n n R R Q77. Consider a radioative nuleus that is travelling at a seed with reset to the lab frae. It eits -ras of frequen v in its rest frae. There is a stationar detetor, (whih is not on the ath of the nuleus) in the lab. If a -ra hoton is eitted when the nuleus is losest to the detetor, its observed frequen at the detetor is 3 v 3 v () v 3 v Ans. : Solution: vv v (If detetor is not in the ath at nuleus) vv v 4 3 H.No. 4 D, Ground Floor, Jia Sarai, Near IIT, Hauz Khas, New Delhi 6 Phone: /

43 NET/JRF (JUNE-7) Q78. The Hailtonian for a sste desribed b the generalised oordinate and generalised Ans. : oentu is H where, and are onstants. The orresonding Lagrangian is () Solution: H a. H a a L P H P a P Q79. An inertial observer sees two events E and E haening at the sae loation but 6 s aart in tie. Another observer oving with a onstant veloit v (with reset to the first one) sees the sae events to be 9 s aart. The satial distane between the events, as easured b the seond observer, is aroiatel 3 () 7 H.No. 4 D, Ground Floor, Jia Sarai, Near IIT, Hauz Khas, New Delhi 6 Phone: /

44 Ans. : () Solution:, t 6 t 6, 6 t,? t 9 t t 9 6 v ' v ' t t 6 9 v / v / 6 t t v / v / 5 v v / v v / vt vt v / v / t t Q8. A ball weighing g, released fro a height of 5, bounes erfetl elastiall off a late. The ollision tie between the ball and the late is.5s. The average fore on the late is aroiatel 3 N N () 5 N 4 N Ans. : Solution:.kg. v / se gh v v gh hange in oentu during ollision, v v k. g /se f P 4N t.5 H.No. 4 D, Ground Floor, Jia Sarai, Near IIT, Hauz Khas, New Delhi 6 Phone: /

45 Q8. A solid vertial rod, of length L and ross-setional area A, is ade of a aterial of Young s odulus Y. The rod is loaded with a ass M, and, as a result, etends b a sall aount L in the equilibriu ondition. The ass is then suddenl redued to M /. As a result the rod will undergo longitudinal osillation with an angular frequen YA / ML YA / ML () YA / M L YA / M L Ans. : Fl Solution: Y YA F l A l l For ass, YAl g l YA l For ass g l Equation (i) and (ii) is for equilibriu ondition Change in fore will generate aeleration g YA l YA l F g l l l l l YA l YA l Q8. The Lagrangian of a free relativisti artile (in one diension) of ass is given b L where / d dt. If suh a artile is ated uon b a onstant fore in the diretion of its otion, the hase sae trajetories obtained fro the orresonding Hailtonian are ellises loids () herbolas arabolas Ans. : () Solution: 4. E For onstant fore dp F dt, P Ft P t F H.No. 4 D, Ground Floor, Jia Sarai, Near IIT, Hauz Khas, New Delhi 6 Phone: /

46 u F/ t Ft u u / Ft. d dt F t. Ft F t tdt Ft Ft F F P 4 P F F F P F 4 4, whih is equation of herbola. Q83. A Hailtonian sste is desribed b the anonial oordinate q and anonial oentu. A new oordinate Q is defined as Qt qt t, where t is the tie and is a onstant, that is, the new oordinate is a obination of the old oordinate and oentu at a shifted tie. The new anonial oentu eressed as t qt t qt t q t Ans. : Solution: Given q t q t () Q t q t t If P is new oentu, then QP, For otion, Q, P q, q q, qq, q, q, For otion QP q q,, P t an be H.No. 4 D, Ground Floor, Jia Sarai, Near IIT, Hauz Khas, New Delhi 6 Phone: /

47 ,, i.e. anonial transfor For otion () QP q q For otion QP, q, q Otion () and are orret. But fro translation setr otion is ore suitable. Q84. The energ of a one-diensional sste, governed b the Lagrangian n L k where k and n are two ositive onstants, is E. The tie eriod of osillation satisfies n n n n n n k k E () k E k n Ans. : H J Solution: T J, Tie Period T, where J is Ation variable J H E J Pd E k 4 EV d /n E k n J 4 E k dn /n E k k 4 E E n d Don t tr to solve integration rather tr to ake E indeendent. E k n n n E V E k /n E n J 4 E t dt k, Put k E n t E n d k dt J 4 E k /n n n, where n t dt J T k E E n. n n k n E n n H.No. 4 D, Ground Floor, Jia Sarai, Near IIT, Hauz Khas, New Delhi 6 Phone: /