fiziks Institute for NET/JRF, GATE, IIT JAM, JEST, TIFR and GRE in PHYSICAL SCIENCES CLASSICAL MECHANICS SOLUTIONS

Transcription

1 fizis CLASSICAL MECHANICS SOLUTIONS GATE- Q. For the set of all Lorentz transforations with eloities along the x-axis onsider the two stateents gien below: P: If L is a Lorentz transforation then, L - is also a Lorentz transforation. Q: If L and L are Lorentz transforations then, L L is neessarily a Lorentz transforation. Choose the orret otion (A) P is true and Q is false (B) Both P and Q are true (C) Both P and Q are false (D) P is false and Q is true V x x x, where, λ >. Q. A artile is laed in a region with the otential Then, V (A) x = and x are oints of stable euilibriu (B) x = is a oint of stable euilibriu and (C) x = and x are oints of unstable euilibriu (D) There are no oints of stable or unstable euilibriu V x x V x x x x, x. x x x is a oint of unstable euilibriu V V At x, e (Stable) and At x, e x x (unstable) Head offie fizis, H.No., G.F, Jia Sarai, Near IIT, Hauz Khas, New Delhi 6 Phone: / Branh offie Anand Institute of Matheatis, 8 B/6, Jia Sarai, Near IIT Hauz Khas, New Delhi 6 Website: Eail: fizis.hysis@gail.o

2 fizis Q. A π eson at rest deays into two hotons, whih oe along the x-axis. They are both deteted siultaneously after a tie, t = s. In an inertial frae oing with a eloity V =.6 in the diretion of one of the hotons, the tie interal between the two detetions is (A) 5 s (B) s (C) s (D) s t t.6.6 se, t t.6.6 tt 5se 5se Stateent for Lined Answer Questions 4 and 5: The Lagrangian for a sile endulu is gien by L l gl os Q4. Hailton s euations are then gien by (A) gl sin; (B) gl sin ; l l (C) ; (D) P H gl os l g ; l H P P H glsin ; P l P.Q5. The l Poisson braet between θ and is l (A), (B), (C), (D), g l Head offie fizis, H.No., G.F, Jia Sarai, Near IIT, Hauz Khas, New Delhi 6 Phone: / Branh offie Anand Institute of Matheatis, 8 B/6, Jia Sarai, Near IIT Hauz Khas, New Delhi 6 Website: Eail: fizis.hysis@gail.o

3 fizis P,, where P P. l l l P P. l l GATE- Q6. A artile is oing under the ation of a generalized otentialv,. The agnitude of the generalized fore is (A) (B) (C) (D) () d V V F dt F. Q7. Two bodies of ass and are onneted by a sring onstant. The freueny of the noral ode is (A) / (B) / (C) / (D) / where redue ass Q8. Let (, ) and (P, Q) be two airs of anonial ariables. The transforation Q os, P sin. is anonial for (A) α =, β = / (B) α =, β = (C) α =, β = (D) α = /, β = (d) Q P Q P os os sin sin os sin., Head offie fizis, H.No., G.F, Jia Sarai, Near IIT, Hauz Khas, New Delhi 6 Phone: / Branh offie Anand Institute of Matheatis, 8 B/6, Jia Sarai, Near IIT Hauz Khas, New Delhi 6 Website: Eail: fizis.hysis@gail.o

4 fizis Q9. Two artiles eah of rest ass ollide head-on and sti together. Before ollision, the seed of eah ass was.6 ties the seed of light in free sae. The ass of the final entity is (A) 5 / 4 (B) (C) 5 / (D) 5 / 8 () Fro onseration of energy Sine. 6 5 / GATE- Q. In a entral fore field, the trajetory of a artile of ass and angular oentu L in lane olar oordinates is gien by, os L r where, ε is the eentriity of the artile s otion. Whih one of the following hoie for ε gies rise to a araboli trajetory? ε = ε = () < ε < (d) ε > l os for araboli trajetory l. r Q. A artile of unit ass oes along the x-axis under the influene of a otential, V x xx. The artile is found to be in stable euilibriu at the oint x =. The tie eriod of osillation of the artile is () (d) Head offie fizis, H.No., G.F, Jia Sarai, Near IIT, Hauz Khas, New Delhi 6 Phone: / Branh offie Anand Institute of Matheatis, 8 B/6, Jia Sarai, Near IIT Hauz Khas, New Delhi 6 Website: Eail: fizis.hysis@gail.o 4

5 fizis V x,x x V x xx x xx V x x x x V x 4 V T. x T x Q. A rod of roer length l oriented arallel to the x-axis oes with seed / along the x-axis in the S-frae, where is the seed of the light in free sae. The obserer is also oing along the x-axis with seed / with reset to the S-frae. The length of the rod as easured by the obserer is.5l.48l ().87l (d).97l (d) u x l l.97 l Q. A artile of ass is attahed to fixed oint O by a weightless inextensible string of length a. It is rotating under the graity as shown in the figure. The Lagrangian of the artile is L, a sin ga os where θ and are the a z olar angles. The Hailtonian of the artiles is H ga os a sin g H gaos a sin () H os (d) H os ga a ga a Head offie fizis, H.No., G.F, Jia Sarai, Near IIT, Hauz Khas, New Delhi 6 Phone: / Branh offie Anand Institute of Matheatis, 8 B/6, Jia Sarai, Near IIT Hauz Khas, New Delhi 6 Website: Eail: fizis.hysis@gail.o 5

6 fizis H P P L P P a L P sin ga os P a P and P a Put the alue of and L a sin P a sin H P P P P P P a sin gaos a a sin a a sin H P a P a P P gaos a sin a sin H P P ga os a sin Stateent for Lined Answer Questions 4 and 5: Q4. A artile of ass slides under the graity without frition along the araboli ath y ax axis shown in the figure. Here a is a onstant. y x The Lagrangian for this artile is gien by L L 4a x x x gax gax () (d) L L 4a x x gax x gax (d) Head offie fizis, H.No., G.F, Jia Sarai, Near IIT, Hauz Khas, New Delhi 6 Phone: / Branh offie Anand Institute of Matheatis, 8 B/6, Jia Sarai, Near IIT Hauz Khas, New Delhi 6 Website: Eail: fizis.hysis@gail.o 6

7 fizis Euation of onstrain is gien by yax, K.E T x y y axx T x 4ax x x 4ax V gy gax. Sine artile is oing downward diretion so otential V is negatie. L T V L 4a x x gax Q5. The Lagrange s euation of otion of the artile for aboe uestion is gien by () x gax 4a x x gax 4a xx (d) x gax () d dl dl dt dx dx ( 4 a x ) x 4a xx gax 4a x x gax 4a xx GATE- Q6. In the ost general ase, whih one of the following uantities is NOT a seond order tensor? Stress Strain () Moent of inertia (d) Pressure Strain is not a tensor. Q7. An eletron is oing with a eloity of.85 in the sae diretion as that of a oing hoton. The relatie eloity of the eletron with reset to hoton is ().5 (d).5 Head offie fizis, H.No., G.F, Jia Sarai, Near IIT, Hauz Khas, New Delhi 6 Phone: / Branh offie Anand Institute of Matheatis, 8 B/6, Jia Sarai, Near IIT Hauz Khas, New Delhi 6 Website: Eail: fizis.hysis@gail.o 7

8 fizis Q8. The Lagrangian of a syste with one degree of freedo is gien by L, where and are non-zero onstants. If denotes the anonial oentu onjugate to then whih one of the following stateents is CORRECT? and it is a onsered uantity. () (d) and it is not a onsered uantity. and it is a onsered uantity. and it is not a onsered uantity. (d) L L but Q9. The relatiisti for of Newton s seond law of otion is F d dt F d dt () P d (d) dt F F F / d dt dp d d dt dt dt d d F dt dt Head offie fizis, H.No., G.F, Jia Sarai, Near IIT, Hauz Khas, New Delhi 6 Phone: / Branh offie Anand Institute of Matheatis, 8 B/6, Jia Sarai, Near IIT Hauz Khas, New Delhi 6 Website: Eail: fizis.hysis@gail.o 8

9 fizis Q. Consider two sall blos, eah of ass M, attahed to two idential srings. One of the srings is attahed to the wall, as shown in the figure. The sring onstant of eah sring is. The asses slide along the surfae and the frition is negligible. The freueny of one of the noral odes of the syste is, M () 5 M M M M (d) () T 6 M x x, x x x x x x x x V T ; V x x x x 5 GATE- 4 Q. If the half-life of an eleentary artile oing with seed.9 in the laboratory frae is s, then the roer half-life is s. / s.8 Head offie fizis, H.No., G.F, Jia Sarai, Near IIT, Hauz Khas, New Delhi 6 Phone: / Branh offie Anand Institute of Matheatis, 8 B/6, Jia Sarai, Near IIT Hauz Khas, New Delhi 6 Website: Eail: fizis.hysis@gail.o 9

10 fizis t t, t t 8 = t s Q. Two asses and are attahed to the two ends of a assless sring with fore onstant K. If osillation is Hz..8 f g and K.N /.. 4 4, then the natural angular freueny of 4.8 Hz Q. The Hailton s anonial euation of otion in ters of Poisson Braets are, H;, H H, ; H, () H, ; H, (d), H;, H df f f f.. dt t t t df f H f H f.. dt t df f f, H dt t d d H, dt and, H dt Q4. A bead of ass an slide without frition along a ass less rod et at Head offie fizis, H.No., G.F, Jia Sarai, Near IIT, Hauz Khas, New Delhi 6 Phone: / o 45 with the ertial as shown in the figure. The rod is rotating about the ertial axis with a onstant angular seed. At any instant r is the distane of the bead fro the origin. The ẑ oentu onjugate to r is r r () r o 45 r (d) r xˆ Branh offie Anand Institute of Matheatis, 8 B/6, Jia Sarai, Near IIT Hauz Khas, New Delhi 6 Website: Eail: fizis.hysis@gail.o

11 fizis L ( r r r sin ) gr os euation of onstrain is and it is gien 4 L ( r r ) gr L the oentu onjugate to r is r r = r r Q5. A artile of ass is in a otential gien by V r a ar r r when a and r are ositie onstants. When disturbed slightly fro its stable euilibriu osition it undergoes a sile haroni osillation. The tie eriod of osillation is r r r () (d) a a a 4 r a V r a ar r r for euilibriu V a ar 4 r r r r r V a 4ar a 4ar a 5 5 r r r r r r r V r r T r a Q6. A lanet of ass oes in a irular orbit of radius r in the graitational otential V r r, where is a ositie onstant. The orbit angular oentu of the lanet is r r () r (d) r (d) Head offie fizis, H.No., G.F, Jia Sarai, Near IIT, Hauz Khas, New Delhi 6 Phone: / Branh offie Anand Institute of Matheatis, 8 B/6, Jia Sarai, Near IIT Hauz Khas, New Delhi 6 Website: Eail: fizis.hysis@gail.o

12 fizis V efftie J r r dv effet J = at r r dr r r so J r Q7. Gien that the linear transforation of a generalized oordinate and the orresonding oentu, Q 4a P is anonial, the alue of the onstant a is.5 Q P Q P... 4 a. a.5 Q8. The Hailtonian of artile of ass is gien by H following figure desribes the otion of the artile in hase sae?.whih one of the () (d) (d) Head offie fizis, H.No., G.F, Jia Sarai, Near IIT, Hauz Khas, New Delhi 6 Phone: / Branh offie Anand Institute of Matheatis, 8 B/6, Jia Sarai, Near IIT Hauz Khas, New Delhi 6 Website: Eail: fizis.hysis@gail.o

13 fizis GATE- 5 Q9. A satellite is oing in a irular orbit around the Earth. If T, V and E are its aerage ineti, aerage otential and total energies, resetiely, then whih one of the following otions is orret? V T; E T V T ; E () V Ans.: T T ; E (d) V Fro Virial theore T n V where V n r T ; E T V V n V T r r A and t, r yˆ, resetiely. The ties at whih these eents tae lae in a frae Q. In an inertial frae S, two eents A and B tae lae at t A, r B B S oing with a eloity.6ˆ y with reset to S are gien by t ; A t B t ; A t B () t ; A t B (d) t ; A t B Ans.: ' Veloity of S with reset to S is.6 ta y ' ta for eent A t A, y so t ' B tb y t for eent B tb, y so tb ' A ' Head offie fizis, H.No., G.F, Jia Sarai, Near IIT, Hauz Khas, New Delhi 6 Phone: / Branh offie Anand Institute of Matheatis, 8 B/6, Jia Sarai, Near IIT Hauz Khas, New Delhi 6 Website: Eail: fizis.hysis@gail.o

14 fizis Q. The Lagrangian for a artile of ass at a osition r oing with a eloity is gien by L Cr. V r anonial oentu, then its Hailtonian is gien by Cr V r Cr V r r () V Ans.: L Cr. V r, wherev ris a otential and C is a onstant. If is the where r (d) C r V where L r Crr. V r H r Lr L L rcr r r Cr Cr Cr Cr H r V r Cr Cr H Cr V r Cr Cr H Vr H Cr V r Q. The Hailtonian for a syste of two artiles of asses and at r and r haing eloities and is gien by C H zˆ r r, wrong where r r r C is onstant. Whih one of the following stateents is orret? The total energy and total oentu are onsered Only the total energy is onsered () The total energy and the z - oonent of the total angular oentu are onsered (d) The total energy and total angular oentu are onsered Ans.: () Head offie fizis, H.No., G.F, Jia Sarai, Near IIT, Hauz Khas, New Delhi 6 Phone: / Branh offie Anand Institute of Matheatis, 8 B/6, Jia Sarai, Near IIT Hauz Khas, New Delhi 6 Website: Eail: fizis.hysis@gail.o 4

15 fizis Lagrangian is not funtion of tie so energy is onsere and oonent of r r are Only in ẑ diretion eans otential is syetri under so L z is onsere. Q. A artile of ass. g falls freely in the earth s graitational field with an initial eloity s. If the air exerts a fritional fore of the for, f, then for.5 N s, the eloity (in s ) at tie t. s is (uto two deial laes). (use g s and. 7 e ) Ans.: Data gien is inorret d g dt d g dt d dt g u d g. dt ln g t u..5.5 ln u ln... ln 5u ln 4. ln g u ln g. ln 4 an not be defined. So gien data are not orret. Q4. Consider the otion of the Sun with reset to the rotation of the Earth about its axis. If F and F Co denote the entrifugal and the Coriolis fores, resetiely, ating on the Ans.: Sun, then F is radially outward and F Co F F is radially inward and F Co F () F is radially outward and F Co F (d) F is radially outward and F Co F Head offie fizis, H.No., G.F, Jia Sarai, Near IIT, Hauz Khas, New Delhi 6 Phone: / Branh offie Anand Institute of Matheatis, 8 B/6, Jia Sarai, Near IIT Hauz Khas, New Delhi 6 Website: Eail: fizis.hysis@gail.o 5

16 fizis Q5. A artile with rest ass M is at rest and deays into two artiles of eual rest asses M whih oe along the z axis. Their eloities are gien by.8zˆ.8 z ˆ ().6zˆ.6 zˆ; (d).8 z ˆ Ans.: M M M Fro oentu onseration PP P P P P Fro energy onseration E E E M M M M M GATE-6 Q6. The ineti energy of a artile of rest ass is eual to its rest ass energy. Its oentu in units of, where is the seed of light in auu, is. Ans. :.7 (Gie your answer uto two deial laes E E Head offie fizis, H.No., G.F, Jia Sarai, Near IIT, Hauz Khas, New Delhi 6 Phone: / Branh offie Anand Institute of Matheatis, 8 B/6, Jia Sarai, Near IIT Hauz Khas, New Delhi 6 Website: Eail: fizis.hysis@gail.o 6

17 fizis Q7. In an inertial frae of referene S, an obserer finds two eents ourring at the sae tie at oordinates x and x d A different inertial frae S oes with eloity with reset to S along the ositie x -axis. An obserer in S also noties these two eents and finds the to our at ties t and t and at ositions x and x resetiely. If t t t, x x x and t, x d d () t, x d (d) Ans.: (), whih of the following stateents is true? t, x d d d t, x x x t t ' ' t t ' x t x x t x t ' ' x ' x t t it is gien t, x d ' x x t it is gien t, x d x d ' Q8. The Lagrangian of a syste is gien by sin gl os L l, where, l and g are onstants. Whih of the following is onsered? sin sin () sin (d) sin Head offie fizis, H.No., G.F, Jia Sarai, Near IIT, Hauz Khas, New Delhi 6 Phone: / Branh offie Anand Institute of Matheatis, 8 B/6, Jia Sarai, Near IIT Hauz Khas, New Delhi 6 Website: Eail: fizis.hysis@gail.o 7

18 fizis Ans.: is yli oordinate so onstants. Then sin L l sin is onstant hene, l and g are Q9. A artile of rest ass M is oing along the ositie x -diretion. It deays into two hotons and as shown in the figure. The energy of is GeV and the energy of GeV is.8 GeV. The alue of M (in units of ) is. (Gie your answer uto two deial laes) M 45 6 Ans.:.4 4 M EE.8GeV E E GeV.8GeV os os.gev Head offie fizis, H.No., G.F, Jia Sarai, Near IIT, Hauz Khas, New Delhi 6 Phone: / Branh offie Anand Institute of Matheatis, 8 B/6, Jia Sarai, Near IIT Hauz Khas, New Delhi 6 Website: Eail: fizis.hysis@gail.o 8

19 fizis GATE- 7 Q4. If the Lagrangian d L dt d is odified to L L, whih one dt of the following is TRUE? Both the anonial oentu and euation of otion do not hange Canonial oentu hanges, euation of otion does not hange () Canonial oentu does not hange, euation of otion hanges (d) Both the anonial oentu and euation of otion hange Ans. : For lagrangian d L dt anonial oentu is and d L L euation of otion is gien by dt For langrangian d d L L L Canonial oentu is dt dt d L L Euation of otion dt Q4. Two idential asses of g eah are onneted by a assless sring of sring Ans. : 4.4 onstant N/. The non-zero angular eigenfreueny of the syste is rad/s. (u to two deial laes), where and N / 4.4 Head offie fizis, H.No., G.F, Jia Sarai, Near IIT, Hauz Khas, New Delhi 6 Phone: / Branh offie Anand Institute of Matheatis, 8 B/6, Jia Sarai, Near IIT Hauz Khas, New Delhi 6 Website: Eail: fizis.hysis@gail.o 9

20 fizis Q4. The hase sae trajetory of an otherwise free artile bouning between two hard walls elastially in one diension is a straight line arabola () retangle (d) irle Ans. : () E E Q4. The Poisson braet x, xy y x is eual to Ans. : x y () x (d) y,,,, x xy yx x xy x yx y x x y Q44. An objet traels along the x -diretion with eloity in a frae O. An obserer in a Ans. :.8 u frae O sees the sae objet traelling with eloity 4. The relatie eloity of O with reset to O in units of is.. (u to two deial laes). u x ' x, u 4 Head offie fizis, H.No., G.F, Jia Sarai, Near IIT, Hauz Khas, New Delhi 6 Phone: / x ' x u ' u x Q45. A unifor solid ylinder is released on a horizontal surfae with seed 5 / s without Ans. :. any rotation (sliing without rolling). The ylinder eentually starts rolling without sliing. If the ass and radius of the ylinder are g and resetiely, the final linear eloity of the ylinder is / s. (u to two deial laes). Branh offie Anand Institute of Matheatis, 8 B/6, Jia Sarai, Near IIT Hauz Khas, New Delhi 6 Website: Eail: fizis.hysis@gail.o

21 fizis r ri r r. / se r Q46. A erson weighs w at Earth s north ole and erfet shere of radius 64, the alue w e at the euator. Treating the Earth as a w we is.. (u to two w deial laes). (Tae g s ). Ans. :. g g ge g R w we R w g Put the alue of g / se T 46 R 64 w we Then. w Head offie fizis, H.No., G.F, Jia Sarai, Near IIT, Hauz Khas, New Delhi 6 Phone: / Branh offie Anand Institute of Matheatis, 8 B/6, Jia Sarai, Near IIT Hauz Khas, New Delhi 6 Website: Eail: fizis.hysis@gail.o