Classical Mechanics(net/gate)

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Classical Mechanics(NET/GATE)  1 A particle is placed in a region with the potential V  x   kx 2  x3 , 2 3 where k,  > 0 k (a) x  0 and x  are points of stable equilibrium  k (b) x  0 is a point of stable equilibrium and x  is a point of unstable  equilibrium k (c) x  0 and x  are points of unstable equilibrium  (d) There are no points of stable or unstable equilibrium 1.

A ° at rest decay into two photons, which moves along the x-axis. They are both detected simultaneously after a time, t = 10s,In an inertial frame moving with a velocity v = 0.6 c in the direction of one of the photons, the time interval between the two direction sis (a) 15c (b) 0c (c) 10c (d) 20c 3. A particle is moving under the action of a generalized 1 q potential V  q, q   2 . The magnitude of the generalized force is q 2.

(a)

2 1  q  q3

(b)

2 1  q  q3

(c)

2 q

3

(d)

q q3

Statement Linked answer Questions 4 and 5:

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The

4.

5.

Lagrangian

7.

a

simple

1 L  ml 22  mgl 1  cos   2 Hamliton's equations are then given by p (a) p  mgl sin ;   2 ml p (c) p  m;    m

pendulum

is

given

(b) p  mgl sin ;  

by

p

ml 2 p g (d) p     ;    ml l

The Poission bracket between  and  is g 1 1 (a) ,   1 (b) ,   2 (c) ,   (d) ,   l m ml Two bodies of mass m and 2m are connected by a spring constant k. The frequency of the normal made

 

6.

for

 

 

 

(d) k / 2m (a) 3k / 2m (b) k / m (c) 2k / 3m Let (p, q) and (P, Q) be two pairs of canonical variables. The transformation

Q  q cos p  , P  q sin p 

8.

9.

is canonical for 1 (b)  = 2,  = 2 (c)  = 1,  = 1 (d) (a)   2,   2 1   ,  2 2 Two particles each rest mass m collide head-on and stick together. Before collision, the speed of each mass was 0.6 times the speed of light in free space. The mass of the final entity is (a) 5m/4 (b) 2m (c) 5m/2 (d) 25m/8 A particle of unit mass moves along the x-axis under the influence of a potential, V  x   x  x  2  . The particle is found to be in stable 2

equilibrium at the point x = 2. The time period of oscillation of the particle is  3 (a) (b)  (c) (d) 2 2 2

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10.

11.

A rod of proper length l0 oriented parallel to the x-axis moves with speed 2c/3 along the x-axis in the S-frame, where c is the speed of light in free space. The observer is also moving along the x-axis with speed c/2 with respect to the S-frame. The length of the rod as measured by the observer is (a) 0.35 l0 (b) 0.48 l0 (c) 0.48 l0 (d) 0.97 l0 A particle of mass m is attached to a fixed point O by a weightless inextensible string of length a. It is rotating under the gravity as shown in the figure. The Lagrangian of the particle is 1 L  ,    ma 2  2  sin 2  2  mga cos where  and  are 2 the polar





 p2  2  p    mga cos (a) H  (b) 2ma 2  sin 2   p2  1  2 p    mga cos H 2  2  2ma  sin   1 p 2  p2  mga cos (d) (c) H  2  2ma 1 H p 2  p2  mga cos 2  2ma An electron is moving with a velocity of 0.85 c in the same direction s that of a moving photon. The relative of the electron with respect to photon is (a) c (b) – c (c) 0.15 c (d) –0.15 c The Lagrangian of a system with one degree of freedom  is given by L = q2 + q2, where  and  are non-zero constants. If pq denotes the canonical momentum conjugate to q then which one of the following statements is CORRECT? (a) pq = 2q and it is a conserved quantity. (b) pq = 2q and it is not a conserved quantity. (c) pq = 2 q and it is a conserved quantity. 1





12.

13.





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(d) pq = 2 q and it is a not conserved quantity. Statement for Linked Answer Questions 14 and 15 A particle of mass m slides under the gravity without friction along the parabolic path 2 y = ax , as shown in the figure. Here a is a constant.

14.

The Lagrangian for this particle is given by 1 1 (a) L  mx 2  mgax 2 (b) L  m 1  4a 2 x 2 x 2  mgax 2 2 2 1 1 (c) L  mx 2  mgax 2 (d) L  m 1  4a 2 x 2 x 2  mgax 2 2 2 The Lagarange's equation of motion of the particle for above questions is given by (b) (a) x  2 gax

 

15.





 

m 1  4a 2 x 2 x  2mgax  4ma 2xx 2





(c) m 1  4a 2 x 2 x  2mgax  4ma 2xx 2 16.

(d) x  2 gax

Consider two small blocks, each of mass M, attached to two identical springs. One of the spring is attached to the wall, as shown in the figure. The spring constant of each spring is k. The masses slide along the surface and the friction is negligible. The frequency of one of the normal modes of the system is,

(a)

3 2 2

k (b) M

3 3 k (c) 2 M

3 5 2

k (d) M

3 6 2

k M

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17.

If the half-life of an elementary particle moving with speed 0.9 c in the laboratory frame is 5 × 10–8 s, then the proper half-life is _____ × 10–8 s.

c  3 108 m / s 

18.

19.

Two masses m and 3m are attached to the two end of a massless spring with force constant K. If m = 100 g and K = 0.3 N/m, then the neutral angular frequency of oscillation is _______ Hz. The Hamilton's canonical equation of motion in terms of Poisson Brackets are (a) q  q, H ; p   p, H  (b) q  H , q; p  H , p (c) q  H , p; p  H , p

20.

A bead of mass m can slide without friction along a massless rod kept at 45° with the vertical as shown in the figure. The rod is rotating about the vertical axis with a constant angular speed . At any instant r is the distance of the beam from the origin. The momentum conjugate to r is

1 1 mr (c) mr (d) 2 2 A particle of mass m is in a potential given by (a) mr

21.

(d) q   p, H ; p  q, H 

(b)

2 mr

a ar02 V r     3 r 3r

where a and r0 are positive constants. When disturbed slightly from its stable equilibrium position it undergoes a simple harmonic oscillation. The time period of oscillation is

mr03 (a) 2 2a

mr03 (b) 2 a

2mr03 mr03 (c) 2 (d) 4 a a

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22.

A planet of mass m moves in a circular orbit of radius r0 in the potential k V  r    , where k is a positive constant. The orbit angular momentum r of the planet is (a) 2r0km

23.

24.

25.

(b)

(c) r0km (d)

2r0km

r0km

Given that the linear transformation of a generalized coordinate q and the corresponding p, q = q + 4ap, P = q + 2p is canonical, the value of the constant a is ____ p 2 aq 2  .Which 2m 2 one of the following figure describe the motion of the particle in phase space.

The Hamiltonian of particle of mass m is given by H 

(a)

(b)

(c)

(d)

A satellite is moving in a circular orbit around the Earth. If T, V and E are its average kinetic, average potential and total energies, respectively, then which one of the following options is correct? (a) V  2T ; E  T (b) V  T ; E  0 T T 3T T (d) V  ;E ;E 2 2 2 2 In an inertial frame S, two events A and B take place at  ct A  0, rA  0 and  ctB  0, rB  2 yˆ  , respectively. The times at which

(c) V 

26.

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these events take place in a frame S' moving with a velocity 0.6 cyˆ with

27.

respect to S are given by 3 (a) ct 'A  0 ; ctB'   (b) ct 'A  0 ; ctB'  0 2 3 1 (c) ct 'A  0 ; ct B'  (d) ct 'A  0 ; ct B'  2 2 The Lagrangian for a particle of mass m at a position r moving with a m velocity is given v is given by L  v 2  Cr .v  V  r  , where V (r) is a 2 potential and C is a constant. If pc is the canonical momentum, then its Hamiltonian is given by 1 (a)  pc  Cr   V  r  2m

28.

(b)

1  pc  Cr   V  r  2m

1 pc2  V r  (c) (d)  pc  Cr 2  V  r  2m 2m The Hamiltonial for a system of two particles of masses m1 and m2 at velocities given by r1 and v1 and v2 is r2 having 1 1 C H  m1v12  m2v22  zˆ.  r1  r2  , Where C is constant. Which 2 2 2  r1  r2 

29.

30.

one of the following statement is correct? (a) The total energy and total momentum are conserved (b) Only the total energy is conserved (c) The total energy and the z-component of the total angular momentum are conserved (d) The total energy and total angular momentum are conserved A particle of mass 0.01 kg falls freely in the earth's gravitational field with an initial velocity (0) = 10 ms–1. If the air exerts a frictional force of the form, f = –kv, then for k = 0.05 Nm–1s, the velocity (in ms–1) at time t = 0.2 s is _____ (upto two decimal places). (use g = 10 m/s–2 and e = 2.72) The kinetic energy of a particle of rest mass m0 is equal to its rest mass energy. Its momentum in units of m0c, where c is the speed of light in vaccum, is _____ (Give your answer upto two decimal places)

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31.

In an inertial frame of reference S, an observer finds two events occurring at the same time at coordinates x1 = 0 and x2 = d. A different inertial frame S' moves with velocity v with respect to S along the positive x-axis. An observer in S; also notices these two events and finds them to occur at times t1' and t2' and at positions x1' and x2' respectively. 1 If t '  t2'  t1' , x'  x2'  x1' and   , which of the following 2 v 1 2 c statements is true? d (b) t '  0, x '  (a) t '  0, x '   d



 vd d  vd , x '   d (d) t '  2 , x '  c  c The Lagrangian of a system is given by 1 L  ml 2  2  sin 2  2   mgl cos , where m, l and g are constants.   2 Which of the following is conserved? (c) t ' 

32.

(a)  sin 2 

33.

(b)  sin  (c)

 sin 

(d)



sin 2  A particle of rest mass M is moving along the positive x-direction. It decays into two photons  1 and  1 as shown in the figure. The energy of  1 is 1 GeV and the energy of  2 is 0.82 GeV. The value of M (in units of

GeV c

2

) is ______ . (Give your answer upto two decimal points).

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2

34.

35.

36.

37.

38.

39.

If

the

Lagrangian

1  dq  1 L0  m    m 2q 2 is 2  dt  2

modified

to

 dq  L  L0  aq   , which one of the following is TRUE?  dt  (a) Both the canonical momentum and equation of motion do not change (b) Canonical momentum charges, equation of motion does not change (c) Canonical momentum does not charges, equation of motion change (d)Both the canonical momentum and equation of motion charge Two identical masses of 10 gm each are connected by a massless spring of spring constant 1N/m. The non-zero angular eigenfrequency of the system is ..... rad/s. (up to two decimal places) The Poisson bracket  x, xp y  ypx  is equal to (a) –x (b) y (c) 2px

(d) py c An object travels along the x-direction with velocity with velocity in a 2 frame O. An observer in a frame O' sees the same object travelling with c velocity . The relative velocity of O' with respect to O in units of c is 4 ............. (up to two decimal places). A spaceship is travelling with a velocity of 0.7 c away from a space station. The spaceship ejects a probe with a velocity 0.59 c opposite to its own velocity. A person in the space station would see the probe moving at a speed Xc, where the value of X is ________ (up to three decimal places). A particle moves in one dimension under a potential V  x    x with some non-zero total energy. Which one of the following best describes the particle trajectory in the phase space?

(a)

(b)

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(c)

40.

(d)

For the transformation

Q  2qe1 2a cos p, P  2qea 1 sin p 41.

(where a is a constant) to be canonical, the value of a is ____. Consider a transformation from one set of generalized coordinate and momentum (q, p) to another set (Q, P) denoted by,

Q  pq s ; P  q r where s and r are constants. The transformation is canonical if (a) s = 0 and r = 1(b) s = 2 and r = –1(c) s = 0 and r = –1(d) s = 2 and r = 1 42.

p2  kqt where q and p 2m are the generalized coordinate and momentum, respectively, t is time and k is a constant. For the initial condition, q  0 and p  0 at

The Hamiltonian for a particle of mass m is H 

t  0, q  t   t a . The value of a is ______ 43.

A ball bouncing of a rigid floor is described by the potential energy function V  x   mgx for x  0  for x  0



Which of the following schematic diagrams best represents the phase space plot of the ball?

(a )

(b )

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(c )

44.

(d )

ap 2q 4   2 , where  and  are 2 q parameters with appropriate dimensions, and q and p are the generalized coordinate and momentum, respectively. The corresponding Lagrangian L  q, q  is

Consider the Hamiltonial H  q, p  

1 q2  (a)  2 q 4 q 2

1 q2  1 q2  1 q2  (b)  (c)  (d)    q4 q2 2 q 4 q 2 2 q 4 q 2

45.

Two spaceships A and B, each of the same rest length L, are moving in 4c 3c the same direction with speeds and , respectively, where c is the 5 5 speed of light. As measured by B, the time taken by A to completely overtake B [see figure below] in units of L/c (to the nearest integer) is ________

46.

Two events, one on the earth and the other one on the Sun, occur simultaneously in the earth's frame. The time difference between the two events as seen by an observer in a spaceship moving with velocity 0.5 c in the earth's frame along the line joining the earth to the Sum is t, where c is the speed of light. Given that light travels from the sum to the earth in 8.3 minutes in the earth's frame, the value of t in minutes (rounded off to two decimal places) is _____

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47.

48.

b A particle of unit mass moving in a potential V  x   ax 2  2 , where a x and b are positive constants. The angular frequency of small oscillations about the minimum of the potential is (b) 8a (c) 8a / b (d) 8b / a (a) 8b The Hamiltonian of a system with n degrees of freedom is given by H  q1,...qn ; p1,... pn ; t  , with an explicit dependence on the time t. Which of the following is correct? (a) Different phase trajectories cannot intersect each other (b) H always represents the total energy of the system and is a constant of the motion (c) The equations qi  H / pi , pi  H / pi are not valid since H has

49.

50.

explicit time dependence (d) Any initial volume element in phase space remains unchanged in magnitude under time evolution. The Lagrangian of a particle of charge e and mass m in applied electric 1 and magnetic fields is given by L  mv 2  eA.v  e , where A and  2 are the vector and scalar potentials corresponding to the magnetic and electric fields, respectively. Which of the following statement is correct? (a) The canonically conjugate momentum of the particle is given by p  mv p2 e  A. p  e (b) The Hamiltonian of the particle is given by H  2m m (c) L remains unchanged under a gauge transformation of the potentials (d) Under a gauge transformation of the potentials, L changes by the total time derivative The Hamiltonian of a particle of unit mass moving in the xy-plane is given to be: 1 1 H  xpx  yp y  x 2  y 2 in suitable units. The initial values are given 2 2 1 1 to be  x  0 y  0   1, 1 and px  0  , p y  0    ,  . During the 2 2





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51.

motion, the curve traced out by the particles in the xy-plane and the pxpy-plane are (a) both straight lines (b) a straight line and a hyperbola respectively (c) a hyperbola and an ellipse, respectively (d) both hyperbolas A double pendulum consist of two point masses m attached by strings of length l as shown in figure: The kinetic energy of the pendulum is

1 2 2 ml 1  22    2 1 (b) ml 2  212  22  212 cos 1  2    2 1 (c) ml 2 12  222  212 cos 1  2    2 1 (d) ml 2  212  22  212 cos 1  2    2 The potential of a diatomic molecule as a function of the distance r a b between the atoms is given by V  r    6  12 . The value of the r r potential at equilibrium separation between the atoms is

(a)

52.

(a) 4a 2 / b

(b) 2a 2 / b

(c) a2 / 2b (d) a2 / 4b

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53.

A particle of mass m moves inside a bowl. If the surface of the bowl is 1 given by the equation z  a x 2  y 2 , where a is a constant, the 2 Lagrangian of the particle is 1 1 (a) m r 2  r 2 2  gar 2 (b) m  1  a 2r 2 r 2  r 2 2   2  2 1 1 (c) m r 2  r 2 2  r 2 sin 2  2  gar 2 (d) m  1  a 2r 2 r 2  r 2 2   2 2 



 

54.















The trajectory on the zpz -plane (phase-space trajectory) of a ball bouncing perfectly elastically off a hard surface at z  0 is given by approximately (neglect friction):

55.

(a)

(b)

(c)

(d)

The bob of a simple pendulum, which undergoes small oscillations, is immersed in water. Which of the following figures best represents the phase space diagram for the pendulum?

(a)

(b)

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(c)

56.

57.

Two events separated by a (spatial) distance 9 × 109m, are simultaneous in one inertial frame. The time interval between these two events in a frame moving with a constant speed 0.8 c (where the speed of light c  3 108 m / s ) is (a) 60s (b) 40s (c) 20s (d) 0s If the Lagrangian of a particle moving in one dimensions is given by L

58.

59.

(d)

x2  V  x  the Hamiltonian is 2x

x2 p2 1 2 1  V  x  V  x (b) (c) x  V  x  (d) (a) xp  V  x  2x 2x 2 2 What is proper time interval between the occurrence of two events if in one inertial frame events are separated by 7.5 × 108 m and 6.5 s apart? (a) 6.50 s (b) 6.00 s (c) 5.75 s(d) 5.00 s Three particles of equal mass (m) are connected by two identical massless spring of stiffness constant (K) as shown in the figure.

If x1, x2 and x3 denote the horizontal displacement of the masses from their respective equilibrium positions the potential energy of the system is 1 1 (a) K x12  x22  x32 (b) K  x12  x22  x32  x2  x1  x3   2 2  1 1 (c) K  x12  2 x22  x32  2 x2  x1  x3  (d) K  x12  2 x22  2 x2  x1  x3    2  2 





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60.

The Hamiltonian of a simple pendulum consisting of a mass m attached to a massless string of length l is H 

p2 2ml 2

 mgl 1  cos  . If L denotes

dL is: dt g g 2g p sin  (a)  (b)  p sin 2 (c) p cos (d) lp2 cos l l l Which of the following set of phase-space trajectories is not possible for a particle obeying Hamiltan's equation motion? the Lagrangian, the value of

61.

(a)

(b)

(c)

(d)

62.

The muon has mass 105 MeV/c2 and mean life time 2.2 s in its rest frame. The mean distance traversed by a muon of energy 315 MeV before decaying is approximately, (a) 3× 105 km (b) 2.2 cm (c) 6.6 m (d) 1.98 km

63.

The area of a disc in its rest frame S is equal to 1 (in some units). The disc will appear distorted to an observer O moving with a speed m with respect to S along the plane of the disc. The area of the disc measured in the rest frame of the observer O is (c is the speed of light in vacuum)

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64.

65.

66.

1

1

1

 u2   u2  2  u2  2  u2  (a) 1  2  (b) 1  2  (c) 1  2  (d)  1  2   c   c   c   c          A system is government by the Hamiltonian 1 2 1 2 H   px  ay    px  bx  2 2 where a and b are constants and px, py are momenta conjugate to x and y respectively. for what values of a and b will the quantities  px  3 y  and  px  2 x  be conserved? (a) a = –3, b = 2 (b) a = 3, b = –2(c) a = 2, b = –3(d) a = –2, b = 3 The Lagrangian of a particle of mass m moving in one dimension is given by 1 L  mx 2  bx 2 where b is a positive constant. The coordinate of the particle x (t) at time t is given by: (in following c1 an c2 are constants) b 2 t  c1t  c2 (a)  (b) c1t  c2 2m  bt   bt   bt   bt  (c) c1 cos    c2 sin   (d) c1 cosh    c2 sinh   m m m m Let A, B and C be functions of phase space variables (coordinates and moment of a mechanical system). If {,} represents the Poisson bracket, the value of  A,B, C   A, B, C is given by (a) 0

67.

68.

(b) B,C, A

(c)  A,C, B (d) C, A,, B

z2 A particle moves in a potential V  x  y  . Which component(s) of 2 the angular momentum is/are constant(s) of motion? (a) none (b) Lx , Ly and LZ (c) only Lx and L y (d) only LZ 2

2

The Hamiltonian of a relativistic particle of rest mass m and momentum p is given by H 

p 2  m2  V  x  , in units in which the speed of light c

= 1. The corresponding Lagrangian is (a) L  m 1  x 2  V  x 

(b) L  m 1  x 2  V  x 

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1 (d) L  mx 2  V  x  2 Consider a particle of mass m attached to two identical springs each of length l and spring constant k (see the figure). The equilibrium configuration is the one where the springs are unscratched. There are no other external forces on the system. If the particle is given a small displacement along the x-axis which of the following describes the equation of motion for small oscillations?

(c) L  1  mx 2  V  x  69.

(a) mx 

70.

kx 2 0 (b) mx  kx  0 (c) mx  2kx  0 (d) mx  l

kx3

0 l2 The time period of a simple pendulum under the influence of the acceleration due to gravity g is T. The bob is subjected to an additional

acceleration of magnitude

3g in the horizontal direction. Assuming

small oscillations, the mean position and time period of oscillation, respectively, of the bob will be (a) 0° to the vertical and

3T (b) 30° to the vertical and T/2

(c) 60° to the vertical and T/ 71.

2 (d) 0° to the vertical and T / 3

A particle of mass m and coordinate q has the Lagrangian 1  L  mq 2  qq 2 , where  is a constant. The Hamiltaonian for the 2 2 system is given by p 2  qp 2 (a)  2m 2m 2

(c) 72.

p2 (b) 2  m  q 

 qp 2 p2  2m 2  m   q  2

(d)

pq 2

The coordinates and momenta xi , pi  i  1, 2, 3 of a particle satisfy the canonical

Poisson

bracket

relations

xi , p j   ij .

If

C1  x2 p3  x3 p2 and C2  x1 p2  x2 p1 are constants of motion, and if

C3  C1, C2  x1 p3  x3 p1 , then

(a) C2 , C3  C1 and C3 , C1  C2 (b) C2 , C3  C1 and C3 , C1  C2 (c) C2 , C3  C1 and C3 , C1  C2 (d) C2 , C3  C1 and C3 , C1  C2

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73.

A canonical transformation relates the old coordinates (q, p) to the new ones (Q, P) by the relations Q  q 2 and P  p / 2q . The corresponding

74.

time independent generating function is (a) P/q2 (b) q2p (c) q2/p (d) qP2 The equation of motion of a system described by the time-depending 1  Lagrangian L  e t  mx 2  V  x   is 2 

dV dV 0 0 (b) mx   mx  dx dx dV dV 0 0 (c) mx   mx  (d) mx  dx dx A particle of mass m is moving in the potential 1 1 V  x    ax 2  bx 2 where a, b are positive constants. The frequency 2 4 of small oscillations about a point of stable equilibrium is (a) a / m (b) 2a / m (c) 3a / m (d) 6a / m The Hamiltonian of a classical particle moving in one dimension is (a) mx   mx 

75.

76.

p2   q 4 where a is a positive constant and p and q are its H 2m momentum and position respectively. Given that its total energy E  E0 the available volume of phase space depends on E0 as 3

77.

(a) E0 4

(b) E0

(c)

E0

(d) is independent of E0

A

mechanical

system

is

described

by

the

Hamiltonian

P2 1  m 2q 2 . As a result of the canonical transformation H ( q, p )  2m 2 Q generated by F (q, Q)  , the Hamiltonian in the new coordinate Q q and momentum P becomes 1 2 2 m 2 2 Q P  Q (a) 2m 2

1 2 2 m 2 2 Q P  P (b) 2m 2

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1 2 m 2 2 P  Q (c) 2m 2

78.

79.

A particle moves in two dimensions on the ellipse x2  4 y 2  8. at a particular instant it is at the point (x, y) = (2, 1) and the x-component of its velocity is 6 (in suitable units). Then the y-component of its velocity is (a) -3 (b) -2 (c) 1 (d) 4 Consider three inertial frames of reference A, B and C. the frame B c moves with a velocity with respect to A, and C moves with a velocity 2 c with respect to B in the same direction. The velocity of C as measured 10 in A is

3c 4c (b) (c) 7 7 If the lagrangian of a dynamical system 1 L  mx 2  mx 2  mxy, then its Hamiltonian is 2 1 1 2 p y (a) H  px p y  (b) m 2m 1 1 2 p y (d) (c) H  px p y  m 2m (a)

80.

81.

1 2 4 m 2 2 Q P  P (d) 2m 2

c 3c (d) 7 7 in two dimensions is

1 1 2 px p y  p x m 2m 1 1 2 H  px p y  p x m 2m H

A particle of mass m moves in the one dimensional potential V (x)





x3 



x 4 where  ,   0 . One of the equilibrium points is x = 0. The

3 4 angular frequency of small oscillations about the other equilibrium point is.  2   (a) (b) (c) (d) m 3m 12m 24m 82.

A particle of unit mass moves in the x y-plane in such a way that x(t) = y (t) and y (t) = –x (t). We can conclude that it is in a conservative forcefield which can be derived from the potential. 1 1 (a) ( x 2  y 2 ) (b) ( x 2  y 2 ) (c) x  y (d) x  y 2 2

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83.

84.

85.

Let q and p be the canonical coordinate and momentum of a dynamical system. Which of the following transformations is canonical? 1 2 1 2 q and P1 p 1. Q1  2 2 1 1 2. Q2   p  q  and P2  p  q  2 2 (a) neither 1 nor 2 (b) both 1 and 2 (c) only 1 (d) only 2 Which of the following figures is schematic representation of the phase space trajectories (i.e., contours of constant energy) of a particle moving 1 1 in a one-dimensional potential V  x   x 2  x 4 2 4

(a )

(b )

(c)

(d )

The Lagrangian of a system is given by 1 5  L  mq12  2mq22  k  q12  2q22  2q1q2  2 4  where m and k are positive constants. The frequencies of its normal modes are

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86.





k 3k 5k 6k k k k 13  73 (c) , , , (b) (d) 2m 2m m 2m m 2m m A canonical transformation  p, q    P, Q  is performed on the

(a)

Hamiltonian

p2 1  m 2q 2 via H 2m 2

the

generating

function,

1 F  m 2 cot Q . If Q  0   0 , which of the following graphs shows 2 sehematrically the dependence of Q  t  on t?

87.

(a)

(b)

(c)

(d)

The Lagrangian of a particle moving in a plane s given in Cartesian coordinates as

L  xy  x2  y 2

88.

In polar coordinates the expression for the canonical momentum pr (conjugate to the radial coordinate r) is (a) r sin   r cos (b) r cos  r sin  (c) r 2cos  r sin 2 (d) r 2sin   r cos2 Let (x, t) and (x', t') be the coordinate systems used by the observers O and O' respectively. Observer O' moves with a velocity v =c along their common positive x-axis. If x  x  ct and x  x  ct are the linear combinations of the coordinates, the Lorentz transformation relating O and O' takes the form x   x x   x and x '   (a) x '   1  2 1  2

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1  1  x and x '  x 1  1 

(b) x ' 

x   x x   x and x '   (c) x '   1  2 1  2 1  1  x and x '  x 1  1 

(d) x '  89.

90.

The Hamiltonian of a system with generalized coordinate and momentum (q, p) is 2 2 H = p q . A solution of the Hamiltonian equation of motion is (in the following A and B are constants) A A (b) p  Ae2 At , q  e2 At (a) p  Be2 At , q  e2 At B B 2 2 A A (c) p  Ae At , q  e  At (d) p  2 Ae A t , q  e A t B B A canonical transformation  q, p    Q, P  is made through the generating function F  q, P   q 2 P on the Hamiltonian

H  q, P  

p2





q4

4 2 q where a and b are constants. The equation of motion for (Q, P) are P 4P  Q (b) Q  (a) Q  and P    Q and P  2   2

P2 2P  Q (c) Q  and P  (d) Q  and P    Q  2  The Lagrangian of a system moving in three dimensions is 1 1 1 2 L  mx12  m x22  x32  kx12  k  x2  x3  2 2 2 The independent constant of motion is/are (a) energy along (b) only energy, one component of the linear momentum and one component of the angular momentum (c) only energy, one component of the linear momentum (d) only energy, one component of the angular momentum P

91.





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92.

A realistic particle moves with a constant velocity v with respect to the laboratory frame, In time t, measured in the rest frame of the particle, the distance that it travels in the laboratory frame is (a) v

(b)

c 1

93.

v2

(c) v 1 

v2

(d)

c2

v v2

1 2 c2 c A particle in two dimensions is in a potential V  x, y   x  2 y . Which of

the following (apart from the total energy of the particle) is also a constant of motion? (b) px  2 p y (c) px  2 p y (d) p y  2 px (a) p y  2 px 94.

95.

The dynamics of a particle governed by the Lagrangian 1 1 L  mx2  kx 2  kxxt describes 2 2 (a) an undamped simple harmonic oscillator (b) a damped harmonic oscillator with a time varying damping factor (c) an undamped harmonic oscillator with a time dependent frequency (d) a free particle The parabolic coordinates  ,   are related to the Cartesian coordinates





1 2 2  ,  . The Lagrangian of a two2 dimensional simple harmonic oscillator of mass m and angular frequency  is 1 (a) m  2   2   2  2   2   m  1 1   (b) m  2   2   2   2   2  2   2  m 4   1 1   (c) m  2   2  2   2   2  m 2   (x, y) by x   ,  and y 



















1 1   m  2   2  2   2   2  m 4   The Hamiltonian for a system described by the generalised coordinate x and generalised momentum p is (d)

96.





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p2 1   2 x2 H  ax p  2 1  2 x  2 2

where ,  and  are constants. The corresponding Lagrangian is 1 1 1 1 x 2   2 x 2  ax 2 x (a)   ax2 1  2 x    2 x 2 (b) 2 1  2  x  2 2 2 (c) 97.

98.

99.







 1  2 x   12  2 x2 (d) 2 1 12 x  x2  12  2 x2  ax2 x

1 2 x  a2 x 2

2

An inertial observer sees two events E1 and E2 happening at the same location but 6 s apart in time. Another observer moving with a constant velocity v (with respect to the first one) sees the same events to be 9 s apart. The spatial distance between the events, as measured by the second observer, is approximately (a) 300 m (b) 1000 m (c) 2000 m(d) 2700 m A ball weight 100 gm, released from a height of 5m, bounces perfectly elastically off a plate. The collision time between the ball and the plate is 0.5 s. The average force on the plate is approximately (a) 3N (b) 2N (c) 5N (d) 4N The Lagrangian of a free relativistic particle (in one dimension) of mass

m is given by L  m 1  x 2 where x  dx / dt . If such a particle is acted upon by a constant force in the direction of its motion, the phase space trajectories obtained from the corresponding Hamiltnian are (a) ellipse (b) cycloids (c) hyperbolas(d) parabolas 100. A Hamiltonian system is described by the canonical coordinate q and canonical momentum p. A new coordinate Q is defined as Q  t   q  t     p  t    , where t is the time and  is a constant, this is, the new coordinate is a combination of the fold coordinate and momentum at a shifted time. The new canonical momentum P  t  can be expressed as (a) p  t     q  t   

(b) p  t     q t   

1 1 (d)  p  t     q t     p  t     q t    2 2 101. The energy of a one-dimensional system, governed by the Lagrangian (c)

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1 1 L  mx 2  kx 2n 2 2 where k and n are two positive constants, is E0. The time period of oscillation  satisfies 

1 n

1 1 n 2n E 2n (c) 0



1 n2 2 n E 2 n (d) 0



1 1 n 2n E 2n 0

 k  k 102. A light signal travels from a point A to a point B, both within a glass slab that is moving with uniform velocity (in the same direction as the light) with speed 0.3 c with respect to an external observer. If the refractive index of the slab is 1.5, then the observer will measure the speed of the signal as (a) 0.67 c (b) 0.81 c (c) 0.97 c (d) c 103. Consider a set of particles which interact by a pair potential (a)   k

(b)   k



V  ar 6 where r is the inter-particle separation and a > 0 is a constant. If a system of such particles has reached virial equilibrium, the ratio of the kinetic to the total energy of the system is 1 3 1 2 (b) (c) (d) (a) 3 3 2 4 104. A particle moves in one dimension in a potential V  x   k 2 x4   2 x2 where k and  are constants. Which of the following curves best describes the trajectories of this system in phase space?

(a)

(b)

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(c)

105. Let

(d)

 x, p  be

the generalized coordinate and momentum of a

Hamiltonian system. If new variables  X , P  are defined by X  xa sin h (p) and P = x cos h (p), where   and  are constants, then the conditions for it to be a canonical transformation, are 1 1 (a)      1 and      1 2 2 1 1 (b)     1 and     1 2 2 1 1 (c)      1 and      1 2 2 1 1 (d)     1 and     1 2 2 106. Two particles A and B move with relativistic velocities of equal magnitude v, but in opposite directions, along the x-axis of an inertial frame of reference. The magnitude of the velocity A, as seen from the rest frame of B, is (a)

2v  v2  1  2   c 

(b)

2v  v2   1  2   c 

(c) 2v

cv (d) cv

2v 1

v2 c2

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107. Which of the following figure best describes the trajectory of a particle a a moving in a repulsive central potential V  r   V  r   (a > 0 is a r r constant)

(a)

(b)

(c)

(d)

108. A Particle moves in the one-dimensional potential V  x   ax6 , where a > 0 is a constant. If the total energy of the particles is E, its time period in a periodic motion is proportional to (a) E

1

3

(b) E

1

1

2

(c) E

1

3

(d) E

2

1 1 109. A particle of mass m, kept in potential V  x    kx 2   x 4 (where k 2 4 and  are positive constants), undergoes small oscillations about an equilibrium point. The frequency of oscillations is (a)

1 2

2 m

(b)

1 2

k m

(c)

1 2

2k 1 (d) m 2

 m

xp 2 1  kx , where 110. The Hamiltonial of a one-dimensional system is H  2m 2 m and k are positive constants. The corresponding Eular-Lagrange equation for the system is

(a) mx  k  0

(b) mx  2 x  kx2  0

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(c) 2mx  mx  kx2  0 (d) mmx  2mx2  kx2  0 111. A particle of mass m, moving along the x-direction, experiments a damping force  v 2 , where  is a constant and v is its instantaneous sped. If the speed at t = 0 is v0, the speed of time t is (a) v0e



 v0t m

v0

(b)

(c)

mv0 (d) m   v0t

2v0

 v 0t  v  1  ln 1  0t  1 e t m   112. The time period of a particle of mass m. undergoes small oscillations x around x = 0, in the potential V  V0 cosh   , is L

mL2 (a)  V0

mL2 (b) 2 2V0

mL2 2mL2 (c) 2 (d) 2 V0 V0

113. The motion of a particle in one dimension is described by the 2  1   dx  Langrangian L      x 2  in suitable units. The value of the action  2   dt    along the classical path from x  0 at t  0 to x  x0 at t  t0 , is

(a)

x02

x02 1 2 1 2 (b) x0 tan t0 (c) x0 cot t0 (d) 2 2 2cos 2 t0

2sin 2 t0

114. The Hamiltonian of a classical one-dimension harmonic oscillator is 1 H  p 2  x 2 , in suitable units. The total time derivative of the 2









dynamical variable p  2 x is (a)

2 p  x (b) p  2 x (c) p  2 x

(d) x  2 p

115. A relativistic particle of mass m and charge e is moving in a uniform electric field of strength . Starting from rest at t = 0, how much time will c it take to reach the speed ? 2 3 mc mc mc 1 mc (a) (b) (c) 2 (d) 2 e e e 3 e

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