• ## Liquid methane spotted on Titan moon of Saturn

1. The lagrangian of a particle of mass “m” is Where V,W and ω are constants. The conserved quantities are ( )

2. The lagraThe lagrangian of a charged particle moving in the presence of electric and magnetic fields is given by, ( )
(a) ½mv² – q∅ + q V.A
(b) ½mv² + q∅ – q V.A
(c) ½mv² + q∅ + q V.A
(d) ½mv² – q(∅ + V.A)

Here , ∅ is scalar potential and A is vector potential

3. Identify the correct statements: ( )
I. The physical quantities that are required to uniquely and completely define the position (or) configuration of the system in space at an instant are
called Generalized co-ordinates.
II. The space in which each dimension corresponds to a generalized co-ordinate is always called co-ordinate space of the system.
III. The space in which each dimension corresponds to a generalized co-ordinate is sometimes called configuration space of the system.
IV. The dimension of configuration space is always equal to the number of degrees of freedom.

4. Three particles of mass ‘m’ each situated at x (t) , x (t) and x (t)1 2 3 respectively are connected by two springs of spring constant ‘K’ and unstretched
length ‘l’. The system is free to oscillate only in one dimension along the straight line joining all the three particles. The lagrangian of the system is    5. Identify the correct statements : ( )
I. The number of physical quantities that are required to uniquely and completely define the position (or) configuration of the system is called degrees
of freedom.
II. The number of degrees of freedom of the system consisting of a cylinder rolling on a stationary inclined plane is two.
III. The number of degrees of freedom of a double pendulum is two. (Assume inextensible string)
IV. The number of degrees of freedom of a conical pendulum and a spherical pendulum are two.

6. If “T” is kinetic energy, obtain the correct statements. ( ) if the forces are non-conservative if the forces are non-conservative if the forces are non-conservative if the forces are non-conservative.

7. A particle is moving under the action of a generalized potential V(q, q′) = (1+ q′)/q² The magnitude of the generalized force is( )
(a) 2(1+ q′)/q³

(b) 2(1- q’)/q³

(c)  2/q³

(d)q’/q²

8. I. The motion of a cylinder rolling on a rough inclined plane is an example of Holonomic, Sceleronomous, bilateral, conservative constrained motion.
II. The motion of a bob suspended from a vertical spring is an example of Non-Holonomic, Rheonomic, Bilateral and conservative constrained motion.
III. The motion of a uniform plane disk rolling on a 2-D plane surface is an example of Non-Holonomic, Sceleronomous, Bilateral and conservative
constrained motion.
IV. The motion of ideal gas molecules in an expanding container is an example of Non-Holonomic, Rheonomic, Bilateral and dissipative constrained
motion. ( )

9. The lagrangian of a particle of mass ‘m’ moving in one dimension is where ‘α ‘ and ‘k’ are positive constants. The
equation of motion of the particle is  10. For a closed system, if the space is homogeneous, then the conserved quantity is ( )

11. The generalized momentum for a charged particle moving in the presence of an electric and magnetic fields is given by( )

12. If “U” is the generalized potential, and if the forces are non-conservative then    13. The lagrangian for a double pendulum is given by, L =    14. For a closed system, if the time is homogeneous, then the conserved quantity is

15. Identify the true statements : ( )

16. For a closed system, if the space is isotropic, then the conserved quantity is ( )

17. The lagrangian of a free particle in spherical polar co-ordinates is given by L = m/2(r’²+r²θ’²+r²sin²θ∅’²). The quantity that is conserved is  18. The lagrangian for the rolling motion of a cylinder of mass ‘m’ and radius “R” on a stationary inclined plane of angle ‘ α’ is given by, L =  19. The lagrangian of a system is given by, L = q’²/2+ qq’ – q²/2. It describes the motion of

20. The expression for generalized force is    21. The Lagrange’s equations of motion are . I. L = T – V always ; T is K.E and V is P.E
II. qi is  Cartesian co-ordinates sometimes.
III. qi is generalized velocities.

22. If “T” is kinetic energy, obtain the correct statements. ( ) if the forces are non-conservative if the forces are non-conservative if the forces are non-conservative if the forces are non-conservative.

23. If the kinetic energy ‘T’ of a system is expressed interms of generalized co-ordinates qi and the generalized velocities . is valid

24. If a generalized co-ordinate is cyclic in Lagrangian, then ( )
(a) no explicit dependence of both co-ordinate and velocity in the lagrangian.

(b) no explicit dependence of co-ordinate in the lagrangian.
(c) the generalized momentum corresponding to cyclic co-ordinate is conserved.

(d) both (b) and (c) are correct.

25. Choose the correct statement : ( )
(a)The dimensions of Pq is always that of energy.

(b) The dimentions of Pq is always that of Angular momentum.
(c) If the generalized co-ordinate is “ Theta ”, then the generalized momentum is Angular momentum.

(d) All of the above three are correct.

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