GATE PH topic:classical-mechanics PYQs & Answers — Page 3

Questions 21–30 of 51. Imported answers require independent review unless marked reviewed.

A uniform block of mass

slides on a smooth horizontal bar. Another mass

is connected to it by an inextensible string of length

of negligible mass, and is constrained to oscillate in the X-Y plane only. Neglect the sizes of the masses. The number of degrees of freedom of the system is two and the generalized coordinates are chosen as

and

, as shown in the figure.

and

are the generalized momenta corresponding to

and

, respectively, then the correct option(s) is(are)

Your answerSelect all correct options. You can choose more than one.

is conserved

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The time derivative of a differentiable function

is added to a Lagrangian

such that

where

are the generalized coordinates, generalized velocities and time, respectively. Let

be the generalized momentum and

the Hamiltonian associated with

. If

and

are those associated with

, then the correct option(s) is(are)

Your answerSelect all correct options. You can choose more than one.

Both

and

satisfy the Euler-Lagrange’s equations of motion

is conserved, then

is necessarily conserved

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A particle is moving in a central force field given by

, where

is the unit vector pointing away from the center of the field. The potential energy of the particle is given by

Your answerChoose one option.

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A small disc is suspended by a fiber such that it is free to rotate about the fiber axis (see figure). For small angular deflections, the Hamiltonian for the disc is given by

where

is the moment of inertia and

is the restoring torque per unit deflection. The disc is subjected to angular deflections (

) due to thermal collisions from the surrounding gas at temperature

and

is the momentum conjugate to

. The average and the root-mean-square angular deflection,

and

, respectively are

Your answerChoose one option.

and

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Let

be the momentum conjugate to the generalized coordinate

. If the transformation

is canonical, then

_____.

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If a particle is moving along a sinusoidal curve, the number of degrees of freedom of the particle is ___.

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Consider the Lagrangian

, where

and

are constants. If

and

are the momenta conjugate to the coordinates

and

respectively, then the Hamiltonian is

Your answerChoose one option.

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Which one of the following matrices does NOT represent a proper rotation in a plane?

Your answerChoose one option.

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A uniform magnetic field

exists in an inertial frame

. A perfect conducting sphere moves with a constant velocity

with respect to this inertial frame. The rest frame of the sphere is

(see figure). The electric and magnetic fields in

and

are related as

The induced surface charge density on the sphere (to the lowest order in

) in the frame

Your answerChoose one option.

maximum along

uniform over the sphere

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The potential energy of a particle of mass

is given by

The angular frequency of small oscillations of the particle about

Your answerChoose one option.

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GATE PH topic:classical-mechanics PYQs

Question 44

Question 46

Question 10

Question 15

Question 23

Question 25

Question 31

Question 32

Question 33

Question 39