GATE PH topic:lagrangian-formulation PYQs & Answers

Questions 1–10 of 14. Imported answers require independent review unless marked reviewed.

On a horizontal plane, a projectile of mass

is launched from the ground with speed

at an angle

with the horizontal. In addition to the gravitational force (

), it also experiences a drag force

, where

is its velocity and

is a constant. It hits the ground at a distance

from the point of launch with its velocity making an angle

with the horizontal, as shown schematically in the figure. Then which of the following options is correct?

Your answerChoose one option.

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A projectile of mass

is launched from the ground with the initial speed

at an angle

from the horizontal. Take the ground to be horizontal. Ignoring the drag, the magnitude of Hamilton’s action

for the particle from the beginning till it hits the ground is

. The value of

(rounded off to two decimal places) is ____

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The Lagrangian

with the generalized coordinate

is transformed to

. Consider the following statements: (i) Expression for the canonical momentum does not change. (ii) The equation of the motion does not change. Which of the following options is correct for the above statements?

Your answerChoose one option.

Both (i) and (ii) are correct.

Both (i) and (ii) are not correct.

(i) is correct and (ii) is not correct.

(i) is not correct and (ii) is correct.

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Two frames

(solid lines) and

(dashed lines) with common origin are shown in the figure below. Frame

is inertial while

is rotating about the common

-axis. There is a point mass fixed at

on the

-axis of the

frame. The magnitude of the centrifugal force and the Coriolis force experienced by the mass in the

frame is

and

, respectively. Which of the following options is correct for these forces?

Your answerChoose one option.

and

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A symmetric rigid body has moment of inertia

about its principal axes 1, 2, and 3, respectively, with

and

. It is rotating in space with no torque on it so that its angular momentum

is constant. Let

be the components of its angular velocity along the principal axes 1, 2, and 3, respectively. Which of the following quantities is/are constant during the motion of this rigid body?

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

Angle between axis 2 and

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Two 1 kg blocks are connected to two massless springs of spring constants 8 N.m

and 4 N.m

. The system is kept on a frictionless horizontal floor with one end of a spring attached to a wall (see figure below). They are performing oscillatory motion along the

-axis with the normal mode frequencies

and

(

). The ratio

(rounded off to two decimal places) is _____.

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A 15 cm long scale is held horizontally with one of its ends on the edge of a 1 m high table and the other end resting on one’s index finger. As the finger is removed (see figure below), the scale starts rotating about its end on the table. After 0.1 s, during which it has rotated by a negligibly small angle but has gained a rotational speed as it leaves the table and falls vertically towards the ground. When its centre of mass has fallen by 0.5 m, it has rotated by an angle

. The value of

in degrees (rounded off to one decimal place) is _____

(

m.s

)

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The figure shows a system of two equal masses

and three massless horizontal springs with spring constants

. Ignore gravity. The masses can move only in the horizontal direction and there is no dissipation. If

and

(all in appropriate units), the frequencies of the normal modes of the system in the same system of units are

Your answerChoose one option.

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The Lagrangian of a particle of mass

and charge

moving in a uniform magnetic field of magnitude

that points in the

direction, is given by

where

are the components of its velocity

. If

denote the conjugate momenta in the

directions and

is the Hamiltonian, which of the following option(s) is/are correct?

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

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A bead is constrained to move along a long, massless, frictionless horizontal rod parallel to the

axis. The rod itself is moving vertically upward along the

direction against gravity with a constant speed, starting from

, and remains horizontal. The conjugate momenta are denoted by

and the Hamiltonian by

. Which of the following option(s) is/are correct?

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

is the total energy of the system and is conserved

is the total energy of the system and is not conserved

is not the total energy of the system, but it is conserved

is not the total energy of the system and is not conserved

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GATE PH topic:lagrangian-formulation PYQs

How to practice topic:lagrangian-formulation

Question 16

Question 28

Question 46

Question 49

Question 55

Question 64

Question 65

Question 38

Question 51

Question 52