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QUESTION: 1

Consider a satellite in a circular orbit around the earth. Then properties of the satellite depend on the radius of the orbit r as follows: speed time period angular momentum and kinetic energy

Solution:

For a satellite in a circular orbit

QUESTION: 2

The phase space diagram for free particle motion

Solution:

QUESTION: 3

For the Lagrangian

find equation of motion

Solution:

**Correct Answer :- d**

**Explanation :** L = (x^{2}+ x,2y^{2})/2 - kx,2y^{2}/2

d/dt(dL/dx) - dL/dx = 0

=> d/dt(x) - xy^{2} + kxy^{2} = 0

=> x = xy^{2} - kxy^{2}

d/dt(dL/dy) - dL/dy = 0

d/dt(x^{2}y) + kx^{2}y = 0

=> x^{2}y + 2xy + kx^{2}y = 0

xy = -2xy - kxy

QUESTION: 4

The value of is

Solution:

From the Poisson's bracket

QUESTION: 5

A bullet is fired horizontally in the north direction with a speed u at θ latitude. It hits a target 1 meter away. Calculate the Coriolis acceleration.

Solution:

*Answer can only contain numeric values

QUESTION: 6

If a spring of spring constant k is stretched by x0, the work done is W0. Now, a second spring of spring constant 3k is stretched by x_{0}/3. The work done is The value of a is _____ (Correct upto one decimal place).

Solution:

For first spring,

∴

*Answer can only contain numeric values

QUESTION: 7

The moment of inertia of pairs of solid sphere each having mass m radius r kept in contact about a tangent passing through the point of contact is βmr^{2}. The value of β is ______ . (Upto one decimal place)

Solution:

*Answer can only contain numeric values

QUESTION: 8

The moment of inertia tensor of a rigid body is given by

Moment of inertia about an axis is _______.

Solution:

Moment of inertia about

*Answer can only contain numeric values

QUESTION: 9

A uniform chain of length L and mass M is lying on a smooth table and one third of its length is hanging vertically down over the edge of the table. The work required to pull the hanging part on the table is The value of α is _________

Solution:

QUESTION: 10

The Hamiltonian of a system is given by

It describes the motion of

Solution:

Equations of motion are

*Answer can only contain numeric values

QUESTION: 11

The muon is an unstable particle that spontaneously decay into an electron and two neutrinos if the number of muons at t = 0 is N_{0}, is the mean life time of muon. Suppose muons move at speed 0.95c, the number of muons remain after travelling a distance of 3.0 km is _______ (in terms of N_{0}) (upto three decimal places)

Solution:

S frome attached to the earth and S'frome to be rest frome of muon

*Answer can only contain numeric values

QUESTION: 12

A comet in an orbit about the sun has a velocity 10km/sec at aphelion and 80km/sec at perihelion if the earth's velocity in a circular orbit is 30 km/sec and the radius of its orbit is 1.5 x 10^{8}km, the aphelion distance R_{a} for comet is ____ x 10^{8}km. (Nearest integer)

Solution:

Let v be velocity of the earth, R the radius of the earth

applying conservation of energy and the angular momentum of comet

QUESTION: 13

Find Lagrangian of the pendulum of mass m and length R attached to a spring, the other end of which is fixed at the bottom as shown in the figure. The length of the undeformed spring is l

Solution:

Lagrangian, L = T - V

QUESTION: 14

The kinetic energy of a particle in terms of coordinate r and q = sin θ, where r & θ are polar coordinate is

Solution:

Kinetic energy of particle

QUESTION: 15

Consider the At wood's machine. Let x and y be the vertical position of the middle mass and right mass, respectively with upward taken to be positive. The conjugate momenta, P_{x} and P_{y} respectively, are

Solution:

If the right two masses move up by x and y then the left mass move down by (x+y)/2

∴ Lagrangian

QUESTION: 16

If I_{ij} is the tensor of inertia of a solid sphere, x^{2}+y^{2}+z^{2} = a^{2}, of mass M in the first octant, then

Solution:

In first quadrant, dm = pdx dy dz

QUESTION: 17

A train with proper length L moves with speed 5c/13 with respect to the ground. A ball is thrown from the back of the train to the front. The speed of the ball with respect to the train is c/3. As viewed by someone on the ground how much time does the ball spend in the air and how far does it travel.

Solution:

QUESTION: 18

The force of interaction between a particle of mass m1 and a second particle of mass m_{2} separated by a distance r is given by an attractive gravitational force and a repulsive force that is proportional to r-3, with probability constant C,

the angular frequency of small oscillation about the stable equilibrium position is

Solution:

*Answer can only contain numeric values

QUESTION: 19

A disk of mass m and radius R is attached to a spring of constant K as shown in the figure. The disk rolls and forth without slipping. The angular frequency of the motion of the disk is The value of α is ______.

Solution:

*Answer can only contain numeric values

QUESTION: 20

The particle of mass m and angular momentum L^{2} = 10mV_{0}R^{2} moves in a potential

The radius of the stable circular orbit is _____R.

Solution:

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