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JEE Mains | PYQ

If \( V \) is the gravitational potential due to sphere of uniform density on it’s surface, then it’s value at the center of sphere will be :

Given below are two statements:
Statement I: Rotation of the earth shows effect on the value of acceleration due to gravity \( (g) \).
Statement II: The effect of rotation of the earth on the value of ‘\( g \)’ at the equator is minimum and that at the pole is maximum.
In the light of the above statements, choose the correct answer from the options given below:

Given below are two statements: one is labelled as Assertion A and the other is labelled as Reason R.
Assertion A: Earth has atmosphere whereas moon doesn't have any atmosphere.
Reason R: The escape velocity on moon is very small as compared to that on earth.
In the light of the above statements, choose the correct answer from the options given below:

The acceleration due to gravity at height \( h \) above the earth if \( h << R \) (Radius of earth) is given by:

Two planets A and B of radii \( R \) and \( 1.5R \) have densities \( \rho \) and \( \dfrac{\rho}{2} \) respectively. The ratio of acceleration due to gravity at the surface of B to A is:

Given below are two statements :
Statements I : For a planet, if the ratio of mass of the planet to its radius increases, the escape velocity from the planet also increases.
Statement II : Escape velocity is independent of the radius of the planet.
In the light of above statements, choose the most appropriate answer from the options given below :

Assuming the earth to be a sphere of uniform mass density, the weight of a body at a depth \( d = \dfrac{R}{2} \) from the surface of earth, if its weight on the surface of earth is \( 200 \, \text{N} \), will be: (Given \( R \) = radius of earth)

At a certain depth “\( d \)” below surface of earth, value of acceleration due to gravity becomes four times that of its value at a height \( 3R \) above earth surface. Where \( R \) is Radius of earth (Take \( R = 6400 \, \text{km} \)). The depth \( d \) is equal to:

Two satellites of masses \( m \) and \( 3m \) revolve around the earth in circular orbits of radii \( r \) and \( 3r \) respectively. The ratio of orbital speeds of the satellites respectively is:

If earth has a mass nine times and radius twice to that of a planet \( P \). Then \( \frac{v_e}{3}\sqrt{x}\, \text{ms}^{-1} \) will be the minimum velocity required by a rocket to pull out of gravitational force of \( P \), where \( v_e \) is escape velocity on earth. The value of \( x \) is:

Given below are two statements:
Statements I: Acceleration due to gravity is different at different places on the surface of earth.
Statements II: Acceleration due to gravity increases as we go down below the earth’s surface.
In the light of the above statements, choose the correct answer from the options given below:

A space ship of mass \( 2 \times 10^{4} \, \text{kg} \) is launched into a circular orbit close to the earth surface. The additional velocity to be imparted to the space ship in the orbit to overcome the gravitational pull will be: (if \( g = 10 \, \text{m} / \text{s}^{2} \) and radius of earth \( = 6400 \, \text{km} \)).

\( T \) is the time period of simple pendulum on the earth’s surface. Its time period become \( x \, T \) when taken to a height \( R \) (equal to earth’s radius) above the earth’s surface. Then, the value of \( x \) will be:

A planet has double the mass of the earth. Its average density is equal to that of the earth. An object weighing \( W \) on earth will weigh on that planet:

Two satellites A and B move round the earth in the same orbit. The mass of A is twice the mass of B. The quantity which is same for the two satellites will be:

The time period of a satellite of earth is \( 24 \, \text{hours} .\) If the separation between the earth and the satellite is decreased to only fourth of the previous value, then its new time period will become.

The weight of a body at the surface of earth is \( 18 \, \text{N} \). The weight of the body at an altitude of \( 3200 \, \text{km} \) above the Earth’s surface is (given, radius of earth \( R_e = 6400 \, \text{km} \) ):

The weight of a body on the earth is \( 400 \, \text{N} \). Then weight of the body when taken to a depth half of the radius of the earth will be:

A planet having mass \( 9Me \) and radius \( 4R_e \), where \( Me \) and \( Re \) are mass and radius of earth respectively, has escape velocity in \( \text{km/s} \) given by: 
(Given escape velocity on earth \( V_e = 11.2 \times 10^{3} \, \text{m/s} \))

The ratio of escape velocity of a planet to the escape velocity of earth will be: Given: Mass of the planet is \( 16 \) times mass of earth and radius of the planet is \( 4 \) times the radius of earth.

The time period of a satellite, revolving above earth’s surface at a height equal to \( R \), will be:
(Given: \( g = \pi^2 \, \text{m/s}^2 \), \( R \) = radius of earth).

Two identical particles each of mass '\( m \)' go round a circle of radius \( a \) under the action of their mutual gravitational attraction. The angular speed of each particle will be:

A body is released from a height equal to the radius \( (R) \) of the earth. The velocity of the body when it strikes the surface of the earth will be:
(Given \( g \) = acceleration due to gravity on the earth.)

Given below are two statements:
Statement-I: If \( E \) be the total energy of a satellite moving around the earth, then its potential energy will be \( \dfrac{E}{2} \).
Statement-II: The kinetic energy of a satellite revolving in an orbit is equal to the half the magnitude of total energy \( E \).
In the light of the above statements, choose the most appropriate answer from the options given below.

Every planet revolves around the sun in an elliptical orbit:
A. The force acting on a planet is inversely proportional to square of distance from sun.
B. Force acting on planet is inversely proportional to product of the masses of the planet and the sun.
C. The centripetal force acting on the planet is directed away from the sun.
D. The square of time period of revolution of planet around sun is directly proportional to cube of semi-major axis of elliptical orbit.
Choose the correct answer from the options given below:

Choose the incorrect statement from the following:

An object is allowed to fall from a height \( R \) above the earth, where \( R \) is the radius of earth. Its velocity when it strikes the earth's surface, ignoring air resistance, will be:

Given below are two statements: one is labelled as Assertion A and the other is labelled as Reason R.
Assertion A: A pendulum clock when taken to Mount Everest becomes fast.
Reason R: The value of \( g \) (acceleration due to gravity) is less at Mount Everest than its value on the surface of earth.
In the light of the above statements, choose the most appropriate answer from the options given below:

If the gravitation field in the space is given as \( \left( -\frac{K}{r^{2}} \right) \). Taking the reference point to be at \( r = 2 \, \text{cm} \) with gravitational potential \( V = 10 \, \text{J/kg} \). Find the gravitational potential at \( r = 3 \, \text{cm} \) in SI unit. (Given, that \( K = 6 \, \text{cm/kg} \))