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A metal wire of uniform mass density having length \( L \) and mass \( M \) is bent to form a semicircular arc and a particle of mass \( m \) is placed at the centre of the arc. The gravitational force on the particle by the wire is:

A body of mass \( m \) is projected with velocity \( \lambda v_{e} \) in vertically upward direction from the surface of the earth into space. It is given that \( v_{e} \) is escape velocity and \( \lambda < 1 \). If air resistance is considered to be negligible, the maximum height from the centre of earth, to which the body can go, will be: (\( R \): radius of earth)

The weight of a body on the surface of the earth is \( 100 \, \text{N} \). The gravitational force on it when taken at a height, from the surface of earth, equal to one-fourth the radius of the earth is:

A body of mass \( m \) is taken from earth surface to the height \( h \) equal to twice the radius of earth \( (R_e) \), the increase in potential energy will be: \( (g = \) acceleration due to gravity on the surface of Earth\() \)

An astronaut takes a ball of mass \( m \) from earth to space. He throws the ball into a circular orbit about earth at an altitude of \( 318.5 \, \text{km} \). From earth's surface to the orbit, the change in total mechanical energy of the ball is \( x \times \frac{G M_{e} m}{21 R_{e}} \).
The value of \( x \) is: (take \( R_{e} = 6370 \, \text{km} \))

The orbital angular momentum of a satellite is \( L \), when it is revolving in a circular orbit at height \( h \) from earth surface. If the distance of satellite from the earth centre is increased by eight times to its initial value, then the new angular momentum will be:

Acceleration due to gravity on the surface of earth is \( 'g' \). If the diameter of earth is reduced to one third of its original value and mass remains unchanged, then the acceleration due to gravity on the surface of the earth is __________ \( g \).

The approximate height from the surface of earth at which the weight of the body becomes \( \frac{1}{3} \) of its weight on the surface of earth is: [Radius of earth \( R = 6400 \, \text{km} \) and \( \sqrt{3} = 1.732 \)]

The escape velocities of two planets \( A \) and \( B \) are in the ratio \( 1:2 \). If the ratio of their radii respectively is \( 1:3 \), then the ratio of acceleration due to gravity of planet \( A \) to the acceleration of gravity of planet \( B \) will be:

Two satellites \( S_1 \) and \( S_2 \) are revolving in circular orbits around a planet with radius \( R_1 = 3200 \, \text{km} \) and \( R_2 = 800 \, \text{km} \) respectively. The ratio of speed of satellite \( S_1 \) to the speed of satellite \( S_2 \) in their respective orbits would be \( \frac{1}{x} \) where \( x = \) ______________.

A body weight \( W \), is projected vertically upwards from earths surface to reach a height above the earth which is equal to nine times the radius of earth. The weight of the body at that height will be:

If a body of mass \( 1 \, \text{kg} \) falls on the earth from infinity, it attains velocity \( (v) \) and kinetic energy \( (k) \) on reaching the surface of earth. The values of \( v \) and \( k \) respectively are _________.
(Take radius of earth to be \( 6400 \, \text{km} \) and \( g = 9.8 \, \text{m} / \text{s}^2 \) )

Earth has mass \( 8 \) times and radius \( 2 \) times that of a planet. If the escape from the earth is \( 11.2 \,\text{km/s} \), the escape velocity in km/s from the planet will be:

A particle is released from height \( S \) above the surface of the earth. At certain height its kinetic energy is three times its potential energy. The height from the surface of the earth and the speed of the particle at that instant are respectively.

Two particles of equal mass \( m \) move in a circle of radius \( r \) under the action of their mutual gravitational attraction. The speed of each particle will be:

A body is projected vertically upwards from the surface of earth with a velocity equal to one third of escape velocity. The maximum height attained by the body will be: (Take radius of earth \( = 6400 \, \text{km} \) and \( g = 10 \, \text{ms}^{-2} \))

The time period of a satellite in a circular orbit of radius \( R \) is \( T \). The period of another satellite in a circular orbit of radius \( 9R \) is:

A planet \( P_{1} \) is moving around the star of mass \( 2M \) in the orbit of radius \( R \). Another planet \( P_{2} \) is moving around another star of mass \( 4M \) in a orbit of radius \( 2R \). Ratio of time periods of revolution of \( P_{2} \) and \( P_{1} \) is ______.

Given below are two statements: one is labelled as Assertion (A) and the other is labelled as Reason (R).
Assertion (A): The angular speed of the moon in its orbit about the earth is more than the angular speed of the earth in its orbit about the sun.
Reason (R): The moon takes less time to move around the earth than the time taken by the earth to move around the sun.
In the light of the above statements, choose the correct answer from the options given below:

The escape velocity from a spherical planet \( A \) is \( 10 \, \text{km/s} \). The escape velocity from another planet \( B \) whose density and radius are \( 10\% \) of those of planet \( A \), is _______ \( \text{m/s} \).

The masses and radii of the earth and moon are \( (M_1, R_1) \) and \( (M_2, R_2) \) respectively. Their centres are at a distance \( r \) apart. Find the minimum escape velocity for a particle of mass \( m \) to be projected from the middle of these two masses:

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:

Consider a binary star system of star A and star B with masses \( m_{A} \) and \( m_{B} \) revolving in a circular orbit of radii \( r_{A} \) and \( r_{B} \) respectively. If \( T_{A} \) and \( T_{B} \) are the time period of star A and star B, respectively, then:

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 height of any point \( P \) above the surface of earth is equal to diameter of earth. The value of acceleration due to gravity at point \( P \) will be: (Given \( g \) = acceleration due to gravity at the surface of earth).

Two satellites revolved around a plane in coplanar circular orbits in anticlockwise direction. Their period of revolutions are \( 1 \) hour and \( 8 \) hours respectively. The radius of the orbit of nearer satellite is \( 2 \times 10^3 \, \text{km} \). The angular speed of the farther satellite as observed from the nearer satellite at the instant when both the satellites are closet if \( \frac{\pi}{x} \, \text{rad} \, \text{h}^{-1} \) where \( x \) is:

Escape velocity of a body from earth is \\( 11.2 \\, \\text{km/s} \\). If the radius of a planet be one-third the radius of earth and mass be one-sixth that of earth, the escape velocity from the planet is:

The escape velocity of a body on a planet ‘A’ is \( 12 \, \text{km s}^{-1} \). The escape velocity of the body on another planet ‘B’, whose density is four times and radius is half of the planet ‘A’, is:

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

Two planets \( A \) and \( B \) having masses \( m_{1} \) and \( m_{2} \) move around the sun in circular orbits of \( r_{1} \) and \( r_{2} \) radii respectively. If angular momentum of \( A \) is \( L \) and that of \( B \) is \( 3L \), the ratio of time period \( \left( \frac{T_A}{T_B} \right) \) is:

If \( R_E \) be the radius of Earth, then the ratio between the acceleration due to gravity at a depth \( r \) below and a height \( r \) above the earth surface is: (Given: \( r < R_E \))

The height in terms of radius of the earth \( (R) \), at which the acceleration due to gravity becomes \( \frac{g}{9} \), where \( g \) is acceleration due to gravity on earth's surface, is

A satellite is launched into a circular orbit of radius \(' R' \) around the earth. A second satellite is launched into an orbit of radius \( 1.03R \). The time period of revolution of the second satellite is larger than the first one approximately by:

Four spheres each of mass \( m \) form a square of side \( d \) (as shown in figure). A fifth sphere of mass \( M \) is situated at the centre of square. The total gravitational potential energy of the system is:

To project a body of mass \( m \) from earth’s surface to infinity, the required kinetic energy is (assume, the radius of earth is \( R_E \), \( g \) = acceleration due to gravity on the surface of earth):

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:
Statement I: Acceleration due to earth’s gravity decreases as you ‘up’ or ‘down’ from earth’s surface.
Statement II: Acceleration due to earth’s gravity is same at a height \( h \) and depth \( d \) from earth’s surface, if \( h = d \).
In the light of above statements, choose the most appropriate answer form the options given below:

Three identical spheres of mass \( m \), are placed at the vertices of an equilateral triangle of length \( a \). When released, they interact only through gravitational force and collide after a time \( T = 4 \) seconds. If the sides of the triangle are increased to length \( 2a \) and also the masses of the spheres are made \( 2m \), then they will collide after __________ seconds.

A small point of mass \( m \) is placed at a distance \( 2R \) from the centre \( 'O' \) of a big uniform solid sphere of mass \( M \) and radius \( R \). The gravitational force on \( 'm' \) due to \( M \) is \( F_1 \). A spherical part of radius \( \frac{R}{3} \) is removed from the big sphere as shown in the figure and the gravitational force on \( m \) due to remaining part of \( M \) is found to be \( F_2 \). The value of ratio \( F_1 : F_2 \) is:

If the radius of earth shrinks by \( 2\% \) while its mass remains same. The acceleration due to gravity on the earth’s surface will approximately.

Two satellites \( A \) and \( B \), having masses in the ratio \( 4:3 \), are revolving in circular orbits of radii \( 3r \) and \( 4r \) respectively around the earth. The ratio of total mechanical energy of \( A \) to \( B \) is:

Given below are two statements: One is labelled as Assertion A and the other is labelled as Reason R.
Assertion A: If we move from poles to equator, the direction of acceleration due to gravity of earth always points towards the center of earth without any variation in its magnitude.
Reason R: At equator, the direction of acceleration due to the gravity is towards the center of earth.
In the light of above statements, choose the correct answer from the options given below.

A satellite revolving around a planet in stationary orbit has time period \( 6 \) hours. The mass of planet is one-fourth the mass of earth. The radius orbit of planet is: (Given = Radius of geo-stationary orbit for earth is \( 42 \times 10^4 \, \text{km} \))

A light planet is revolving around a massive star in a circular orbit of radius \( R \) with a period of revolution \( T \). If the force of attraction between planet and star is proportional to \( R^{-3/2} \) then chose the correct option:

A \( 90 \, \text{kg} \) body placed at \( 2R \) distance from surface of earth experience gravitational pull of:
\( (R = \) Radius of earth, \( g = 10 \, \text{m s}^{-2} ) \)

Two planets \( A \) and \( B \) of equal mass are having their period of revolutions \( T_{A} \) and \( T_{B} \) such that \( T_{A}=2T_{B} \). These planets are revolving in the circular orbits of radii \( r_{A} \) and \( r_{B} \) respectively. Which out of the following would be the correct relationship of their orbits?

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)

If a satellite orbiting the Earth is \( 9 \) times closer to the Earth than the Moon, what is the time period of rotation of the satellite? Given rotational time period of Moon \( = 27 \, \text{days} \) and gravitational attraction between the satellite and the moon is neglected.

The mass of the moon is \( \dfrac{1}{144} \) times the mass of a planet and its diameter is \( \dfrac{1}{16} \) times the diameter of a planet. If the escape velocity on the planet is \( v \), the escape velocity on the moon will be:

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:

The time period of a satellite revolving around earth in a given orbit is \( 7 \ \text{hours} \). If the radius of orbit is increased to three times its previous value, then approximate new time period of the satellite will be:

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:

The percentage decrease in the weight of a rocket, when taken to a height of \( 32 \, \text{km} \) above the surface of earth will be:
(Radius of earth = \( 6400 \, \text{km} \))

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 \( = 6400 \, \text{km} \)).

Two planets, \(A\) and \(B\) are orbiting a common star in circular orbits of radii \(R_A\) and \(R_B\), respectively, with \(R_B = 2R_A\). The planet \(B\) is \(4\sqrt{2}\) times more massive than planet \(A\). The ratio \( \left( \frac{L_B}{L_A} \right) \) of angular momentum \( (L_B) \) of planet \(B\) to that of planet \(A\) \( (L_A) \) is closest to integer _______.

The distance between Sun and Earth is \( R \). The duration of year if the distance between Sun and Earth becomes \( 3R \) will be:

\( T \) 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:

Correct formula for height of a satellite from earths surface is:

Four particles each of mass \( M \), move along a circle of radius \( R \) under the action of their mutual gravitational attraction as shown in figure. The speed of each particle is:

A satellite of mass \( \dfrac{M}{2} \) is revolving around earth in a circular orbit at a height of \( \dfrac{R}{3} \) from earth surface. The angular momentum of the satellite is \( M \sqrt{\dfrac{GMR}{x}} \). The value of \( x \) is __________, where \( M \) and \( R \) are the mass and radius of earth, respectively. \( (G \) is the gravitational constant).

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:

Given below are two statements: one is labelled as Assertion (A) and other is labelled as Reason (R).
Assertion (A): The radius vector from the Sun to a planet sweeps out equal areas in equal intervals of time and thus areal velocity of planet is constant.
Reason (R): For a central force field the angular momentum is a constant. In the light of the above statements, choose the most appropriate answer from the option given below:

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:

If the acceleration due to gravity experienced by a point mass at a height \( h \) above the surface of earth is same as that of the acceleration due to gravity at a depth \( \alpha h \) (\( h \ll R_e \)) from the earth surface. The value of \( \alpha \) will be _________. (use \( R_e = 6400 \, \text{km} \))

A body is projected vertically upwards from the surface of earth with a velocity sufficient enough to carry it to infinity. The time taken by it to reach height \( h \) is _______ \( \text{s} \).

A satellite of \( 10^3 \, \text{kg} \) mass is revolving in circular orbit of radius \( 2R \). If \( \frac{10^4 R}{6} \, \text{J} \) energy is supplied to the satellite, it would revolve in a new circular orbit of radius: (use \( g = 10 \, \text{m/s}^2 \), \( R \) = radius of earth)

A person whose mass is \( 100 \, \text{kg} \) travels Earth to Mars in a spaceship. Neglect all other objects in sky and take acceleration due to gravity on the surface of the Earth and Mars as \( 10 \, \text{m/s}^2 \) and \( 4 \, \text{m/s}^2 \) respectively. Identify from the below figure, the curve that fits best for the weight of the passenger as a function of time:
 

If one wants to remove all the mass of the earth to infinity in order to break it up completely. The amount of energy that needs to be supplied will be \( \dfrac{x}{5} \dfrac{GM^{2}}{R} \) where \( x \) is ______
 (Round off to the Nearest Integer)
\( M \) is the mass of earth, \( R \) is the radius of earth, \( G \) is the gravitational constant

A satellite is launched into a circular orbit of radius \( R \) around earth, while a second satellite is launched into a circular orbit of radius \( 1.02R \). The percentage difference in the time periods of the two satellites is:

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 acceleration due to gravity on the surface of earth is \( g \). If the diameter of earth reduces to half of its original value and mass remains constant, then acceleration due to gravity on the surface of earth would be :

Inside a uniform spherical shell:
(a) The gravitational field is zero.
(b) The gravitational potential is zero.
(c) The gravitational field is same everywhere.
(d) The gravitation potential is same everywhere.
(e) All of the above
Choose the most appropriate answer from the options given below:

If the angular velocity of earth’s spin is increased such that the bodies at the equator start floating, the duration of the day would be approximately:
[Take \( g = 10 \, \text{m s}^{-2} \), the radius of earth, \( R = 6400 \times 10^3 \, \text{m} \). Take \( \pi = 3.14 \)]

A body of mass \( (2M) \) splits into four masses \( (m, \, M - m, \, m, \, M - m) \), which are rearranged to form a square as shown in the figure. The ratio of \( \dfrac{M}{m} \) for which, the gravitational potential energy of the system becomes maximum is \( x : 1 \). The value of \( x \) is ______.

Assuming the earth to be a sphere of uniform mass density, a body weighed \( 300 \, \text{N} \) on the surface of earth. How much it would weigh at \( R/4 \) depth under surface of earth?

Match List I with List II:

(Where \( a \) = radius of planet orbit, \( r \) = radius of planet, \( M \) = mass of Sun, \( m \) = mass of planet)
Choose the correct answer from the options given below:

Four identical particles of mass \( m \) are kept at the four corners of a square. If the gravitational force exerted on one of the masses by the other masses is \( \left( \frac{2\sqrt{2}+1}{32} \right)\frac{G m^2}{L^2} \), the length of the sides of the square is:

Two objects of equal masses placed at certain distance from each other attracts each other with a force of \( F \). If one-third mass of one object is transferred to the other object, then the new force will be:

Suppose two plants (spherical in shape) of radii \( R \) and \( 2R \), but mass \( M \) and \( 9M \) respectively have a centre to centre separation \( 8R \) as shown in the figure. A satellite of mass '\( m \)' is projected from the surface of the planet of mass \( 'M' \) directly towards the centre of the second planet. The minimum speed \( 'v' \) required for the satellite to reach the surface of the second planet is \( \sqrt{\dfrac{a}{7} \, \dfrac{GM}{7 \, R}} \) then the value of \( 'a' \) is ________.
[Given : The two planets are fixed in their position]

The radius in kilometer to which the present radius of earth \( (R = 6400 \, \text{km}) \) to be compressed so that the escape velocity is increased \( 10 \) times is ______.

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.

An object is taken to a height above the surface of earth at a distance \( \dfrac{5}{4} R \) from the centre of the earth. Where radius of earth, \( R = 6400 \, \text{km} \). The percentage decrease in the weight of the object will be:

An object is kept at rest at a distance of \( 3R \) above the earth's surface where \( R \) is earth's radius. The minimum speed with which it must be projected so that it does not return to earth is:
(Assume \( M \) = mass of earth, \( G \) = Universal gravitational constant)

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:

The gravitational potential at a point above the surface of earth is \( -5.12 \times 10^{7} \, \text{J} / \text{kg} \) and the acceleration due to gravity at that point is \( 6.4 \, \text{m} / \text{s}^{2} \). Assume that the mean radius of earth to be \( 6400 \, \text{km} \). The height of this point above the earth’s surface is:

At what distance above and below the surface of the earth a body will have same weight. (take radius of earth as \( R \))

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.)

A body weighs \( 49 \, \text{N} \) on a spring balance at the north pole. What will be its weight recorded on the same weighing machine, if it is shifted to the equator?
[Use \(g= \frac{GM}{R^2} = 9.8 \, \text{m s}^{-2} \) and radius of earth, \( R = 6400 \, \text{km} \)]

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.

The length of a seconds pendulum at a height \( h = 2R \) from earth surface will be:
(Given \( R \) = Radius of earth and acceleration due to gravity at the surface of earth, \( g = \pi^2 \, \text{ms}^{-2} \))

Three masses \( 200 \,\text{kg} \), \( 300 \,\text{kg} \) and \( 400 \,\text{kg} \) are placed at the vertices of an equilateral triangle with sides \( 20 \,\text{m} \). They are rearranged on the vertices of a bigger triangle of side \( 25 \,\text{m} \) and with the same centre. The work done in this process ________ \( \text{J} \).
\( \left(\text{Gravitational constant } G = 6.7 \times 10^{-11} \, \text{N m}^{2}/\text{kg}^{2}\right) \)

A satellite of mass \( 1000 \, \text{kg} \) is launched to revolve around the earth in an orbit at a height of \( 270 \, \text{km} \) from the earth's surface. Kinetic energy of the satellite in this orbit is __________ \( \times 10^{10} \, \text{J} \).
(Mass of earth \( = 6 \times 10^{24} \, \text{kg} \) , Radius of earth \( = 6.4 \times 10^{6} \, \text{m} \) , Gravitational constant \( = 6.67 \times 10^{-11} \, \text{N m}^{2} \text{kg}^{-2} \) )

Given below are two statements: one is labelled as Assertion A and the other is labelled as Reason R.
Assertion A: The kinetic energy needed to project a body of mass \( m \) from earth surface to infinity is \( \dfrac{1}{2}mgR \), where \( R \) is the radius of earth.
Reason R: The maximum potential energy of a body is zero when it is projected to infinity from earth surface.
In the light of the above statements, choose the correct answer from the options given below:

A mass of \( 50\,\text{kg} \) is placed at the centre of a uniform spherical shell of mass \( 100\,\text{kg} \) and radius \( 50\,\text{m} \). If the gravitational potential at a point, \( 25\,\text{m} \) from the centre is \( V \) kg/m. The value of \( V \) is:

Given below are two statements:
Statement I: The law of gravitation holds good for any pair of bodies in the universe.
Statement II: The weight of any person becomes zero when the person is at the centre of the earth.
In the light of the above statements, choose the correct 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:

A planet takes \( 200 \, \text{days} \) to complete one revolution around the Sun. If the distance of the planet from Sun is reduced to one fourth of the original distance, how many days will it take to complete one revolution:

Choose the incorrect statement from the following:

Two identical particles of mass \( 1 \, \text{kg} \) each go round a circle of radius \( R \), under the action of their mutual gravitational attraction. The angular speed of each particle is:

A geostationary satellite is orbiting around an arbitary planet ‘\( P \)’ at a height of \( 11R \) above the surface of ‘\( P \)’, \( R \) being the radius of ‘\( P \)’. The time period of another satellite in hours at a height of \( 2R \) from the surface of ‘\( P \)’ is ________. ‘\( P \)’ has the time period of \( 24 \) hours.

Assume there are two identical simple pendulum clocks. Clock-\( 1 \) is placed on the earth and Clock-\( 2 \) is placed on a space station located at a height \( h \) above the earth surface. Clock-\( 1 \) and Clock-\( 2 \) operate at time periods \( 4\,\text{s} \) and \( 6\,\text{s} \) respectively. Then value of \( h \) is: \( \big(\text{consider radius of earth } R_{E}=6400\,\text{km} \text{ and } g \text{ on earth } 10\,\text{m}/\text{s}^{2}\big) \)

Two satellite \( A \) and \( B \) go round a planet in circular orbits having radii \( 4R \) and \( R \) respectively. If the speed of \( A \) is \( 3v \), the speed of \( B \) will be;

The minimum and maximum distances of a planet revolving around the Sun are \( x_1 \) and \( x_2 \). If the minimum speed of the planet on its trajectory is \( v_0 \) then its maximum speed will be

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:

The variation of acceleration due to gravity \( (g) \) with distance \( (r) \) from the center of the earth is correctly represented by: (Given \( R \) = radius of earth)

An object of mass \( 1\,\text{kg} \) is taken to a height from the surface of earth which is equal to three times the radius of earth. The gain in potential energy of the object will be
[If, \( g=10\,\text{m s}^{-2} \) and radius of earth \( =6400\,\text{km} \)]

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:

Consider a planet in some solar system which has a mass double the mass of earth and density equal to the average density of earth. If the weight of an object on earth is \( W \), the weight of the same object on that planet will be:

Initially a satellite of \( 100 \, \text{kg} \) is in a circular orbit of radius \( 1.5 R_E \). This satellite can be moved to a circular orbit of radius \( 3 R_E \) by supplying \( \alpha \times 10^{6} \, \text{J} \) of energy. The value of \( \alpha \) is ______________.
\( \big(\text{Take Radius of Earth } R_E = 6 \times 10^{6} \, \text{m} \text{ and } g = 10 \, \text{m}/\text{s}^{2} \big) \)

Net gravitational force at the center of a square is found to be \( F_1 \) when four particles having mass \( M, 2M, 3M \) and \( 4M \) are placed at the four corners of the square as shown in figure and it is \( F_2 \) when the positions of \( 3M \) and \( 4M \) are interchanged. The ratio \( \frac{F_1}{F_2} \) is \( \frac{\alpha}{\sqrt{5}} \). The value of \( \alpha \) is __________.

Three identical particles \( AB=BC=13\,\text{m} \). The gravitational force on a fourth particle \( P \) of the same mass is \( F \), when placed at a distance \( 13\,\text{m} \) from the particle \( B \) on the perpendicular bisector of the line \( AC \). The value of \( F \) will be approximately:

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 \( K = 6 \, \text{cm/kg} \))