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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:

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.

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:

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

The distance between Sun and Earth is \( R \). The duration of year if the distance between Sun and Earth becomes \( 3R \) 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 \( R_e = 6400 \, \text{km} \))

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:

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:

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, \( g = \pi^2 \, \text{ms}^{-2} \))

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.

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: \( \big(\text{consider radius of earth } R_{E}=6400\,\text{km} \text{ and } g \text{ on earth } 10\,\text{m}/\text{s}^{2}\big) \)

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

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: