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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:

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 \( R_{e} = 6370 \, \text{km} \))

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:

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:

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:

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):

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, \( g = 10 \, \text{m s}^{-2} ) \)

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:

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

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)

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 :

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:

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 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:

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;

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JEE Main 2024 4th April Evening Shift

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