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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:

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:

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:

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:

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: \( r < R_E \))

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 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 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:

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 ______.

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 ______.

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

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:

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.

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

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:

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JEE Main 2021 27th July Evening Shift

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: