Gravitation

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?

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 :

The escape velocity of a body on a planet 'A' is 12 kms$^{-1}$. 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 time period of a satellite revolving around earth in a given orbit is 7 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 radii of two planets A and B are in the ratio $2:3$. Their densities are $3\rho$ and $5\rho$ respectively. The ratio of their acceleration due to gravity is :

Three identical particles A, B and C of mass 100 kg each are placed in a straight line with AB = BC = 13 m. The gravitational force on a fourth particle P of the same mass is F, when placed at a distance 13 m from the particle B on the perpendicular bisector of the line AC. The value of F will be approximately :

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

An object is taken to a height above the surface of earth at a distance $\dfrac54$R from the centre of the earth. Where radius of earth, R = 6400 km. The percentage decrease in the weight of the object will be :

The percentage decrease in the weight of a rocket, when taken to a height of 32 km above the surface of earth will, be :

(Radius of earth = 6400 km)

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 km and g = 10 ms$^{-2}$)

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 :

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, then the maximum height from the centre of earth, to which the body can go, will be :

(R : radius of earth)

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 :

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 s and 6 s respectively. Then the value of $h$ is -

(consider radius of earth $R_E$ = 6400 km and g on earth 10 m/s$^2$)

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 αh (h << R$_e$) from the earth surface. The value of α will be _______.

(use R$_e$ = 6400 km)

An object of mass 1 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 m s$^{-2}$ and radius of earth = 6400 km]

The time period of X planet of our solar system is 8 years. The distance of earth from the sun is $1.5\times10^{11}$ m. The distance of the X planet from the sun will be :

The distance between two stars of masses $3M_S$ and $6M_S$ is $9R$. Here $R$ is the mean distance between the centers of the Earth and the Sun, and $M_S$ is the mass of the Sun. The two stars orbit around their common center of mass in circular orbits with period $nT$, where $T$ is the period of Earth's revolution around the Sun.

The value of $n$ is ____.

Two stars of masses m and 2m at a distance d rotate about their common centre of mass in free space. The period of revolution is :

Four identical particles of equal masses 1 kg made to move along the circumference of a circle of radius 1 m under the action of their own mutual gravitational attraction. The speed of each particle will be :