Gravitation

If a body of mass 1 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 km and $g=9.8\ \mathrm{m/s^2}$)

The height in terms of radius of the earth ($R$), at which the acceleration due to gravity becomes $\frac g9$, where $g$ is acceleration due to gravity on earth's surface, is ______.

When one moves from a point 16 km below the earth's surface to a point 16 km above the earth's surface. The change in g is approximately $\alpha$ %. The value of $\alpha$ is _____. (Take radius of the earth = 6400 km.)

A body of mass $m$ is taken from the surface of earth to a height equal to twice the radius of earth ($R_e$). The increase in potential energy will be ______. ($g$ is acceleration due to gravity at the surface of earth)

Given below are two statements. One is labelled as Assertion (A) and the other is labelled as Reason (R). Assertion (A) : A simple pendulum is taken to a planet of mass and radius, 4 times and 2 times, respectively, than the Earth. The time period of the pendulum remains same on earth and the planet. Reason (R) : The mass of the pendulum remains unchanged at Earth and the other planet. In the light of the above statements, choose the correct answer from the options given below :

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 days and gravitational attraction between the satellite and the moon is neglected.

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

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.

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