The acceleration due to gravity at a height of $(\sqrt{2}-1) \mathrm{R}$ from the surface of the Earth is

(Acceleration due to gravity on the surface of the Earth $=10 \mathrm{~ms}^{-2}$ and $R$ is radius of the Earth)

The potential energy of a satellite of mass ' $m$ ' revolving around the Earth at a height of $R_e$ from the surface of the Earth is

( $R_e=$ Radius of Earth, $\mathrm{g}=$ acceleration due to gravity)

The time period of a simple pendulum on the surface of the Earth is $T$. If the pendulum is taken to a height equal to half of the radius of the Earth, then its time period is

If the escape velocity of a body from the surface of the Earth is $11.2 \mathrm{~km} \mathrm{~s}^{-1}$, then the orbital velocity of a satellite in an orbit which is at a height equal to the radius of the Earth is

An artificial satellite is revolving around a planet of radius $R$ in a circular orbit of radius ' $a$ '. If the time period of revolution of the satellite. $T \propto a^{3 / 2} g^x R^y$, then the values of $x$ and $y$ are respectively

[ $g=$ acceleration due to gravity]

A mass of $6 \times 10^{24} \mathrm{~kg}$ is to be compressed in the form of a solid sphere such that the escape velocity from its surface is $3 \times 10^4 \mathrm{~ms}^{-1}$. The radius of the sphere is

(Universal gravitational constant $=6.66 \times 10^{-11} \mathrm{~N} \mathrm{~m}^2 \mathrm{~kg}^{-2}$ )

Two solid spheres each of radius ' $R$ ' made of same material are placed in contact with each other. If the gravitational force acting between them is $F$, then

If the angular velocity of a planet about its axis is halved, the distance of the stationary satellite of this planet from the centre of the planet becomes $2^n$ times the initial distance. Then, the value of ' $n$ ' is

An infinite number of objects each 1 kg mass are placed on the $X$-axis on both sides of $x=0$ at $\pm 1 \mathrm{~m}$, $\pm 2 \mathrm{~m}, \pm 4 \mathrm{~m}, \pm 8 \mathrm{~m} \ldots \ldots$ and so on. The magnitude of the resultant gravitational potential (in SI units) at $x=0$ is

( $G=$ Universal gravitational constant)

Statement (A) Two artificial satellites revolving in the same circular orbit have same period of revolution.

Statement (B) The orbital velocity is inversely proportional to the square root of radius of the orbit.

Statement (C) The escape velocity of the body is independent of the altitude of the point of projection.

A uniform solid sphere of radius $$R$$ produces a gravitational acceleration of $$a_0$$ on its surface. The distance of the point from the centre of the sphere where the gravitational acceleration becomes $$\frac{a_0}{4}$$ is

A projectile is thrown straight upward from the earth's surface with an initial speed $$v=\alpha v_e$$ where $$\alpha$$ is a constant and $$v_e$$ is the escape speed. The projectile travels upto a height 800 km from earth's surface, before it comes to rest. The value of the constant $$\alpha$$ is (radius of the earth $$=6400 \mathrm{~km}$$)

If the Earth stops rotating in its orbit aboutthe sun, there will be variation in the weightof our bodies at

At what depth below surface of the Earth, the acceleration due to gravity will be half of its value that at $$1600 \mathrm{~km}$$ above the surface of the Earth?

The gravitational potential energy is maximum at

A geostationary satellite is taken to a neworbit, such that its distance from centre ofthe earth is doubled. Then, find the timeperiod of this satellite in the new orbit.

The distance through which one has to digthe Earth from its surface, so as to reach thepoint where the acceleration due to gravity isreduced by 40% of that at the surface of theEarth, is (radius of Earth is 6400 km)

Infinite number of masses each of 3kg areplaced along a straight line at the distancesof 1 m, 2m, 4m, 8m, ...... from a point O onthe same line. If G is the universal gravitational constant, then the magnitude ofgravitational field intensity at O is

A particle is kept on the surface of a uniformsphere of mass 1000 kg and radius 1 m. Thework done per unit mass against thegravitational force between them is

[G = 6.67 $$\times$$ 10$$^{-11}$$ Nm$$^2$$ kg$$^{-2}$$]