Gravitation

A particle of mass $m$, and angular momentum $\ell$ is moving in a circular orbit of radius $r_0$ under the influence of an attractive force $\vec F(r)=-\dfrac{k}{r^2}\hat r$. Keeping its angular momentum unchanged, the particle is displaced radially by a small distance $\delta r\ll r_0$, due to which its radial distance varies periodically. The corresponding time period is:

A planet ($P_1$) is moving around the star of mass $2M$ in the orbit of radius $R$. Another planet ($P_2$) is moving around another star of mass $4M$ in a orbit of radius $2R$. Ratio of time periods of revolution of $P_2$ and $P_1$ is ________.

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 $g=9.8$ m/s²)

The height in terms of radius of the earth ($R$), at which the acceleration due to gravity becomes $\dfrac g9$, 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)

Consider a star of mass $m_2$ kg revolving in a circular orbit around another star of mass $m_1$ kg with $m_1 \gg m_2$. The heavier star slowly acquires mass from the lighter star at a constant rate of $\gamma$ kg/s. In this transfer process, there is no other loss of mass. If the separation between the centers of the stars is $r$, then its relative rate of change $\dfrac{1}{r}\dfrac{dr}{dt}$ (in s$^{-1}$) is given by:

A geostationary satellite above the equator is orbiting around the earth at a fixed distance $r_1$ from the center of the earth. A second satellite is orbiting in the equatorial plane in the opposite direction to the earth's rotation, at a distance $r_2$ from the center of the earth, such that $r_1 = 1.21\,r_2$. The time period of the second satellite as measured from the geostationary satellite is $\dfrac{24}{p}$ hours. The value of $p$ is _____.

A particle of mass $m$ is under the influence of the gravitational field of a body of mass $M\ (\gg m)$. The particle is moving in a circular orbit of radius $r_0$ with time period $T_0$ around the mass $M$. Then, the particle is subjected to an additional central force, corresponding to the potential energy $V_c(r) = m\alpha/r^3$, where $\alpha$ is a positive constant of suitable dimensions and $r$ is the distance from the center of the orbit. If the particle moves in the same circular orbit of radius $r_0$ in the combined gravitational potential due to $M$ and $V_c(r)$, but with a new time period $T_1$, then $(T_1^2 - T_0^2)/T_1^2$ is given by

[$G$ is the gravitational constant.]

Two satellites P and Q are moving in different circular orbits around the Earth (radius $R$). The heights of P and Q from the Earth surface are $h_P$ and $h_Q$, respectively, where $h_P = R/3$. The accelerations of P and Q due to Earth's gravity are $g_P$ and $g_Q$, respectively. If $g_P/g_Q = 36/25$, what is the value of $h_Q$?

Two spherical stars $A$ and $B$ have densities $\rho_A$ and $\rho_B$, respectively. $A$ and $B$ have the same radius, and their masses $M_A$ and $M_B$ are related by $M_B = 2M_A$. Due to an interaction process, star $A$ loses some of its mass, so that its radius is halved, while its spherical shape is retained, and its density remains $\rho_A$. The entire mass lost by $A$ is deposited as a thick spherical shell on $B$ with the density of the shell being $\rho_A$. If $v_A$ and $v_B$ are the escape velocities from $A$ and $B$ after the interaction process, the ratio $\dfrac{v_B}{v_A} = \sqrt{\dfrac{10n}{15^{1/3}}}$. The value of $n$ is _____.

The approximate height from the surface of earth at which the weight of the body becomes $\dfrac13$ of its weight on the surface of earth is :

[Radius of earth $R=6400$ km and $\sqrt3=1.732$]

The distance between Sun and Earth is R. The duration of year if the distance between Sun and Earth becomes 3R will be :

The height of any point P above the surface of earth is equal to diameter of earth. The value of acceleration due to gravity at point P will be : (Given g = acceleration due to gravity at the surface of earth).

Two satellites $S_1$ and $S_2$ are revolving in circular orbits around a planet with radius $R_1=3200$ km and $R_2=800$ km respectively. The ratio of speed of satellite $S_1$ to the speed of satellite $S_2$ in their respective orbits would be $\dfrac1x$ where $x=$ ______.

The variation of acceleration due to gravity (g) with distance (r) from the center of the earth is correctly represented by :

(Given R = radius of earth)

The elongation of a wire on the surface of the earth is $10^{-4}$ m. The same wire of same dimensions is elongated by $6\times10^{-5}$ m on another planet. The acceleration due to gravity on the planet will be ______ ms$^{-2}$. (Take acceleration due to gravity on the surface of earth = 10 ms$^{-2}$)

Given below are two statements: One is labelled as Assertion A and the other is labelled as Reason R.

Assertion A: If we move from poles to equator, the direction of acceleration due to gravity of earth always points towards the center of earth without any variation in its magnitude.

Reason R: At equator, the direction of acceleration due to the gravity is towards the center of earth.

In the light of above statements, choose the correct answer from the options given below

Given below are two statements :

Statement I : The law of gravitation holds good for any pair of bodies in the universe.

Statement II : The weight of any person becomes zero when the person is at the centre of the earth.

In the light of the above statements, choose the correct answer from the options given below.

Four spheres each of mass $m$ form a square of side $d$ (as shown in figure). A fifth sphere of mass $M$ is situated at the centre of square. The total gravitational potential energy of the system is :