Gravitation

Consider two satellites S$_1$ and S$_2$ with periods of revolution 1 hr. and 8 hr. respectively revolving around a planet in circular orbits. The ratio of angular velocity of satellite S$_1$ to the angular velocity of satellite S$_2$ is :

Consider a spherical gaseous cloud of mass density $\rho(r)$ in free space where $r$ is the radial distance from its center. The gaseous cloud is made of particles of equal mass $m$ moving in circular orbits about the common center with the same kinetic energy $K$. The force acting on the particles is their mutual gravitational force. If $\rho(r)$ is constant in time, the particle number density $n(r) = \rho(r)/m$ is

[$G$ is universal gravitational constant]

A planet of mass $M$, has two natural satellites with masses $m_1$ and $m_2$. The radii of their circular orbits are $R_1$ and $R_2$ respectively. Ignore the gravitational force between the satellites. Define $v_1$, $L_1$, $K_1$ and $T_1$ to be, respectively, the orbital speed, angular momentum, kinetic energy and time period of revolution of satellite 1; and $v_2$, $L_2$, $K_2$ and $T_2$ to be the corresponding quantities of satellite 2. Given $m_1/m_2 = 2$ and $R_1/R_2 = 1/4$, match the ratios in List-I to the numbers in List-II.

A rocket is launched normal to the surface of the Earth, away from the Sun, along the line joining the Sun and the Earth. The Sun is $3 \times 10^5$ times heavier than the Earth and is at a distance $2.5 \times 10^4$ times larger than the radius of the Earth. The escape velocity from Earth’s gravitational field is $v_e = 11.2\ \mathrm{km\,s^{-1}}$. The minimum initial velocity $(v_S)$ required for the rocket to be able to leave the Sun-Earth system is closest to

(Ignore the rotation and revolution of the Earth and the presence of any other planet)

The variation of acceleration due to gravity g with distance d from centre of the earth is best represented by (R = Earth's radius) :

If the Earth has no rotational motion, the weight of a person on the equator is W. Determine the speed with which the earth would have to rotate about its axis so that the person at the equator will weigh $\dfrac34$W. Radius of the Earth is 6400 km and $g=10$ m/s$^2$.

The mass density of a spherical body is given by $\rho(r)=\dfrac kr$ for r ≤ R and ρ(r) = 0 for r > R, where r is the distance from the centre.

The correct graph that describes qualitatively the acceleration, a, of a test particle as a function of r is :

A satellite is revolving in a circular orbit at a height '$h$' from the earth's surface (radius of earth R ; h << R). The minimum increase in its orbital velocity required, so that the satellite could escape from the earth's gravitational field, is close to : (Neglect the effect of atmosphere.)

Figure shows elliptical path abcd of a planet around the sun S such that the area of triangle csa is $\dfrac14$ the area of the ellipse. (See figure) With db as the semimajor axis, and ca as the semiminor axis. If $t_1$ is the time taken for planet to go over path abc and $t_2$ for path taken over cda then :

An astronaut of mass m is working on a satellite orbiting the earth at a distance h from the earth's surface. The radius of the earth is R, while its mass is M. The gravitational pull $F_G$ on the astronaut is :

A bullet is fired vertically upwards with velocity $v$ from the surface of a spherical planet. When it reaches its maximum height, its acceleration due to the planet's gravity is 1/4th of its value at the surface of the planet. If the escape velocity from the planet is $v_{esc}=v\sqrt N$, then the value of $N$ is (ignore energy loss due to atmosphere)

A large spherical mass $M$ is fixed at one position and two identical point masses $m$ are kept on a line passing through the centre of $M$ (see figure). The point masses are connected by a rigid massless rod of length $l$ and this assembly is free to move along the line connecting them. All three masses interact only through their mutual gravitational interaction. When the point mass nearer to $M$ is at a distance $r=3l$ from $M$, the tension in the rod is zero for $m=k\left(\dfrac{M}{288}\right)$. The value of $k$ is

A spherical body of radius $R$ consists of a fluid of constant density and is in equilibrium under its own gravity. If $P(r)$ is the pressure at $r\,(r<R)$, then the correct option(s) is(are)

From a solid sphere of mass M and radius R, a spherical portion of radius $\dfrac R2$ is removed, as shown in the figure. Taking gravitational potential V = 0 at r = ∞, the potential at the centre of the cavity thus formed is :

(G = gravitational constant)

A very long (length L) cylindrical galaxy is made of uniformly distributed mass and has radius R (R<<L). A star outside the galaxy is orbiting the galaxy in a plane perpendicular to the galaxy and passing through its centre. If the time period of star is T and its distance from the galaxy's axis is r, then :

Which of the following most closely depicts the correct variation of the gravitation potential V(r) due to a large planet of radius R and uniform mass density ? (figures are not drawn to scale)

A planet of radius $R=\frac1{10}\times$(radius of Earth) has the same mass density as Earth. Scientists dig a well of depth $\frac R5$ on it and lower a wire of the same length and of linear mass density $10^{-3}\ \mathrm{kg\,m^{-1}}$ into it. If the wire is not touching anywhere, the force applied at the top of the wire by a person holding it in place is (take the radius of Earth $=6\times10^6$ m and the acceleration due to gravity on Earth is $10\ \mathrm{m\,s^{-2}}$)

Four particles, each of mass M and equidistant from each other, move along a circle of radius R under the action of their mutual gravitational attraction. The speed of each particle is :

India's Mangalyan was sent to the Mars by launching it into a transfer orbit EOM around the sun. It leaves the earth at E and meets Mars at M. If the semi-major axis of Earth's orbit is $a_e=1.5\times10^{11}$ m, that of Mar's orbit $a_m=2.28\times10^{11}$ m, taken Kepler's laws give the estimate of time for Mangalyan to reach Mars from Earth to be close to :

A capillary tube is immersed vertically in water and the height of the water column is $x$. When this arrangement is taken into a mine of depth $d$, the height of the water column is $y$. If $R$ is the radius of earth, the ratio $\dfrac xy$ is :