Gravitation

A particle of mass m, and angular momentum ℓ is moving in a circular orbit of radius r0 under the influence of an attractive force $\vec{F}(r)=-\frac{k}{r^2} \hat{r}$. Keeping its angular momentum unchanged, the particle is displaced radial...

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 ...

Consider a star of mass m2 kg revolving in a circular orbit around another star of mass m1 kg with m1 \gg m2. The heavier star slowly acquires mass from the lighter star at a constant rate of $\gamma$ kg/s. In this transfer process, there i...

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...

Two satellites $\mathrm{P}$ and $\mathrm{Q}$ are moving in different circular orbits around the Earth (radius $R$ ). The heights of $\mathrm{P}$ and $\mathrm{Q}$ from the Earth surface are $h_{\mathrm{P}}$ and $h_{\mathrm{Q}}$, respectively...

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 $M_{B}=2 M_{A}$. Due to an interaction process, star $A$ loses som...

The distance between two stars of masses 3MS and 6MS is 9R. Here R is the mean distance between the centers of the Earth and the Sun, and MS is the mass of the Sun. The two stars orbit around their common center of mass in circular orbits w...

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 ...

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},{...

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 t...

A large spherical mass M is fixed at one position and two identical 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 mo...

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 \({\left( {{1 \over 4}} \right)^{th}}\) of its value at the sur...

A planet of radius R = \({1 \over {10}} \times\) (radius of Earth) has the same mass density as Earth. Scientists dig a well of
depth \({R \over 5}\)
on it and lower a wire of the same length and of linear mass density 10-3 kg m-1 into it....

Two bodies, each of mass M, are kept fixed with a separation \(2L\). A particle of mass m is projected from
the midpoint of the line joining their centres, perpendicular to the line. The gravitational constant is G. The
correct statement(s)...

Two spherical planets P and Q have the same uniform density r, masses MP and MQ and surface
areas A and 4A respectively. A spherical planet R also has uniform density r and its mass is (MP
+ MQ). The escape velocities from the planets P, Q ...

A satellite is moving with a constant speed ‘V’ in a circular orbit about the earth. An object of mass ‘m’ is
ejected from the satellite such that it just escapes from the gravitational pull of the earth. At the time of its
ejection, the ki...

A binary star consists of two stars A (mass 2.2Ms) and B (mass 11Ms), where Ms is the mass of the sun. They are separated by distance d and are rotating about their centre of mass, which is stationary. The ratio
of the total angular momentu...

Gravitational acceleration on the surface of a planet is \({{\sqrt 6 } \over {11}}g\), where \(g\) is the gravitational acceleration on
the surface of the earth. The average mass density of the planet is \({2 \over 3}\)
times that of the ea...

A thin uniform annular disc (see figure) of mass M has outer radius 4R and inner radius 3R. The work required to take a unit mass from point P on its axis to infinity is

Column II shows five systems in which two objects are labelled as X and Y. Also in each case a point P is shown. Column I gives some statements about X and/or Y. Match these statements to the appropriate system(s) from Column II:

.tg {bor...

STATEMENT - 1
An astronaut in an orbiting space station above the Earth experiences weightlessness.
and
STATEMENT - 2
An object moving around the Earth under the influence of Earth's gravitational force is in a state of 'free-fall'.

A spherically symmetric gravitational system of particles has a mass density
\(\rho = \left\{ {\matrix{ {{\rho _0}} & {for} & {r \le R} \cr 0 & {for} & {r > R} \cr } } \right.\)
Where \(\rho_0\) is a constant. A test mass can u...

Some physical quantities are given in Column I and some possible SI units in which these quantities may be expressed are given in Column II. Match the physical quantities in Column I with the units in Column II and indicate your answer by d...

A system of binary stars of masses $m_{\mathrm{A}}$ and $m_{\mathrm{B}}$ are moving in circular orbits of radii $r_{\mathrm{A}}$ and $r_R$, respectively. If $\mathrm{T}_A$ and $\mathrm{T}_B$ are the time periods of masses $m_A$ and $m_B$ re...