Gravitation practice · Mixed difficulty

Every planet revolves around the sun in an elliptical
orbit:

A. The force acting on a planet is inversely
proportional to square of distance from sun.

B. Force acting on planet is inversely proportional
to product of the masses of the planet and the
sun.

C. The Centripetal force acting on the planet is
directed away from the sun.

D. The square of time period of revolution of
planet around sun is directly proportional to
cube of semi-major axis of elliptical orbit.

Choose the correct answer from the options given
below:

Two identical satellites are at $R$ and $7R$ away from earth surface, the wrong statement is

Assertion: For circular orbits, the law of periods is ${T}^2 \propto {r}^3$, where ${M}$ is the mass of sun and $r$ is the radius of orbit.
Reason: The square of the period $T$ of any planet about the sun is proportional to the cube of the semi-major axis $a$ of the orbit.

A satellite of mass 200 kg is orbiting around the earth at a height of 400 km above its surface. The energy required to send it out of the earth’s gravitational influence is (Mass of earth $= 6 \times {10^{24}}kg$, Radius of earth = 6000 km, $G = 6.67 \times {10^{ - 11}}\,N{m^2}\,k{g^{ - 2}}$)

The masses and radii of the earth and moon are ${M_{1,}}{R_1}$ and ${M_{2,}}{R_2}$ respectively, their centres are distance $d$ apart. The minimum velocity with which a particle of mass $m$ should be projected from a point midway between their centres so that it escapes to infinity is

The radius of a planet is twice the radius of earth. Both have almost equal average mass-densities. If $V_P$ and $V_E$ are escape velocities of the planet and the earth, respectively, then

Two bodies of masses ${m_1}\,and\,{m_2}$ are initially at rest at infinite distance apart. They are then allowed to move towards each other under mutual gravitational attraction. Their relative velocity of approach at a separation distance $r$ between them is

In the following four periods
(i) Time of revolution of a satellite just above the earth’s surface $\left( {{T_{st}}} \right)$ , (ii) Period of oscillation of mass inside the tunnel bored along the diameter of the earth $\left( {{T_{ma}}} \right)$ , (iii) Period of simple pendulum having a length equal to the earth’s radius in a uniform field of $9.8N/kg\left( {{T_{sp}}} \right)$ , (iv) Period of an infinite length simple pendulum in the earth’s real gravitational filed $\left( {{T_{is}}} \right)$

The ratio of orbital velocity of an earth's satellite to the escape velocity of the same earth's satellite is very nearly equal to

Two isolated point masses ${m}$ and ${M}$ are separated by a distance $\ell$. The moment of inertia of the system about an axis passing through a point where gravitational field is zero and perpendicular to the line joining the two masses, is

Two spheres of mass $m$ and $M$ are situated in air and the gravitational force between them is $2 F.$ The space around the masses is now filled with a liquid of specific gravity 4. The gravitational force will now be

A satellite of the earth is revolving in a circular orbit with a uniform speed $v$. If the gravitational force suddenly disappears, the satellite will

Assertion: The magnitude of gravitational potential at the surface of solid sphere is less than that of the centre of sphere.
Reason: Due to solid sphere, gravitational potential is same within the sphere.

The gravitational attraction between the two bodies increases when their masses are

An artificial satellite of mass ‘m’ is revolving round the earth in a circle of radius $R$. Then work done in one revolution is

Six stars of equal mass $m$ are moving about the centre of mass of the system such that they are always on the vertices of a regular hexagon of side length a. Their common time period of revolution around center will be

Assertion: If the earth stops rotating about its axis, the value of the weight of the body at equator will decrease.
Reason: The centripetal force does not act on the body at the equator.

What is the energy required to launch a $m \mathrm{~kg}$ satellite from earth's surface to a circular orbit at an altitude of $7 R$? ($R=$ radius of the earth)

Satellite A and B are revolving around the orbit of earth. The mass of A is 10 times of mass of B.The ratio of time period $\left( {\frac{{{T_A}}}{{{T_B}}}} \right)$ is

The Earth's gravity provides the centripetal force on a orbiting satellite to keep it moving in circle at constant speed. Which statements best explains why the speed of the satellite does not change even through there is a net force exerted on it?

If the angular speed of earth is increased so much that the objects start flying from the equator, then the length of the day will be nearly

A mass $M$ is broken into two parts of masses $m_1$ and $m_2$. How are $m_1$ and $m_2$ related so that force of gravitational attraction between the two parts is maximum?

Satellite A and B are revolving around the orbit of earth. The mass of A is 10 times of mass of B.The ratio of time period $\left( {\frac{{{T_A}}}{{{T_B}}}} \right)$ is

Two identical trains $P$ and $Q$ move with equal speeds on parallel tracks along the equator. $P$ moves from east to west and $Q$ from west to east

A planet revolving in elliptical orbit has
I. a constant velocity of revolution
II. has the least velocity when it is nearest to the sun
III. its areal velocity is directly proportional to its velocity.
IV. areal velocity is inversely proportional to its velocity.
V. to follow a trajectory such that the areal velocity is constant.
Choose the correct answer from the options given below.

A cylindrical vessel partially filled with water is rotated about its vertical central axis. It’s surface will

The length of a seconds pendulum if it is placed at height $2 R$ ($R$ : radius of earth) is $\frac{10}{x \pi^2}$ metres. Find $x$.

Three particles each of mass $m$ are kept at the vertices of an equilateral triangle of side $L$. The gravitational field at, centre due to these particles is

Gas escapes from the surface of a planet because it acquires an escape velocity. The escape velocity will depend on which of the following factors :
I. Mass of the planet
II. Mass of the particle escaping
III. Temperature of the planet
IV. Radius of the planet
Select the correct answer from the codes given below :

If suppose moon is suddenly stopped and then released (given radius of moon is one-fourth the radius of earth)and the acceleration of moon with respect to earth is $0.0027\,\mathrm{m}{\mathrm{s}^{ - 2}}$), then the acceleration of the moon just before striking the earth’s surface is (Take $g = 10m{s^{ - 2}})$