Gravitation practice · Mixed difficulty

If orbital velocity of planet is given by $v = {G^a}{M^b}{R^c}$, then

The value of acceleration due to gravity at the surface of earth

A satellite is moving around the earth in a circular orbit. The following statements are given

(i)It is moving with a constant velocity.

(ii) It suffers no acceleration.

(iii) Its angular momentum w.r.t. the earth remains conserved.

(iv) Its distance from centre must be equal to $\sqrt{2}$ times of earth's radius.

The correct option is

A uniform spherical shell gradually shrinks maintaining its shape. The gravitational potential at the centre

Assertion: Earth does not retain hydrogen molecules and helium atoms in its atmosphere, but does retain much heavier molecules, such as oxygen and nitrogen.
Reason: Lighter molecules in the atmosphere have translational speed that is greater or closer to escape speed of earth.

The ratio of the accelerations due gravity on two planets $P_1 \& P_2$ is $k_1$. The ratio of their respective radii is $k_2$. The ratio of their respective escape velocities is

Gravitational potential on the surface of earth is (M=mass of the earth, R= radius of earth)

Orbital velocity of an artificial does not depend upon

A space ship moves from earth to moon and back. The greatest energy required for the space ship is to overcome the difficulty in

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 acceleration due to gravity on a planet is same as that on earth and its radius is 16 times that of earth. What will be the value of escape velocity on that planet if it is ${V_e}$ on earth.

A particle is moving in a circle of radius R with constant speed v, if radius is double then its centripetal force to keep the same speed should be

Statement-1: For the planets orbiting around the sun, angular speed, linear speed, KE change with time, but angular momentum remains constant.
Statement-2: No external torque is acting on the rotating planet. So its angular momentum is constant.

The centripetal acceleration of a body moving in a circle of radius 100 m with a time period of 2 s will be

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

Assertion: The work done to project a body of mass $m$ from the earth's surface (radius $R$ ) to infinity is $m g R$.

Reason: The required work done is equal to change in potential energy of the mass, which is equal to $G M m / R$.

If ${v_e}\,and\,{v_o}$ represent escape velocity and orbital velocity of a satellite corresponding to a circular orbit of radius R, then

Statement-1: A spherical shell produces no gravitational field anywhere.
Statement-2: The field due to various mass elements cancels out, everywhere inside and outside the shell.

Assertion: Kepler's second law can be understood by conservation of angular momentum principle
Reason: Kepler's second law is related with areal velocity which can further be proved to be based on conservation of angular momentum as $(d A / d t)=\left(r^2 \omega\right)/2$

Two electrons are revolving around a nucleus at distances ' $r$ ' and ' $4 r$ '. The ratio of their periods is

Assertion: The value of acceleration due to gravity does not depend upon mass of the body
Reason: Acceleration due to gravity is a constant quantity

The gravitational force between two point masses ${m_1}\,and\,{m_2}$ at separation $r$ is given by $F = k\frac{{{m_1}{m_2}}}{{{r^2}}}$. The constant $k$

The areal velocity of a planet of mass 10 kg is $15\,{m^2}/s$ it angular momentum is $\left[ {\mathrm{in}\,\,Kg\,{m^2}/s} \right]$

For a geostationary satellite the angular speed in units of radian per minute is

Statement-1: Period of revolution of satellite in circular orbit around earth is inversely proportional to its orbital speed.
Statement-2: Period of revolution in uniform circular motion is given by $T=\frac{2 \pi r}{v}$ where $r$ is radius of orbit and $v$ is speed.

Statement 1: Value of acceleration due to gravity is same at all the points inside earth assuming it to be made up of uniform density.
Statement 2: Value of gravitational field increases as we go towards centre in a uniform spherical shell.

If the earth is at one-fourth of its present distance from the sun, the duration of the year will be

The distance of a geostationary satellite from the centre of the earth [Radius R= 6400 km] is nearest to

A satellite is revolving around the earth at a height $h$. The satellite explodes into two equal parts, one part has zero velocity after the explosion, and the other moves in the same direction with double the orbital speed of the original satellite. The maximum height reached by the part with double the velocity is (Neglect the gravitational effects between the fragments)

Time speed of revolution of a nearest satellite around a planet of radius R is T. Period of revolution around another planet, whose radius is 3R but having same density is