Gravitation practice · Mixed difficulty

The escape velocity of a projectile from the earth is approximately

Hubble's law is related with

The escape velocity of a particle from the surface of the earth is given by

According to Kepler, the period of revolution of a planet ($T$) and its mean distance from the sun ($r$) are related by the equation

Statement I: If ice on the pole melts, length of the day will shorten.

Statement II: Ice will flow towards equator and will result in the decrease in moment of inertia of earth resulting in decrease in frequency of rotation of earth.

Assertion: When a planet moves in elliptical orbit around sun, its angular momentum about sun remains conserved.
Reason: Total energy of the planet remains conserved

(A): The angular velocity of a planets orbiting around the sun increases when they are nearest to the sun.
(R): The angular momentum of body is proportional to angular velocity.

For elliptical orbits, in the equation $T^2=\left(\frac{4 \pi^2}{{GM}_s}\right) R^3, R$ refers to

Given that centripetal force $F = - k/{r^2}$. The total energy is

Given below are two statements:

Statement I: If $E$ be the total energy of a satellite moving around the earth, then its potential energy will be $\frac{E}{2}$.

Statement II: The kinetic energy of a satellite revolving in an orbit is equal to the half the magnitude of total energy $E$.


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

Which of the following statement is always correct about a planet revolving around sun?

Assertion: Angular speed of a planet around the sun increases, when it is closer to the sun.

Reason: Total angular momentum of the system remains constant.

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.

Energy required to move an Earth’s satellite of mass m from an orbit of radius R to 5R is (M = mass of earth, R = radius of earth)

Two particles of equal masses are revolving in circular paths of radii ${r_1}\,and\,{r_2}$ respectively with the same speed. The ratio of their centripetal forces is

A cycle wheel of radius 0.4 m completes one revolution in one second then the acceleration of a point on the cycle wheel will be

If the earth stops rotating, the value of $g$ at the equator

A planet is revolving round the sun in an elliptical orbit, If $\mathbf{v}$ is the velocity of the planet when its position vector from the sun is $\mathbf{r}$, then areal velocity of the planet is

Assertion: Satellite is put in an orbit at a height where air resistance is present, then orbital velocity of the satellite will decrease.
Reason: Due to air resistance a lot of heat will be produced which may burn satellite

The mass of the moon is $\frac{1}{8}$ of the earth but the gravitational pull is $\frac{1}{6}$ of the earth. It is due to the fact that

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

Acceleration due to gravity is maximum at (R is the radius of earth)

A pallet of mass 1 g is moving with an angular velocity of 1 $\mathrm{rads}{\mathrm{\;}^{ - 1}}$ along a circle of radius 1 m the centrifugal force is

According to Newton's law of gravitation, the magnitude of the force $F$ on point masses $m_1$ and $m_2$ (say) is given by $|F|=\frac{G m_1 m_2}{r^2}$.
I. This equation is not directly applicable for the gravitational force between extended object (like the earth) and a point mass.
II. This equation can be applied to find the gravitational force between an extended object and a point mass provided $r$ is the distance of the point mass from the geometric centre of the extended objects.
Which of the following statement(s) about this equation is/are correct?

A rocket is intended to leave the Earth’s gravitational field. The
fuel in its main engine is a little less than the amount that is
necessary and an auxiliary engine, (only capable of operating
for a short time) has to be used as well. When is it best to switch
on: the auxiliary engine?

If the angular speed of the earth is doubled, the value of acceleration due to gravity (g) at the north pole

At what height from the surface of earth the total energy of a satellite is equal to its potential energy at a height of 2R from surface of earth…. (R = Radius of earth)

Given mass of the moon is $\frac{1}{{64}}$ of the mass of the earth and corresponding radius is $\frac{1}{{16}}$ of the earth. If the escape velocity on the earth surface is 11.2 km/s, the value of same on the surface of the moon is [km/s]

If gravitational constant is decreasing with time, what will remain unchanged in case of a satellite orbiting around earth

A particle of mass $m$ was transferred from infinity into the centre of the base of a uniform hemisphere of mass $M$ and radius $R.$ What work was performed in the process by the gravitational force exerted on the particle by the hemisphere?