Given below are two statements :

Statement I : A satellite is moving around earth in the orbit very close to the earth surface. The time period of revolution of satellite depends upon the density of earth.

Statement II : The time period of revolution of the satellite is $T=2 \pi \sqrt{\frac{R_e}{g}}$ (for satellite very close to the earth surface), where $R_{\mathrm{e}}$ radius of earth and $g$ acceleration due to gravity. In the light of the above statements, choose the correct answer from the options given below :

The escape velocity from a spherical planet $A$ is $10 \mathrm{~km} / \mathrm{s}$. The escape velocity from another planet $B$ whose density and radius are $10 \%$ of those of planet $A$, is $\_\_\_\_$ $\mathrm{m} / \mathrm{s}$.

Net gravitational force at the center of a square is found to be $F_1$ when four particles having mass $M, 2 M, 3 M$ and $4 M$ are placed at the four corners of the square as shown in figure and it is $F_2$ when the positions of $3 M$ and $4 M$ are interchanged. The ratio $\frac{F_1}{F_2}$ is $\frac{\alpha}{\sqrt{5}}$. The value of $\alpha$ is $\_\_\_\_$ .

Initially a satellite of 100 kg is in a circular orbit of radius $1.5 \mathrm{R}_{\mathrm{E}}$. This satellite can be moved to a circular orbit of radius $3 R_E$ by supplying $\alpha \times 10^6 \mathrm{~J}$ of energy The value of $\alpha$ is $\_\_\_\_$ .

(Take Radius of Earth $R_E=6 \times 10^6 \mathrm{~m}$ and $\mathrm{g}=10 \mathrm{~m} / \mathrm{s}^2$ )

A body of mass $m$ is taken from the surface of earth to a height equal to twice the radius of earth $\left(R_e\right)$. The increase in potential energy will be $\_\_\_\_$ .

( $g$ is acceleration due to gravity at the surface of earth)

When one moves from a point 16 km below the earth's surface to a point 16 km above the earth's surface. The change in g is approximately $\alpha \%$. The value of $\alpha$ is $\_\_\_\_$ .

(Take radius of the earth $=6400 \mathrm{~km}$.)

The height in terms of radius of the earth $(R)$, at which the acceleration due to gravity becomes $\frac{g}{9}$, where $g$ is acceleration due to gravity on earth's surface, is

$\_\_\_\_$ .

If a body of mass 1 kg falls on the earth from infinity, it attains velocity (v) and kinetic energy (k) on reaching the surface of earth. The values of v and k respectively are __________.

(Take radius of earth to be 6400 km and g = 9.8 m/s²)

A planet ($P_1$) is moving around the star of mass $2M$ in the orbit of radius $R$. Another planet ($P_2$) is moving around another star of mass $4M$ in a orbit of radius $2R$. Ratio of time periods of revolution of $P_2$ and $P_1$ is ________.

Given below are two statements: one is labelled as Assertion (A) and the other is labelled as Reason (R).

Assertion (A): The radius vector from the Sun to a planet sweeps out equal areas in equal intervals of time and thus areal velocity of planet is constant.

Reason (R): For a central force field the angular momentum is a constant.

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

An object is kept at rest at a distance of $3 R$ above the earth's surface where $R$ is earth's radius. The minimum speed with which it must be projected so that it does not return to earth is : (Assume $\mathrm{M}=$ mass of earth, $\mathrm{G}=$ Universal gravitational constant)

Given below are two statements : one is labelled as Assertion A and the other is labelled as Reason $\mathbf{R}$

Assertion A : The kinetic energy needed to project a body of mass $m$ from earth surface to infinity is $\frac{1}{2} \mathrm{mgR}$, where R is the radius of earth.

Reason R : The maximum potential energy of a body is zero when it is projected to infinity from earth surface.

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

$ \text { Match the LIST-I with LIST-II } $

Earth has mass 8 times and radius 2 times that of a planet. If the escape velocity from the earth is 11.2 km/s, the escape velocity in km/s from the planet will be:

If a satellite orbiting the Earth is 9 times closer to the Earth than the Moon, what is the time period of rotation of the satellite? Given rotational time period of Moon $=27$ days and gravitational attraction between the satellite and the moon is neglected.

A small point of mass $m$ is placed at a distance $2 R$ from the centre ' $O$ ' of a big uniform solid sphere of mass M and radius R . The gravitational force on ' m ' due to M is $\mathrm{F}_1$. A spherical part of radius $\mathrm{R} / 3$ is removed from the big sphere as shown in the figure and the gravitational force on m due to remaining part of $M$ is found to be $F_2$. The value of ratio $F_1: F_2$ is

A satellite of $10^3 \mathrm{~kg}$ mass is revolving in circular orbit of radius $2 R$. If $\frac{10^4 R}{6} \mathrm{~J}$ energy is supplied to the satellite, it would revolve in a new circular orbit of radius

(use $g=10 \mathrm{~m} / \mathrm{s}^2, R=$ radius of earth)

An astronaut takes a ball of mass $m$ from earth to space. He throws the ball into a circular orbit about earth at an altitude of $318.5 \mathrm{~km}$. From earth's surface to the orbit, the change in total mechanical energy of the ball is $x \frac{\mathrm{GM}_{\mathrm{e}} \mathrm{m}}{21 \mathrm{R}_{\mathrm{e}}}$. The value of $x$ is (take $\mathrm{R}_{\mathrm{e}}=6370 \mathrm{~km})$ :

Two satellite A and B go round a planet in circular orbits having radii 4R and R respectively. If the speed of $\mathrm{A}$ is $3 v$, the speed of $\mathrm{B}$ will be :

Two planets $A$ and $B$ having masses $m_1$ and $m_2$ move around the sun in circular orbits of $r_1$ and $r_2$ radii respectively. If angular momentum of $A$ is $L$ and that of $B$ is $3 \mathrm{~L}$, the ratio of time period $\left(\frac{T_A}{T_B}\right)$ is:

Assuming the earth to be a sphere of uniform mass density, a body weighed $300 \mathrm{~N}$ on the surface of earth. How much it would weigh at R/4 depth under surface of earth ?

To project a body of mass $m$ from earth's surface to infinity, the required kinetic energy is (assume, the radius of earth is $R_E, g=$ acceleration due to gravity on the surface of earth):

A satellite revolving around a planet in stationary orbit has time period 6 hours. The mass of planet is one-fourth the mass of earth. The radius orbit of planet is :

(Given $=$ Radius of geo-stationary orbit for earth is $4.2 \times 10^4 \mathrm{~km}$)

If $\mathrm{G}$ be the gravitational constant and $\mathrm{u}$ be the energy density then which of the following quantity have the dimensions as that of the $\sqrt{\mathrm{uG}}$ :

Match List I with List II :

(Where $\mathrm{a}=$ radius of planet orbit, $\mathrm{r}=$ radius of planet, $\mathrm{M}=$ mass of Sun, $\mathrm{m}=$ mass of planet)

Choose the correct answer from the options given below :

A $90 \mathrm{~kg}$ body placed at $2 \mathrm{R}$ distance from surface of earth experiences gravitational pull of :

($\mathrm{R}=$ Radius of earth, $\mathrm{g}=10 \mathrm{~m} \mathrm{~s}^{-2}$)

Correct formula for height of a satellite from earths surface is :

A metal wire of uniform mass density having length $L$ and mass $M$ is bent to form a semicircular arc and a particle of mass $\mathrm{m}$ is placed at the centre of the arc. The gravitational force on the particle by the wire is :

The mass of the moon is $\frac{1}{144}$ times the mass of a planet and its diameter is $\frac{1}{16}$ times the diameter of a planet. If the escape velocity on the planet is $v$, the escape velocity on the moon will be :

Four identical particles of mass $m$ are kept at the four corners of a square. If the gravitational force exerted on one of the masses by the other masses is $\left(\frac{2 \sqrt{2}+1}{32}\right) \frac{\mathrm{Gm}^2}{L^2}$, the length of the sides of the square is

Escape velocity of a body from earth is $11.2 \mathrm{~km} / \mathrm{s}$. If the radius of a planet be onethird the radius of earth and mass be one-sixth that of earth, the escape velocity from the planet is :

The gravitational potential at a point above the surface of earth is $-5.12 \times 10^7 \mathrm{~J} / \mathrm{kg}$ and the acceleration due to gravity at that point is $6.4 \mathrm{~m} / \mathrm{s}^2$. Assume that the mean radius of earth to be $6400 \mathrm{~km}$. The height of this point above the earth's surface is :

A planet takes 200 days to complete one revolution around the Sun. If the distance of the planet from Sun is reduced to one fourth of the original distance, how many days will it take to complete one revolution :

At what distance above and below the surface of the earth a body will have same weight. (take radius of earth as $R$.)

Given below are two statements : one is labelled as Assertion (A) and the other is labelled as Reason (R).

Assertion (A) : The angular speed of the moon in its orbit about the earth is more than the angular speed of the earth in its orbit about the sun.

Reason (R) : The moon takes less time to move around the earth than the time taken by the earth to move around the sun.

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

The acceleration due to gravity on the surface of earth is $\mathrm{g}$. If the diameter of earth reduces to half of its original value and mass remains constant, then acceleration due to gravity on the surface of earth would be :

Given below are two statements:

Statement I : For a planet, if the ratio of mass of the planet to its radius increases, the escape velocity from the planet also increases.

Statement II : Escape velocity is independent of the radius of the planet.

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

Two planets A and B of radii $\mathrm{R}$ and 1.5 R have densities $\rho$ and $\rho / 2$ respectively. The ratio of acceleration due to gravity at the surface of $\mathrm{B}$ to $\mathrm{A}$ is:

A planet having mass $9 \mathrm{Me}$ and radius $4 \mathrm{R}_{\mathrm{e}}$, where $\mathrm{Me}$ and $\mathrm{Re}$ are mass and radius of earth respectively, has escape velocity in $\mathrm{km} / \mathrm{s}$ given by:

(Given escape velocity on earth $\mathrm{V}_{\mathrm{e}}=11.2 \times 10^{3} \mathrm{~m} / \mathrm{s}$ )

The ratio of escape velocity of a planet to the escape velocity of earth will be:-

Given: Mass of the planet is 16 times mass of earth and radius of the planet is 4 times the radius of earth.

Two satellites $\mathrm{A}$ and $\mathrm{B}$ move round the earth in the same orbit. The mass of $\mathrm{A}$ is twice the mass of $\mathrm{B}$. The quantity which is same for the two satellites will be

A space ship of mass $2 \times 10^{4} \mathrm{~kg}$ is launched into a circular orbit close to the earth surface. The additional velocity to be imparted to the space ship in the orbit to overcome the gravitational pull will be (if $g=10 \mathrm{~m} / \mathrm{s}^{2}$ and radius of earth $=6400 \mathrm{~km}$ ):

If $\mathrm{V}$ is the gravitational potential due to sphere of uniform density on it's surface, then it's value at the center of sphere will be:-

The radii of two planets 'A' and 'B' are 'R' and '4R' and their densities are $\rho$ and $\rho / 3$ respectively. The ratio of acceleration due to gravity at their surfaces $\left(g_{A}: g_{B}\right)$ will be:

The time period of a satellite, revolving above earth's surface at a height equal to $\mathrm{R}$ will be

(Given $g=\pi^{2} \mathrm{~m} / \mathrm{s}^{2}, \mathrm{R}=$ radius of earth)

Given below are two statements:

Statement I : Rotation of the earth shows effect on the value of acceleration due to gravity (g)

Statement II : The effect of rotation of the earth on the value of 'g' at the equator is minimum and that at the pole is maximum.

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

Two satellites of masses m and 3m revolve around the earth in circular orbits of radii r & 3r respectively. The ratio of orbital speeds of the satellites respectively is

Assuming the earth to be a sphere of uniform mass density, the weight of a body at a depth $d=\frac{R}{2}$ from the surface of earth, if its weight on the surface of earth is 200 N, will be:

(Given R = radius of earth)

The acceleration due to gravity at height $h$ above the earth if $h << \mathrm{R}$ (Radius of earth) is given by

The orbital angular momentum of a satellite is L, when it is revolving in a circular orbit at height h from earth surface. If the distance of satellite from the earth centre is increased by eight times to its initial value, then the new angular momentum will be -

Given below are two statements:

Statement I: If $\mathrm{E}$ 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 $\mathrm{E}$.

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

The weight of a body on the earth is $400 \mathrm{~N}$. Then weight of the body when taken to a depth half of the radius of the earth will be:

The weight of a body on the surface of the earth is $100 \mathrm{~N}$. The gravitational force on it when taken at a height, from the surface of earth, equal to one-fourth the radius of the earth is:

Choose the incorrect statement from the following:

Given below are two statements : one is labelled as Assertion $\mathbf{A}$ and the other is labelled as Reason $\mathbf{R}$.

Assertion A : Earth has atmosphere whereas moon doesn't have any atmosphere.

Reason R : The escape velocity on moon is very small as compared to that on earth.

In the light of the above statements, choose the correct answer from the options given below:

A planet has double the mass of the earth. Its average density is equal to that of the earth. An object weighing $\mathrm{W}$ on earth will weigh on that planet:

The escape velocities of two planets $\mathrm{A}$ and $\mathrm{B}$ are in the ratio $1: 2$. If the ratio of their radii respectively is $1: 3$, then the ratio of acceleration due to gravity of planet A to the acceleration of gravity of planet B will be :

For a body projected at an angle with the horizontal from the ground, choose the correct statement.

If earth has a mass nine times and radius twice to that of a planet P. Then $\frac{v_{e}}{3} \sqrt{x} \mathrm{~ms}^{-1}$ will be the minimum velocity required by a rocket to pull out of gravitational force of $\mathrm{P}$, where $v_{e}$ is escape velocity on earth. The value of $x$ is

Given below are two statements:

Statement I: Acceleration due to gravity is different at different places on the surface of earth.

Statement II: Acceleration due to gravity increases as we go down below the earth's surface.

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

At a certain depth "d " below surface of earth, value of acceleration due to gravity becomes four times that of its value at a height $\mathrm{3 R}$ above earth surface. Where $\mathrm{R}$ is Radius of earth (Take $\mathrm{R}=6400 \mathrm{~km}$ ). The depth $\mathrm{d}$ is equal to

If the gravitational field in the space is given as $\left(-\frac{K}{r^{2}}\right)$. Taking the reference point to be at $\mathrm{r}=2 \mathrm{~cm}$ with gravitational potential $\mathrm{V}=10 \mathrm{~J} / \mathrm{kg}$. Find the gravitational potential at $\mathrm{r}=3 \mathrm{~cm}$ in SI unit (Given, that $\mathrm{K}=6 \mathrm{~Jcm} / \mathrm{kg}$)

The time period of a satellite of earth is 24 hours. If the separation between the earth and the satellite is decreased to one fourth of the previous value, then its new time period will become.

Two particles of equal mass '$m$' move in a circle of radius '$r$' under the action of their mutual gravitational attraction. The speed of each particle will be :

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 :

A body of mass is taken from earth surface to the height h equal to twice the radius of earth (R$_e$), the increase in potential energy will be :

(g = acceleration due to gravity on the surface of Earth)

Assume that the earth is a solid sphere of uniform density and a tunnel is dug along its diameter throughout the earth. It is found that when a particle is released in this tunnel, it executes a simple harmonic motion. The mass of the particle is 100 g. The time period of the motion of the particle will be (approximately)

(Take g = 10 m s$^{-2}$ , radius of earth = 6400 km)

Given below are two statements: one is labelled as Assertion A and the other is labelled as Reason R.

Assertion A : A pendulum clock when taken to Mount Everest becomes fast.

Reason R : The value of g (acceleration due to gravity) is less at Mount Everest than its value on the surface of earth.

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

If the distance of the earth from Sun is 1.5 $\times$ 10$^6$ km. Then the distance of an imaginary planet from Sun, if its period of revolution is 2.83 years is :

Given below are two statements:

Statement I : Acceleration due to earth's gravity decreases as you go 'up' or 'down' from earth's surface.

Statement II : Acceleration due to earth's gravity is same at a height 'h' and depth 'd' from earth's surface, if h = d.

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

The weight of a body at the surface of earth is 18 N. The weight of the body at an altitude of 3200 km above the earth's surface is (given, radius of earth $\mathrm{R_e=6400~km}$) :

An object of mass $1 \mathrm{~kg}$ is taken to a height from the surface of earth which is equal to three times the radius of earth. The gain in potential energy of the object will be [If, $\mathrm{g}=10 \mathrm{~ms}^{-2}$ and radius of earth $=6400 \mathrm{~km}$ ]

If the radius of earth shrinks by $2 \%$ while its mass remains same. The acceleration due to gravity on the earth's surface will approximately :

A body of mass $\mathrm{m}$ is projected with velocity $\lambda \,v_{\mathrm{e}}$ in vertically upward direction from the surface of the earth into space. It is given that $v_{\mathrm{e}}$ is escape velocity and $\lambda<1$. If air resistance is considered to be negligible, then the maximum height from the centre of earth, to which the body can go, will be :

(R : radius of earth)

Two satellites $\mathrm{A}$ and $\mathrm{B}$, having masses in the ratio $4: 3$, are revolving in circular orbits of radii $3 \mathrm{r}$ and $4 \mathrm{r}$ respectively around the earth. The ratio of total mechanical energy of $\mathrm{A}$ to $\mathrm{B}$ is :

A body is projected vertically upwards from the surface of earth with a velocity equal to one third of escape velocity. The maximum height attained by the body will be :

(Take radius of earth $=6400 \mathrm{~km}$ and $\mathrm{g}=10 \mathrm{~ms}^{-2}$ )

The percentage decrease in the weight of a rocket, when taken to a height of $32 \mathrm{~km}$ above the surface of earth will, be :

$($ Radius of earth $=6400 \mathrm{~km})$

The length of a seconds pendulum at a height h = 2R from earth surface will be:

(Given R = Radius of earth and acceleration due to gravity at the surface of earth, g = $\pi$² ms^$-$2)

An object is taken to a height above the surface of earth at a distance ${5 \over 4}$ R from the centre of the earth. Where radius of earth, R = 6400 km. The percentage decrease in the weight of the object will be :

Three identical particles $\mathrm{A}, \mathrm{B}$ and $\mathrm{C}$ of mass $100 \mathrm{~kg}$ each are placed in a straight line with $\mathrm{AB}=\mathrm{BC}=13 \mathrm{~m}$. The gravitational force on a fourth particle $\mathrm{P}$ of the same mass is $\mathrm{F}$, when placed at a distance $13 \mathrm{~m}$ from the particle $\mathrm{B}$ on the perpendicular bisector of the line $\mathrm{AC}$. The value of $\mathrm{F}$ will be approximately :

The radii of two planets A and B are in the ratio 2 : 3. Their densities are 3$\rho$ and 5$\rho$ respectively. The ratio of their acceleration due to gravity is :

The time period of a satellite revolving around earth in a given orbit is 7 hours. If the radius of orbit is increased to three times its previous value, then approximate new time period of the satellite will be

The escape velocity of a body on a planet 'A' is 12 kms^$-$1. The escape velocity of the body on another planet 'B', whose density is four times and radius is half of the planet 'A', is :

Water falls from a 40 m high dam at the rate of 9 $\times$ 10⁴ kg per hour. Fifty percentage of gravitational potential energy can be converted into electrical energy. Using this hydroelectric energy number of 100 W lamps, that can be lit, is :

(Take g = 10 ms^$-$2)

Two objects of equal masses placed at certain distance from each other attracts each other with a force of F. If one-third mass of one object is transferred to the other object, then the new force will be :

Two planets A and B of equal mass are having their period of revolutions T_A and T_B such that T_A = 2T_B. These planets are revolving in the circular orbits of radii r_A and r_B respectively. Which out of the following would be the correct relationship of their orbits?

The distance of the Sun from earth is 1.5 $\times$ 10¹¹ m and its angular diameter is (2000) s when observed from the earth. The diameter of the Sun will be :

Four spheres each of mass m from a square of side d (as shown in figure). A fifth sphere of mass M is situated at the centre of square. The total gravitational potential energy of the system is :

Given below are two statements :

Statement I : The law of gravitation holds good for any pair of bodies in the universe.

Statement II : The weight of any person becomes zero when the person is at the centre of the earth.

In the light of the above statements, choose the correct answer from the options given below.

Given below are two statements : One is labelled as Assertion A and the other is labelled as Reason R.

Assertion A : If we move from poles to equator, the direction of acceleration due to gravity of earth always points towards the center of earth without any variation in its magnitude.

Reason R : At equator, the direction of acceleration due to the gravity is towards the center of earth.

In the light of above statements, choose the correct answer from the options given below:

The variation of acceleration due to gravity (g) with distance (r) from the center of the earth is correctly represented by :

(Given R = radius of earth)

The height of any point P above the surface of earth is equal to diameter of earth. The value of acceleration due to gravity at point P will be : (Given g = acceleration due to gravity at the surface of earth).

The distance between Sun and Earth is R. The duration of year if the distance between Sun and Earth becomes 3R will be :

The approximate height from the surface of earth at which the weight of the body becomes ${1 \over 3}$ of its weight on the surface of earth is :

[Radius of earth R = 6400 km and $\sqrt 3 $ = 1.732]