Gravitation · Physics · JEE Main

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 ...

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{~k...

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...

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 ang...

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 ...

To project a body of mass $$m$$ from earth's surface to infinity, the required kinetic energy is (assume, the radius of earth is $$R_E, g=$$ accelerat...

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 orbi...

If $$\mathrm{G}$$ be the gravitational constant and $$\mathrm{u}$$ be the energy density then which of the following quantity have the dimensions as t...

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

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 pl...

A light planet is revolving around a massive star in a circular orbit of radius $\mathrm{R}$ with a period of revolution T. If the force of attraction...

If $\mathrm{R}$ is the radius of the earth and the acceleration due to gravity on the surface of earth is $g=\pi^2 \mathrm{~m} / \mathrm{s}^2$, then t...

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 escap...

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 ...

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-...

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 gravi...

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 dis...

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 mo...

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 rem...

Two identical particles each of mass ' $m$ ' go round a circle of radius $a$ under the action of their mutual gravitational attraction. The angular sp...

A body is released from a height equal to the radius $(\mathrm{R})$ of the earth. The velocity of the body when it strikes the surface of the earth wi...

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 p...

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 ...

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 eart...

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 t...

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}$...

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 impa...

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...

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...

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 ef...

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 satelli...

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 ...

The acceleration due to gravity at height $$h$$ above the earth if $$h

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 satellit...

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 w...

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 ear...

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 h...

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...

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 $$1: 3$$...

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 velocit...

Given below are two statements: Statement I: Acceleration due to gravity is different at different places on the surface of earth. Statement II: Accel...

A body weight $\mathrm{W}$, is projected vertically upwards from earth's surface to reach a height above the earth which is equal to nine times the ra...

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 $$\mathrm{3 R}$$...

An object is allowed to fall from a height $R$ above the earth, where $R$ is the radius of earth. Its velocity when it strikes the earth's surface, ig...

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}...

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 ...

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 pa...

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 su...

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...

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 partic...

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 Mou...

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...

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 I...

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 ...

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 pot...

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 int...

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 ...

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 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...

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 surf...

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 = 640...

Three identical particles $$\mathrm{A}, \mathrm{B}$$ and $$\mathrm{C}$$ of mass $$100 \mathrm{~kg}$$ each are placed in a straight line with $$\mathrm...

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...

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 val...

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 ...

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...

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 trans...

Two planets A and B of equal mass are having their period of revolutions TA and TB such that TA = 2TB. These planets are revolving in the circular orb...

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...

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 grav...

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 ...

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 variation of acceleration due to gravity (g) with distance (r) from the center of the earth is correctly represented by : (Given R = radius of ea...

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 : (Giv...

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 : [R...

Four particles each of mass M, move along a circle of radius R under the action of their mutual gravitational attraction as shown in figure. The speed...

If RE be the radius of Earth, then the ratio between the acceleration due to gravity at a depth 'r' below and a height 'r' above the earth surface is ...

The masses and radii of the earth and moon are (M1, R1) and (M2, R2) respectively. Their centres are at a distance 'r' apart. Find the minimum escape ...

A mass of 50 kg is placed at the centre of a uniform spherical shell of mass 100 kg and radius 50 m. If the gravitational potential at a point, 25 m f...

Inside a uniform spherical shell :(1) the gravitational field is zero(2) the gravitational potential is zero(3) the gravitational field is same everyw...

Two identical particles of mass 1 kg each go round a circle of radius R, under the action of their mutual gravitational attraction. The angular speed ...

The planet Mars has two moons, if one of them has a period 7 hours, 30 minutes and an orbital radius of 9.0 $$\times$$ 103 km. Find the mass of Mars.$...

Consider a planet in some solar system which has a mass double the mass of earth and density equal to the average density of earth. If the weight of a...

The minimum and maximum distances of a planet revolving around the sun are x1 and x2. If the minimum speed of the planet on its trajectory is v0 then ...

A body is projected vertically upwards from the surface of earth with a velocity sufficient enough to carry it to infinity. The time taken by it to re...

A satellite is launched into a circular orbit of radius R around earth, while a second satellite is launched into a circular orbit of radius 1.02 R. T...

Consider a binary star system of star A and star B with masses mA and mB revolving in a circular orbit of radii rA an rB, respectively. If TA and TB a...

A person whose mass is 100 kg travels from Earth to Mars in a spaceship. Neglect all other objects in sky and take acceleration due to gravity on the...

If the angular velocity of earth's spin is increased such that the bodies at the equator start floating, the duration of the day would be approximatel...

The angular momentum of a planet of mass M moving around the sun in an elliptical orbit is $${\overrightarrow L }$$. The magnitude of the areal veloci...

The time period of a satellite in a circular orbit of radius R is T. The period of another satellite in a circular orbit of radius 9R is :

A geostationary satellite is orbiting around an arbitrary planet 'P' at a height of 11R above the surface of 'P', R being the radius of 'P'. The time ...

The maximum and minimum distances of a comet from the Sun are 1.6 $$\times$$ 1012 m and 8.0 $$\times$$ 1010 m respectively. If the speed of the comet ...

A planet revolving in elliptical orbit has :A. a constant velocity of revolution.B. has the least velocity when it is nearest to the sun.C. its areal ...

Find the gravitational force of attraction between the ring and sphere as shown in the diagram, where the plane of the ring is perpendicular to the li...

A solid sphere of radius R gravitationally attracts a particle placed at 3R from its centre with a force F1. Now a spherical cavity of radius $$\left(...

Two satellites A and B of masses 200 kg and 400 kg are revolving round the earth at height of 600 km and 1600 km respectively.If TA and TB are the tim...

Given below are two statements : one is labelled as Assertion A and the other is labelled as Reason R.Assertion A : The escape velocities of planet A ...

A body weights 49N on a spring balance at the north pole. What will be its weight recorded on the same weighing machine, if it is shifted to the equat...

Four identical particles of equal masses 1 kg made to move along the circumference of a circle of radius 1 m under the action of their own mutual grav...

Consider two satellites S1 and S2 with periods of revolution 1 hr. and 8 hr. respectively revolving around a planet in circular orbits. The ratio of a...

Two stars of masses m and 2m at a distance d rotate about their common centre of mass in free space. The period of revolution is :

Two planets have masses M and 16 M and their radii are $$a$$ and 2$$a$$, respectively. The separation between the centres of the planets is 10$$a$$. A...

A satellite is in an elliptical orbit around a planet P. It is observed that the velocity of the satellite when it is farthest from the planet is 6 ti...

The acceleration due to gravity on the earth’s surface at the poles is g and angular velocity of the earth about the axis passing through the pole is ...

The value of the acceleration due to gravity is g1 at a height h = $${R \over 2}$$ (R = radius of the earth) from the surface of the earth. It is agai...

A body is moving in a low circular orbit about a planet of mass M and radius R. The radius of the orbit can be taken to be R itself. Then the ratio of...

On the x-axis and at a distance x from the origin, the gravitational field due a mass distribution is given by $${{Ax} \over {{{\left( {{x^2} + {a^2}}...

The mass density of a planet of radius R varies with the distance r from its centre as $$\rho $$(r) = $${\rho _0}\left( {1 - {{{r^2}} \over {{R^2}}}} ...

A satellite is moving in a low nearly circular orbit around the earth. Its radius is roughly equal to that of the earth’s radius Re . By firing rocket...

The height ‘h’ at which the weight of a body will be the same as that at the same depth ‘h’ from the surface of the earth is (Radius of the earth is R...

The mass density of a spherical galaxy varies as $${K \over r}$$ over a large distance ‘r’ from its centre. In that region, a small star is in a circu...

Planet A has mass M and radius R. Planet B has half the mass and half the radius of Planet A. If the escape velocities from the Planets A and B are vA...

A body A of mass m is moving in a circular orbit of radius R about a planet. Another body B of mass $${m \over 2}$$ collides with A with a velocity wh...

Consider two solid spheres of radii R1 = 1m, R2 = 2m and masses M1 and M2, respectively. The gravitational field due to sphere (1) and (2) are shown. ...

A box weight 196 N on a spring balance at the north pole. Its weight recorded on the same balance if it is shifted to the equator is close to (Take g ...

A satellite of mass m is launched vertically upwards with an initial speed u from the surface of the earth. After it reaches height R (R = radius of t...

The ratio of the weights of a body on the Earth’s surface to that on the surface of a planets is 9 : 4. The mass of the planet is $${1 \over 9}$$ th o...

A spaceship orbits around a planet at a height of 20 km from its surface. Assuming that only gravitational field of the planet acts on the spaceship, ...

The value of acceleration due to gravity at Earth's surface is 9.8 ms–2. The altitude above its surface at which the acceleration due to gravity decre...

A test particle is moving in a circular orbit in the gravitational field produced by a mass density $$\rho (r) = {K \over {{r^2}}}$$ . Identify the c...

A solid sphere of mass 'M' and radius 'a' is surrounded by a uniform concentric spherical shell of thickness 2a and mass 2M. The gravitational field a...

A rocket has to be launched from earth in such a way that it never returns. If E is the minimum energy delivered by the rocket launcher, what should b...

Four identical particles of mass M are located at the corners of a square of side 'a'. What should be their speed if each of them revolves under the i...

Two satellites, A and B, have masses m and 2m respectively. A is in a circular orbit of radius R, and B is in a circular orbit of radius 2R around the...

A straight rod of length L extends from x = a to x = L + a. The gravitational force it exerts on a point mass 'm' at x = 0, if the mass per unit lengt...

A satellite of mass M is in a circular orbit of radius R about the centre of the earth. A meteorite of the same mass, falling towards the earth, colli...

A satellite is revolving in a circular orbit at a height h form the earth surface, such that h < < R where R is the earth. Assuming that the eff...

Two stars of masses 3 $$ \times $$ 1031 kg each, and at distance 2 $$ \times $$ 1011 m rotate in a plane about their common centre of mass O. A meteor...

A satellite is moving with a constant speed v in circular orbit around the earth. An object of mass ‘m’ is ejected from the satellite such that it jus...

The energy required to take a satellite to a height 'h' above Earth surface (radius of Earth = 6.4 $$ \times $$ 103 km) is E1 and kinetic energy requi...

Suppose that the angular velocity of rotation of earth is increased. Then, as a consequence :

A particle is moving with a uniform speed in a circular orbit of radius R in a central force inversely proportional to the nth power of R. If the peri...

A body of mass m is moving in a circular orbit of radius R about a planet of mass M. At some instant, it splits into two equal masses. The first mass ...

Take the mean distance of the moon and the sun from the earth to be $$0.4 \times {10^6}$$ km and $$150 \times {10^6}$$ km respectively. Their masses a...

The mass density of a spherical body is given by $$\rho $$ (r) = $${k \over r}$$ for r $$ \le $$ R and $$\rho $$ (r) = 0 for r > R, where r is the...

If the Earth has no rotational motion, the weight of a person on the equator is W. Determine the speed with which the earth would have to rotate about...

The variation of acceleration due to gravity $$g$$ with distance d from centre of the earth is best represented by (R = Earth’s radius):

Figure shows elliptical path abcd of a planet around the sun S such that the area of triangle csa is $${1 \over 4}$$ the area of the ellipse. (See fig...

A satellite is revolving in a circular orbit at a height $$'h'$$ from the earth's surface (radius of earth $$R;h < < R$$). The minimum increas...

From a solid sphere of mass $$M$$ and radius $$R,$$ a spherical portion of radius $$R/2$$ is removed, as shown in the figure. Taking gravitational pot...

Four particles, each of mass $$M$$ and equidistant from each other, move along a circle of radius $$R$$ under the action of their mutual gravitational...

What is the minimum energy required to launch a satellite of mass $$m$$ from the surface of a planet of mass $$M$$ and radius $$R$$ in a circular orbi...

The mass of a spaceship is $$1000$$ $$kg.$$ It is to be launched from the earth's surface out into free space. The value of $$g$$ and $$R$$ (radius of...

Two bodies of masses $$m$$ and $$4$$ $$m$$ are placed at a distance $$r.$$ The gravitational potential at a point on the line joining them where the g...

The height at which the acceleration due to gravity becomes $${g \over 9}$$ (where $$g=$$ the acceleration due to gravity on the surface of the earth)...

A planet in a distant solar system is $$10$$ times more massive than the earth and its radius is $$10$$ times smaller. Given that the escape velocity ...

This question contains Statement - $$1$$ and Statement - $$2$$. of the four choices given after the statements, choose the one that best describes the...

If $${g_E}$$ and $${g_M}$$ are the accelerations due to gravity on the surfaces of the earth and the moon respectively and if Millikan's oil drop exp...

A particle of mass $$10$$ $$g$$ is kept on the surface of a uniform sphere of mass $$100$$ $$kg$$ and radius $$10$$ $$cm.$$ Find the work to be done a...

The change in the value of $$g$$ at a height $$h$$ above the surface of the earth is the same as at a depth $$d$$ below the surface of earth. When bot...

Average density of the earth

A satellite of mass $$m$$ revolves around the earth of radius $$R$$ at a height $$x$$ from its surface. If $$g$$ is the acceleration due to gravity on...

If $$g$$ is the acceleration due to gravity on the earth's surface, the gain in the potential energy of an object of mass $$m$$ raised from the surfac...

The time period of an earth satellite in circular orbit is independent of

Suppose the gravitational force varies inversely as the nth power of distance. Then the time period of a planet in circular orbit of radius $$R$$ arou...

The time period of satellite of earth is $$5$$ hours. If the separation between the earth and the satellite is increased to $$4$$ times the previous v...

The escape velocity for a body projected vertically upwards from the surface of earth is $$11$$ $$km/s.$$ If the body is projected at an angle of $${4...

Two spherical bodies of mass $$M$$ and $$5M$$ & radii $$R$$ & $$2R$$ respectively are released in free space with initial separation between t...

If suddenly the gravitational force of attraction between Earth and a satellite revolving around it becomes zero, then the satellite will

The kinetic energy needed to project a body of mass $$m$$ from the earth surface (radius $$R$$) to infinity is

Energy required to move a body of mass $$m$$ from an orbit of radius $$2R$$ to $$3R$$ is

The escape velocity of a body depends upon mass as

If the radius of earth is reduced to three-fourth of its present value without change in its mass then value of duration of the day of earth will be _...

A simple pendulum is placed at a place where its distance from the earth's surface is equal to the radius of the earth. If the length of the string is...

If the earth suddenly shrinks to $$\frac{1}{64}$$th of its original volume with its mass remaining the same, the period of rotation of earth becomes $...

If the acceleration due to gravity experienced by a point mass at a height h above the surface of earth is same as that of the acceleration due to gra...

Two satellites S1 and S2 are revolving in circular orbits around a planet with radius R1 = 3200 km and R2 = 800 km respectively. The ratio of speed of...

Two satellites revolve around a planet in coplanar circular orbits in anticlockwise direction. Their period of revolutions are 1 hour and 8 hours resp...

A body of mass (2M) splits into four masses (m, M $$-$$ m, m, M $$-$$ m}, which are rearranged to form a square as shown in the figure. The ratio of $...

Suppose two planets (spherical in shape) in radii R and 2R, but mass M and 9M respectively have a centre to centre separation 8 R as shown in the figu...

The radius in kilometer to which the present radius of earth (R = 6400 km) to be compressed so that the escape velocity is increased 10 times is _____...

If one wants to remove all the mass of the earth to infinity in order to break it up completely.The amount of energy that needs to be supplied will be...

In the reported figure of earth, the value of acceleration due to gravity is same at point A and C but it is smaller than that of its value at point B...

The initial velocity vi required to project a body vertically upward from the surface of the earth to reach a height of 10R, where R is the radius of ...

A ball is dropped from the top of a 100 m high tower on a planet. In the last $${1 \over 2}s$$ before hitting the ground, it covers a distance of 19 m...

An asteroid is moving directly towards the centre of the earth. When at a distance of 10R (R is the radius of the earth) from the earths centre, it ha...

An astronaut of mass m is working on a satellite orbiting the earth at a distance h from the earth’s surface. The radius of the earth is R, while its ...