Gravitation — JEE Main PYQs

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 $g = 9.8 \text{ m/s}^2$)

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 _______.

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 km.)

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

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

A satellite of mass 1000 kg is launched to revolve around the earth in an orbit at a height of 270 km from the earth's surface. Kinetic energy of the satellite in this orbit is _________ $\times10^{10}$ J.

(Mass of earth $= 6\times10^{24}$ kg, Radius of earth $= 6.4\times10^{6}$ m, Gravitational constant $= 6.67\times10^{-11}~\mathrm{Nm^2kg^{-2}}$)

Three identical spheres of mass m, are placed at the vertices of an equilateral triangle of length $a$. When released, they interact only through gravitational force and collide after a time T = 4 seconds. If the sides of the triangle are increased to length $2a$ and also the masses of the spheres are made $2m$, then they will collide after __________ seconds.

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

An object is kept at rest at a distance of 3R 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 M = mass of earth, G = Universal gravitational constant)

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 :

A small point of mass m is placed at a distance 2R 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 $F_1$. A spherical part of radius 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

[Figure: solid sphere of mass M with centre O; a small spherical cavity carved out between O and the surface along the line OP; point mass m at P with OP = 2R]

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 satellite of mass $\frac{M}{2}$ is revolving around the earth in a circular orbit at a height of $\frac{R}{3}$ from the surface of the earth. The angular momentum of the satellite is $M\sqrt{\frac{GMR}{x}}$. The value of $x$ is __________, where M and R are the mass and radius of the earth, respectively.

(G is the gravitational constant)

A satellite is launched into a circular orbit of radius 'R' around the earth. A second satellite is launched into an orbit of radius 1.03 R. The time period of revolution of the second satellite is larger than the first one approximately by

Acceleration due to gravity on the surface of earth is 'g'. If the diameter of earth is reduced to one third of its original value and mass remains unchanged, then the acceleration due to gravity on the surface of the earth is ______ g.

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

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 m is placed at the centre of the arc. The gravitational force on the particle by the wire is :

A 90 kg body placed at 2R distance from surface of earth experiences gravitational pull of :

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

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

Match List I with List II :

List I	List II
(A) Kinetic energy of planet	(I) $-\frac{GMm}{a}$
(B) Gravitation Potential energy of sun-planet system	(II) $\frac{GMm}{2a}$
(C) Total mechanical energy of planet	(III) $\frac{Gm}{r}$
(D) Escape energy at the surface of planet for unit mass object	(IV) $-\frac{GMm}{2a}$

(Where a = radius of planet orbit, r = radius of planet, M = mass of Sun, m = mass of planet)

Choose the correct answer from the options given below :

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 $4.2 \times 10^4\ \mathrm{km}$)

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):

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

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

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

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 km. From earth's surface to the orbit, the change in total mechanical energy of the ball is $x\dfrac{GM_e m}{21R_e}$. The value of $x$ is (take $R_e = 6370$ km) :

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

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

A light planet is revolving around a massive star in a circular orbit of radius R with a period of revolution T. If the force of attraction between planet and star is proportional to $R^{-3/2}$ then choose the correct option :

The acceleration due to gravity on the surface of earth is 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 :

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

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

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

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 $\left(\frac{2\sqrt{2}+1}{32}\right)\frac{Gm^2}{L^2}$, the length of the sides of the square 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 :