Gravitation · JEE Main previous year questions

Match List I with List II List-I: A. AM Broadcast B. FM Broadcast C. Television D. Satellite Communication List-II: I. 88-108 MHz II. 540-1600 kHz III. 3.7-4.2 GHz IV. 54MHz - 890MHz Choose the correct answer from the options given below :

1. A.A-IV, B-III, C-I, D-II
2. B.A-II, B-I, C-IV, D-III
3. C.A-I, B-III, C-II, D-IV
4. D.A-II, B-III, C-I, D-IV

If the acceleration due to gravity experienced by a point mass at a height $\mathrm{h}$ above the surface of earth is same as that of the acceleration due to gravity at a depth $\alpha \mathrm{h}(\mathrm{h}\ll {\mathrm{R}}_{\mathrm{e}})$ from the earth surface. The value of $\alpha$ will be ____. (use ${\mathrm{R}}_{\mathrm{e}}=6400\mathrm{km}$)

A balloon has mass of $10\text{ g}$ in air. The air escapes from the balloon at a uniform rate with velocity $4.5\text{ cm/s}$. If the balloon shrinks in $5\text{ s}$ completely. Then, the average force acting on that balloon will be (in dyne).

1. A.3
2. B.9
3. C.12
4. D.18

1. A.$11.6\text{ m/s}$
2. B.$10.8\text{ m/s}$
3. C.$17.8\text{ m/s}$
4. D.$14.4\text{ m/s}$

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

1. A.$9:16$
2. B.$16:9$
3. C.$1:1$
4. D.$4:3$

A body of mass $\mathrm{m}$ is projected with velocity $\lambda v_e$ in vertically upward direction from the surface of the earth into space. It is given that $v_e$ 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 : ($\mathrm{R}$ : radius of earth)

1. A.$\frac{\mathrm{R}}{1+{\lambda }^2}$
2. B.$\frac{\mathrm{R}}{1-{\lambda }^2}$
3. C.$\frac{\mathrm{R}}{1-\lambda }$
4. D.$\frac{{\lambda }^2\mathrm{R}}{1-{\lambda }^2}$

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 :

1. A.$9:4$
2. B.$9:8$
3. C.$9:10$
4. D.$2:5$

Water falls from a $40\text{ m}$ high dam at the rate of $9\times 10^4\text{ kg}$ per hour. Fifty percentage of gravitational potential energy can be converted into electrical energy. Using this hydroelectric energy number of $100\text{ W}$ lamps, that can be lit, is : (Take $g=10{\text{ ms}}^{-2}$)

1. A.$25$
2. B.$50$
3. C.$100$
4. D.$18$

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.

1. A.Both Statement I and Statement II are true
2. B.Both Statement I and Statement II are false
3. C.Statement I is true but Statement II is false
4. D.Statement I is false but Statement II is true

1. A.$-\frac{Gm}{d}\left[(4+\sqrt{2})m+4\sqrt{2}M\right]$
2. B.$-\frac{Gm}{d}\left[(4+\sqrt{2})M+4\sqrt{2}m\right]$
3. C.$-\frac{Gm}{d}\left[3m^2+4\sqrt{2}M\right]$
4. D.$-\frac{Gm}{d}\left[6m^2+4\sqrt{2}M\right]$

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).

1. A.g/2
2. B.g/4
3. C.g/3
4. D.g/9

Two satellites ${\mathrm{S}}_1$ and ${\mathrm{S}}_2$ are revolving in circular orbits around a planet with radius ${\mathrm{R}}_1=3200\mathrm{km}$ and ${\mathrm{R}}_2=800\mathrm{km}$ respectively. The ratio of speed of satellite ${\mathrm{S}}_1$ to be speed of satellite ${\mathrm{S}}_2$ in their respective orbits would be $\frac{1}{x}$ where $x=\underline{}$.

1. A.$\frac{1}{2}\sqrt{\frac{GM}{R(2\sqrt{2}+1)}}$
2. B.$\frac{1}{2}\sqrt{\frac{GM}{R}(2\sqrt{2}+1)}$
3. C.$\frac{1}{2}\sqrt{\frac{GM}{R}(2\sqrt{2}-1)}$
4. D.$\sqrt{\frac{GM}{R}}$

Two satellites revolve around a planet in coplanar circular orbits in anticlockwise direction. Their period of revolutions are 1 hour and 8 hours respectively. The radius of the orbit of nearer satellite is $2\times 10^3\mathrm{km}$. The angular speed of the farther satellite as observed from the nearer satellite at the instant when both the satellites are closest is $\frac{\pi }{x}\mathrm{rad}{\mathrm{h}}^{-1}$ where x is ____.

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 of each particle is :

1. A.$\sqrt{\frac{G}{2R^3}}$
2. B.$\frac{1}{2}\sqrt{\frac{G}{R^3}}$
3. C.$\frac{1}{2R}\sqrt{\frac{1}{G}}$
4. D.$\frac{2G}{R^3}$

The planet Mars has two moons, if one of them has a period 7 hours, 30 minutes and an orbital radius of $9.0\times 10^3\text{ km}$. Find the mass of Mars. $\left\{Given\frac{4{\pi }^2}{G}=6\times 10^{11}{\mathrm{N}}^{-1}{\mathrm{m}}^{-2}\mathrm{k}{\mathrm{g}}^2\right\}$

1. A.$5.96\times 10^{19}\text{ kg}$
2. B.$3.25\times 10^{21}\text{ kg}$
3. C.$7.02\times 10^{25}\text{ kg}$
4. D.$6.00\times 10^{23}\text{ kg}$

The minimum and maximum distances of a planet revolving around the sun are $x_1$ and $x_2$. If the minimum speed of the planet on its trajectory is $v_0$ then its maximum speed will be :

1. A.$\frac{v_0{x}_1^2}{x_2^2}$
2. B.$\frac{v_0{x}_2^2}{x_1^2}$
3. C.$\frac{v_0{x}_1}{x_2}$
4. D.$\frac{v_0{x}_2}{x_1}$

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 an object on earth is W, the weight of the same object on that planet will be :

1. A.2W
2. B.W
3. C.$2^{\frac{1}{3}}$W
4. D.$\sqrt{2}$W

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 :

1. A.9 T
2. B.27 T
3. C.12 T
4. D.3 T

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 velocity is directly proportional to its velocity. D. areal velocity is inversely proportional to its velocity. E. to follow a trajectory such that the areal velocity is constant. Choose the correct answer from the options given below :

1. A.D only
2. B.E only
3. C.C only
4. D.A only

1. A.$\frac{2\sqrt{2}}{3}\cdot \frac{GMm}{R^2}$
2. B.$\frac{\sqrt{8}}{9}\cdot \frac{GMm}{R}$
3. C.$\frac{\sqrt{8}}{27}\cdot \frac{GMm}{R^2}$
4. D.$\frac{1}{3\sqrt{8}}\cdot \frac{GMm}{R^2}$

The initial velocity $v_i$ 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 the earth, may be described in terms of escape velocity $v_e$ such that $v_i=\sqrt{\frac{x}{y}}\times v_e$. The value of x will be ______ .

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 times less than that when it is closest to the planet. The ratio of distances between the satellite and the planet at closest and farthest points is:

1. A.1 : 2
2. B.1 : 3
3. C.1 : 6
4. D.3 : 4

Two planets have masses M and 16 M and their radii are $a$ and $2a$, respectively. The separation between the centres of the planets is $10a$. A body of mass m is fired from the surface of the larger planet towards the smaller planet along the line joining their centres. For the body to be able to reach at the surface of smaller planet, the minimum firing speed needed is :

1. A.$2\sqrt{\frac{GM}{a}}$
2. B.$\sqrt{\frac{G{M}^2}{ma}}$
3. C.$\frac{3}{2}\sqrt{\frac{5GM}{a}}$
4. D.$4\sqrt{\frac{GM}{a}}$

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 ${\mathrm{R}}_{\mathrm{e}}$. By firing rockets attached to it, its speed is instantaneously increased in the direction of its motion so that it become $\sqrt{\frac{3}{2}}$ times larger. Due to this the farthest distance from the centre of the earth that the satellite reaches is R. Value of R is :

1. A.$2{\mathrm{R}}_{\mathrm{e}}$
2. B.$3{\mathrm{R}}_{\mathrm{e}}$
3. C.$4{\mathrm{R}}_{\mathrm{e}}$
4. D.$2.5{\mathrm{R}}_{\mathrm{e}}$