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question_answer1)
Let \[\omega \] be the angular velocity of the earth's rotation about its axis. Assume that the acceleration due to gravity on the earth's surface has the same value at the equator and the poles. An object weighed at the equator gives the same reading as a reading taken at a depth d below earth's surface at a pole\[(d<<R)\]. The value of d is
A)
\[\frac{{{\omega }^{2}}{{R}^{2}}}{g}\] done
clear
B)
\[\frac{{{\omega }^{2}}{{R}^{2}}}{2g}\] done
clear
C)
\[\frac{2{{\omega }^{2}}{{R}^{2}}}{g}\] done
clear
D)
\[\frac{\sqrt{Rg}}{g}\] done
clear
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question_answer2)
In a gravitational field, at a point where the gravitational potential is zero
A)
the gravitational field is necessarily zero done
clear
B)
the gravitational field is not necessarily zero done
clear
C)
any value between one and infinite done
clear
D)
None of these done
clear
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question_answer3)
The gravitational field strength due to a solid sphere (mass M, radius R) varies with distance r from centre as
A)
B)
C)
D)
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question_answer4)
A particle of mass M is situated at the centre of a spherical shell of same mass and radius a. The gravitational potential at a point situated at \[\frac{a}{2}\] distance from the centre, will be:
A)
\[-\frac{3GM}{a}\] done
clear
B)
\[-\frac{2GM}{a}\] done
clear
C)
\[-\frac{GM}{a}\] done
clear
D)
\[-\frac{4GM}{a}\] done
clear
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question_answer5)
Assuming the radius of the earth as R, the change in gravitational potential energy of a body of mass m, when it is taken from the earth's surface to a height 3R above its surface, is
A)
\[3\text{ }mgR\] done
clear
B)
\[\frac{3}{4}mgR\] done
clear
C)
\[\text{1 }mgR\] done
clear
D)
\[\frac{3}{2}mgR\] done
clear
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question_answer6)
The figure shows a planet in elliptical orbit around the sun 5'. Where is the kinetic energy of the planet maximum?
A)
\[{{P}_{1}}\] done
clear
B)
\[{{P}_{2}}\] done
clear
C)
\[{{P}_{3}}\] done
clear
D)
\[{{P}_{4}}\] done
clear
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question_answer7)
A planet revolves about the sun in elliptical orbit. The areal velocity \[\left( \frac{dA}{dt} \right)\] of the planet is\[4.0\times 1{{0}^{16}} m/s\]. The least distance between planet and the sun is\[2\times {{10}^{12}}m\]. Then the maximum speed of the planet in km/s is
A)
10 done
clear
B)
20 done
clear
C)
30 done
clear
D)
40 done
clear
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question_answer8)
In planetary motion the areal velocity of position vector of a planet depends on angular velocity\[(\omega )\] and the distance of the planet from sun (r). If so, the correct relation for areal velocity is
A)
\[\frac{dA}{dt}\propto \omega r\] done
clear
B)
\[\frac{dA}{dt}\propto {{\omega }^{2}}r\] done
clear
C)
\[\frac{dA}{dt}\propto \omega {{r}^{2}}\] done
clear
D)
\[\frac{dA}{dt}\propto \sqrt{\omega r}\] done
clear
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question_answer9)
The period of moon's rotation around the earth is nearly 29 days. If moon's mass were 2 fold its present value and all other things remain unchanged, the period of moon's rotation would be nearly
A)
\[29\sqrt{2} days\] done
clear
B)
\[29/\sqrt{2} days\] done
clear
C)
\[29\times 2 days\] done
clear
D)
29 days done
clear
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question_answer10)
Which of the following graphs represents the moon planet moving about the sun?
A)
B)
C)
D)
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question_answer11)
If three equal masses m are placed at the three vertices of an equilateral triangle of side \lm then what force acts on a particle of mass 2m placed at the centroid?
A)
\[G{{m}^{2}}\] done
clear
B)
\[2G{{m}^{2}}\] done
clear
C)
Zero done
clear
D)
\[-G{{m}^{2}}\] done
clear
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question_answer12)
Two particles of equal mass 'm' go around a circle of radius R under the action of their mutual gravitational attraction. The speed of each particle with respect to their centre of mass is
A)
\[\sqrt{\frac{Gm}{4R}}\] done
clear
B)
\[\sqrt{\frac{Gm}{3R}}\] done
clear
C)
\[\sqrt{\frac{Gm}{2R}}\] done
clear
D)
\[\sqrt{\frac{Gm}{R}}\] done
clear
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question_answer13)
If suddenly the gravitational force of attraction between Earth and a body revolving around it becomes zero, then the body will
A)
Continue to move in its orbit with same velocity done
clear
B)
Move tangentially to the original orbit with same velocity done
clear
C)
Become stationary in its orbit done
clear
D)
Move towards the earth done
clear
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question_answer14)
A straight rod of length L extends from \[x=a\]to\[x=L+a\]. Find the gravitational force it exerts on a point mass m at x = 0 if the linear density of rod\[\mu =A+B{{x}^{2}}\].
A)
\[Gm\left[ \frac{A}{a}+BL \right]\] done
clear
B)
\[Gm\left[ A\left( \frac{1}{a}-\frac{1}{a+L} \right)+BL \right]\] done
clear
C)
\[Gm\left[ BL+\frac{A}{a+L} \right]\] done
clear
D)
\[Gm\left[ BL+\frac{A}{a} \right]\] done
clear
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question_answer15)
Two concentric uniform shells of mass \[{{M}_{1}}\] and \[{{M}_{2}}\] are as shown in the figure. A particle of mass m is located just within the shell \[{{M}_{2}}\] on its inner surface. Gravitational force on 'm' due to \[{{M}_{1}}\] and \[{{M}_{2}}\] will be
A)
Zero done
clear
B)
\[\frac{G{{M}_{1}}m}{{{b}^{2}}}\] done
clear
C)
\[\frac{G\left( {{M}_{1}}+{{M}_{2}} \right)m}{{{b}^{2}}}\] done
clear
D)
None of these done
clear
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question_answer16)
The figure shows elliptical orbit of a planet m about the sun S. The shaded area SCD is twice the shaded area SAB. If \[{{t}_{1}}\] is the time for the planet to move from C to D and \[{{t}_{2}}\] is the time to move from A to B then
A)
\[{{t}_{i}}=4{{t}_{2}}\] done
clear
B)
\[{{t}_{i}}=2{{t}_{2}}\] done
clear
C)
\[{{t}_{i}}={{t}_{2}}\] done
clear
D)
\[{{t}_{i}}>{{t}_{2}}\] done
clear
View Solution play_arrow
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question_answer17)
If the distance of earth is halved from the sun, then the no. of days in a year will be
A)
365 done
clear
B)
182.5 done
clear
C)
730 done
clear
D)
129 done
clear
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question_answer18)
If the earth is at one-fourth of its present distance from the sun, the duration of the year will be
A)
half the present year done
clear
B)
one-eighth the present year done
clear
C)
one-sixth the present year done
clear
D)
one-tenth the present year done
clear
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question_answer19)
The distance of Neptune and Saturn from the sun is nearly \[1{{0}^{13}} and 1{{0}^{12}}\] meter respectively. Assuming that they move in circular orbits, their periodic times will be in the ratio
A)
10 done
clear
B)
100 done
clear
C)
\[10\sqrt{10}\] done
clear
D)
1000 done
clear
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question_answer20)
A satellite A of mass m is at a distance of r from the surface of the earth. Another satellite B of mass 2m is at a distance of 2r from the earth's centre. Their time periods are in the ratio of
A)
1:2 done
clear
B)
1:16 done
clear
C)
1:32 done
clear
D)
\[1:2\sqrt{2}\] done
clear
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question_answer21)
The time taken by the earth to travel over half its orbit, remote from the sun, separated by the minor axis is about 2 days more than half the year, then the eccentricity of the orbit is
A)
1/30 done
clear
B)
1/60 done
clear
C)
1/15 done
clear
D)
1/70 done
clear
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question_answer22)
For a particle inside a uniform spherical shell, the gravitational force on the particle is
A)
Infinite done
clear
B)
zero done
clear
C)
\[\frac{-G{{m}_{1}}{{m}_{2}}}{{{r}^{2}}}\] done
clear
D)
\[\frac{G{{m}_{1}}{{m}_{2}}}{{{r}^{2}}}\] done
clear
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question_answer23)
Two spheres of masses m and M are situated in air and the gravitational force between them is F. The space around the masses is now filled with a liquid of specific gravity 3. The gravitational force will now be
A)
\[\frac{F}{9}\] done
clear
B)
3F done
clear
C)
F done
clear
D)
\[\frac{F}{3}\] done
clear
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question_answer24)
A mass M is split into two parts m and (M- w), which are then separated by a certain distance. What ratio of m/M maximizes the gravitational force between the two parts?
A)
\[\frac{1}{3}\] done
clear
B)
\[\frac{1}{2}\] done
clear
C)
\[\frac{1}{4}\] done
clear
D)
\[\frac{1}{5}\] done
clear
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question_answer25)
Two stars of mass \[{{\operatorname{m}}_{1}} and {{m}_{2}}\]are parts of a binary system. The radii of their orbits are \[{{r}_{1}} and {{r}_{2}}\]respectively, measured from the C.M. of the system. The magnitude of gravitational force \[{{\operatorname{m}}_{1}}\]exerts on \[{{m}_{2}}\]is
A)
\[\frac{{{m}_{1}}{{m}_{2}}G}{{{\left( {{r}_{1}}+{{r}_{2}} \right)}^{2}}}\] done
clear
B)
\[\frac{{{m}_{1}}G}{{{\left( {{r}_{1}}+{{r}_{2}} \right)}^{2}}}\] done
clear
C)
\[\frac{{{m}_{2}}G}{{{\left( {{r}_{1}}+{{r}_{2}} \right)}^{2}}}\] done
clear
D)
\[\frac{\left( {{m}_{1}}+{{m}_{2}} \right)}{{{\left( {{r}_{1}}+{{r}_{2}} \right)}^{2}}}\] done
clear
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question_answer26)
Two bodies of masses 4 kg and 9 kg are separated by a distance of 60 cm. A 1 kg mass is placed in between these two masses. If the net force on 1 kg is zero, then its distance from 4 kg mass is
A)
26 cm done
clear
B)
30 cm done
clear
C)
28 cm done
clear
D)
24 cm done
clear
View Solution play_arrow
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question_answer27)
If masses of two point objects is doubled and distance between them is tripled, then gravitational force of attraction between them will nearly
A)
Increase by 225% done
clear
B)
decrease by 44% done
clear
C)
Decrease by 56% done
clear
D)
increase by 125% done
clear
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question_answer28)
A central particle M is surrounded by a square array of other particles, separated by either distance d or distance d/2 along the perimeter of the square. The magnitude of the gravitational force on the central particle due to the other particles is
A)
\[\frac{9GMm}{{{d}^{2}}}\] done
clear
B)
\[\frac{5GMm}{{{d}^{2}}}\] done
clear
C)
\[\frac{3GMm}{{{d}^{2}}}\] done
clear
D)
\[\frac{GMm}{{{d}^{2}}}\] done
clear
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question_answer29)
Three particles P, Q and R are placed as per given figure. Masses of P, Q and R are \[\sqrt{3}m,\,\,\sqrt{3}m\] and m respectively. The gravitational force on a fourth particle S of mass m is equal to
A)
\[\frac{\sqrt{3}G{{M}^{2}}}{2{{d}^{2}}}\]in ST direction only done
clear
B)
\[\frac{\sqrt{3}G{{M}^{2}}}{2{{d}^{2}}}\]in SQ direction and \[\frac{\sqrt{3}G{{M}^{2}}}{2{{d}^{2}}}\]in SU direction done
clear
C)
\[\frac{\sqrt{3}G{{M}^{2}}}{2{{d}^{2}}}\] in SQ direction only done
clear
D)
\[\frac{\sqrt{3}G{{M}^{2}}}{2{{d}^{2}}}\]in SQ direction and \[\frac{\sqrt{3}G{{M}^{2}}}{2{{d}^{2}}}\] in ST direction done
clear
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question_answer30)
The acceleration due to gravity on the planet A is 9 times the acceleration due to gravity on planet B. A man jumps to a height of 2m on the surface of A. What is the height of jump by the same person on the planet B?
A)
\[\frac{2}{3}m\] done
clear
B)
\[\frac{2}{9}m\] done
clear
C)
18 m done
clear
D)
6 m done
clear
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question_answer31)
Assume that the acceleration due to gravity on the surface of the moon is 0.2 times the acceleration due to gravity on the surface of the earth. If \[{{\operatorname{R}}_{e}}\]is the maximum range of a projectile on the earth's surface, what is the maximum range on the surface of the moon for the same velocity of projection
A)
\[0.2{{\operatorname{R}}_{e}}\] done
clear
B)
\[2{{\operatorname{R}}_{e}}\] done
clear
C)
\[0.5{{\operatorname{R}}_{e}}\] done
clear
D)
\[5{{\operatorname{R}}_{e}}\] done
clear
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question_answer32)
The height of the point vertically above the earth's surface, at which acceleration due to gravity becomes 1 % of its value at the earth's surface is (Radius of the earth = R)
A)
8 R done
clear
B)
9 R done
clear
C)
10 R done
clear
D)
20 R done
clear
View Solution play_arrow
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question_answer33)
The speed of earth's rotation about its axis is cd. Its speed is increases to x times to make the effective acceleration due to gravity equal to zero at the equator. Then x is:
A)
1 done
clear
B)
8.5 done
clear
C)
17 done
clear
D)
34 done
clear
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question_answer34)
A particle is suspended from a spring and it stretches the spring by 1 cm on the surface of earth. By how much amount the same particle will stretch the same spring at a place 800 km above the surface of earth.
A)
1.59 cm done
clear
B)
2.38 cm done
clear
C)
0.79 cm done
clear
D)
1.38 cm done
clear
View Solution play_arrow
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question_answer35)
The density of a newly discovered planet is twice that of earth. The acceleration due to gravity at the surface of the planet is equal to that at the surface of the earth. If the radius of the earth is R, the radius of the planet would be-
A)
1/2 R done
clear
B)
2 R done
clear
C)
4 R done
clear
D)
¼ R done
clear
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question_answer36)
Imagine a new planet having the same density as that of earth but it is 3 times bigger than the earth in size. If the acceleration due to gravity on the surface of earth is g and that on the surface of the new planet is g?, then
A)
\[g'=g/9\] done
clear
B)
\[g'=27g\] done
clear
C)
\[g=9g\] done
clear
D)
\[g'=3g\] done
clear
View Solution play_arrow
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question_answer37)
If the earth were to rotates faster than its present speed, the weight of an object will
A)
Increase at the equator but remain unchanged at the poles done
clear
B)
Decrease at the equator but remain unchanged at the poles done
clear
C)
Remain unchanged at the equator but decrease at the poles done
clear
D)
Remain unchanged at the equator but increase at the poles done
clear
View Solution play_arrow
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question_answer38)
Four similar particles of mass m are orbiting in a circle of radius r in the same angular direction because of their mutual gravitational attractive force. Then, velocity of a particle is given by
A)
\[{{\left[ \frac{Gm}{r}\left( \frac{1+2\sqrt{2}}{4} \right) \right]}^{\frac{1}{2}}}\] done
clear
B)
\[\sqrt{\frac{Gm}{r}}\] done
clear
C)
\[\sqrt{\frac{Gm}{r}}\left( 1+2\sqrt{2} \right)\] done
clear
D)
\[a{{\left[ \frac{1}{2}\frac{Gm}{r}\left( \frac{1+2\sqrt{2}}{2} \right) \right]}^{\frac{1}{2}}}\] done
clear
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question_answer39)
Consider two solid uniform spherical objects of the same density p. One has radius R and the other has radius 2R. They are in outer space where the gravitational field from other objects are negligible. If they are arranged with their surface touching, what is the contact force between the objects due to their traditional attraction?
A)
\[G{{\pi }^{2}}{{R}^{2}}\] done
clear
B)
\[\frac{128}{81}G{{\pi }^{2}}{{R}^{4}}{{\rho }^{2}}\] done
clear
C)
\[\frac{128}{81}G{{\pi }^{2}}\] done
clear
D)
\[\frac{128}{81}{{\pi }^{2}}{{R}^{4}}G\] done
clear
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question_answer40)
Two spherical bodies of mass M and 5M& radii R & 2R respectively are released in free space with initial separation between their centres equal to 12 R. If they attract each other due to gravitational force only, then the distance covered by the smaller body just before collision is
A)
2.5 R done
clear
B)
4.5 R done
clear
C)
7.5 R done
clear
D)
1.5 R done
clear
View Solution play_arrow
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question_answer41)
From a sphere of mass M and radius R, a smaller sphere of radius \[\frac{R}{2}\] is carved out such that the cavity made in the original sphere is between its centre and the periphery (See figure). For the configuration in the figure where the distance between the centre of the original sphere and the removed sphere is 3R, the gravitational force between the two sphere is
A)
\[\frac{41\,G{{M}^{2}}}{3600\,{{R}^{2}}}\] done
clear
B)
\[\frac{41\,G{{M}^{2}}}{450\,{{R}^{2}}}\] done
clear
C)
\[\frac{59\,G{{M}^{2}}}{450\,{{R}^{2}}}\] done
clear
D)
\[\frac{G{{M}^{2}}}{225\,{{R}^{2}}}\] done
clear
View Solution play_arrow
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question_answer42)
The value of acceleration due to gravity on moving from equator to poles will
A)
Decrease done
clear
B)
increase done
clear
C)
Remain same done
clear
D)
become half done
clear
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question_answer43)
The weight of a body at the centre of the earth is
A)
Zero done
clear
B)
Infinite done
clear
C)
Same as on the surface of earth done
clear
D)
None of these done
clear
View Solution play_arrow
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question_answer44)
Which of the following graphs shows the correct variation of acceleration due to gravity with the height above the earth's surface?
A)
B)
C)
D)
None of these done
clear
View Solution play_arrow
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question_answer45)
At what height from the ground will the value of g be the same as that in 10 km deep mine below the surface of earth?
A)
20 km done
clear
B)
10 km done
clear
C)
15 km done
clear
D)
5 km done
clear
View Solution play_arrow
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question_answer46)
The ratio of radii of earth to another planet is 21 3 and the ratio of their mean densities is 4/5. If an astronaut can jump to a maximum height of 1.5 m on the earth, with the same effort, the maximum height he can jump on the planet is
A)
1 m done
clear
B)
0.8 m done
clear
C)
0.5 m done
clear
D)
125 m done
clear
View Solution play_arrow
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question_answer47)
The value of 'g' at a particular point is\[9.8 m/{{s}^{2}}\]. Suppose the earth suddenly shrinks uniformly to half its present size without losing any mass. The value of 'g? at the same point (assuming that the distance of the point from the centre of the earth does not shrink) will now be
A)
\[4.9\,\,m/se{{c}^{2}}\] done
clear
B)
\[3.1\,\,m/se{{c}^{2}}\] done
clear
C)
\[9.8\,\,m/se{{c}^{2}}\] done
clear
D)
\[19.6\,\,m/se{{c}^{2}}\] done
clear
View Solution play_arrow
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question_answer48)
R is the radius of the earth and co is its angular velocity and \[{{\operatorname{g}}_{p}}\]is the value of g at the poles. The effective value of g at the latitude \[\lambda = 60{}^\circ \] will be equal to
A)
\[{{g}_{p}}-\frac{1}{4}R{{\omega }^{2}}\] done
clear
B)
\[{{g}_{p}}-\frac{3}{4}R{{\omega }^{2}}\] done
clear
C)
\[{{g}_{p}}-R{{\omega }^{2}}\] done
clear
D)
\[{{g}_{p}}+\frac{1}{4}R{{\omega }^{2}}\] done
clear
View Solution play_arrow
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question_answer49)
What should be the velocity of rotation of earth due to rotation about its own axis so that the weight of a person becomes \[\frac{2}{3}\] of the present weight at the equator? Equatorial radius of the earth is R
A)
\[{{\left( \frac{2g}{3R} \right)}^{\frac{1}{2}}}\] done
clear
B)
\[{{\left( \frac{g}{3R} \right)}^{\frac{1}{2}}}\] done
clear
C)
\[{{\left( \frac{g}{7R} \right)}^{\frac{1}{2}}}\] done
clear
D)
\[{{\left( \frac{g}{5R} \right)}^{\frac{1}{2}}}\] done
clear
View Solution play_arrow
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question_answer50)
If the radius of the earth were to shrink by 1 %, with its mass remaining the same, the acceleration due to gravity on the earth's surface would
A)
Decrease by 1% done
clear
B)
decrease by 2% done
clear
C)
Increase by 1% done
clear
D)
increase by 2% done
clear
View Solution play_arrow
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question_answer51)
While approaching a planet circling a distant star, an astronaut determines the radius of a planet is half of that of the earth. After landing on its surface, he finds its acceleration due to gravity is twice as that of the earth. The ratio of the mass of planet and that of the earth.
A)
1 :2 done
clear
B)
2:3 done
clear
C)
3:4 done
clear
D)
4:5 done
clear
View Solution play_arrow
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question_answer52)
The depth d at which the value of acceleration due to gravity becomes \[\frac{1}{n}\] times the value at the surface of the earth, is [R = radius of the earth]
A)
\[\frac{R}{n}\] done
clear
B)
\[R\left( \frac{n-1}{n} \right)\] done
clear
C)
\[\frac{R}{{{n}^{2}}}\] done
clear
D)
\[R\left( \frac{n}{n+1} \right)\] done
clear
View Solution play_arrow
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question_answer53)
The radii of two planets are respectively \[{{R}_{1}}\] and \[{{R}_{2}}\] their densities are respectively\[{{\rho }_{1}} and {{\rho }_{2}}\]. The ratio of the accelerations due to gravity at their surfaces is
A)
\[{{g}_{1}}:{{g}_{2}}=\frac{{{\rho }_{1}}}{{{R}_{1}}^{2}}:\frac{{{\rho }_{2}}}{{{R}_{2}}^{2}}\] done
clear
B)
\[{{g}_{1}}:{{g}_{2}}={{R}_{1}}{{R}_{2}}:{{\rho }_{1}}{{\rho }_{2}}\] done
clear
C)
\[{{g}_{1}}:{{g}_{2}}={{R}_{1}}{{R}_{2}}:{{R}_{2}}{{\rho }_{1}}\] done
clear
D)
\[{{g}_{1}}:{{g}_{2}}={{R}_{1}}{{\rho }_{1}}:{{R}_{2}}{{\rho }_{2}}\] done
clear
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question_answer54)
The change in the value of acceleration of earth towards sun, when the moon comes from the position of solar eclipse to the position on the other side of earth in line with sun is: (mass of the moon\[=7.36\times 1{{0}^{22}} kg\], radius of the moon's orbit\[= 3.8\times 1{{0}^{8}} m\]).
A)
\[6.73\times 1{{0}^{-5}} m/{{s}^{2}}\] done
clear
B)
\[6.73\times 1{{0}^{-3}} m/{{s}^{2}}\] done
clear
C)
\[6.73\times 1{{0}^{-2}} m/{{s}^{2}}\] done
clear
D)
\[6.73\times 1{{0}^{-4}} m/{{s}^{2}}\] done
clear
View Solution play_arrow
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question_answer55)
Which graph correctly presents the variation of acceleration due to gravity with the distance from the centre of the earth (radius of the earth\[={{\operatorname{R}}_{E}}\])
A)
B)
C)
D)
View Solution play_arrow
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question_answer56)
Let g be the acceleration due to gravity at earth's surface and K be the rotational kinetic energy of the earth. Suppose the earth's radius decreases by 2% keeping all other quantities same, then
A)
g decreases by 2% and K decreases by 4% done
clear
B)
g decreases by 4% and K increases by 2% done
clear
C)
g increases by 4% and K decreases by 4% done
clear
D)
g decreases by 4% and K increases by 4% done
clear
View Solution play_arrow
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question_answer57)
Two concentric spherical shells are as shown in figure. Choose the correct statement given below.
A)
Potential at A is greater than B done
clear
B)
Gravitational field at A is less than B done
clear
C)
As one moves from C to D then potential remains constant done
clear
D)
As one moves from D to A then gravitational field decreases done
clear
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question_answer58)
The potential energy of a rock, having mass m and rotating at a height of\[3.2\times {{10}^{6}}m\] from the earth surface, is
A)
\[-6\,\,mg{{R}_{e}}\] done
clear
B)
\[-0.67\,\,mg{{R}_{e}}\] done
clear
C)
\[-0.99\,\,mg{{R}_{e}}\] done
clear
D)
\[-0.33\,\,mg{{R}_{e}}\] done
clear
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question_answer59)
Intensity of the gravitational field inside the hollow spherical shell is
A)
Variable done
clear
B)
minimum done
clear
C)
Maximum done
clear
D)
zero done
clear
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question_answer60)
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 surface of the earth to a height equal to the radius 'R' of the earth is
A)
\[\frac{1}{4}mgr\] done
clear
B)
\[\frac{1}{2}mgr\] done
clear
C)
\[2\,mgr\] done
clear
D)
\[mgr\] done
clear
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question_answer61)
Radius of moon is 1/4 times that of earth and mass is 1/81 times that of earth. The point at which gravitational field due to earth becomes equal and opposite to that of moon, is (Distance between centres of earth and moon is 60R, where R is radius of earth)
A)
5.75 R from centre of moon done
clear
B)
16 R from surface of moon done
clear
C)
53 R from centre of earth done
clear
D)
54 R from centre of earth done
clear
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question_answer62)
Let V and E denote the gravitational potential and gravitational field at a point. It is possible to have
A)
(a)\[V=O\text{ }and\text{ }E=0\] done
clear
B)
\[\operatorname{V}=0 and E\ne 0\] done
clear
C)
\[\operatorname{V}\ne 0 and E=0\] done
clear
D)
All of the above done
clear
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question_answer63)
A uniform spherical shell gradually shrinks maintaining its shape. The gravitational potential at the centre
A)
Increases done
clear
B)
decreases done
clear
C)
remains constant done
clear
D)
cannot say done
clear
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question_answer64)
The gravitational potential at the centre of a square of side 'a? and four equal masses (m each) placed at the comers of a square is
A)
Zero done
clear
B)
\[4\sqrt{2}\frac{Gm}{a}\] done
clear
C)
\[-4\sqrt{2}\frac{Gm}{a}\] done
clear
D)
\[-4\sqrt{2}\frac{G{{m}^{2}}}{a}\] done
clear
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question_answer65)
Three identical stars, each of mass M, form an equilateral triangle (stars are positioned at the comers) that rotates around the centre of-the triangle. The system is isolated and edge length of the triangle is L. The amount of work done, that is required to dismantle the system is:
A)
(a)\[\frac{3G{{M}^{2}}}{L}\] done
clear
B)
\[\frac{3G{{M}^{2}}}{2L}\] done
clear
C)
\[\frac{3G{{M}^{2}}}{4L}\] done
clear
D)
\[\frac{G{{M}^{2}}}{2L}\] done
clear
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question_answer66)
Inside a uniform sphere of density p there is a spherical cavity whose centre is at a distance \[\ell \]from the centre of the sphere. Find the strength F of the gravitational field inside the cavity at the point P.
A)
\[\frac{4}{3}G\pi \rho \vec{\ell }\] done
clear
B)
\[\frac{1}{3}G\pi \rho \vec{\ell }\] done
clear
C)
\[\frac{2}{3}G\pi \rho \vec{\ell }\] done
clear
D)
\[\frac{1}{2}G\pi \rho \vec{\ell }\] done
clear
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question_answer67)
The work done required to increase the separation distance from \[{{\operatorname{x}}_{1}} to {{x}_{1}}+d\]between two masses \[{{\operatorname{m}}_{1}} and {{m}_{2}}\]is
A)
\[\frac{G{{m}_{1}}{{m}_{2}}d}{{{x}_{1}}\left( {{x}_{1}}+d \right)}\] done
clear
B)
\[\frac{G{{m}_{1}}{{m}_{2}}{{x}_{1}}}{\left( {{x}_{1}}+d \right)d}\] done
clear
C)
\[\frac{-G{{m}_{1}}{{m}_{2}}{{x}_{1}}}{\left( {{x}_{1}}+d \right)}\] done
clear
D)
none of these done
clear
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question_answer68)
Gravitational field at the centre of a semicircle formed by a thin wire AB of mass m and length \[\ell \]is
A)
\[\frac{Gm}{{{\ell }^{2}}}\operatorname{along}+x-axis\] done
clear
B)
\[\frac{Gm}{\pi {{\ell }^{2}}}\operatorname{along}+y-axis\] done
clear
C)
\[\frac{2\pi Gm}{{{\ell }^{2}}}\operatorname{along}+x-axis\] done
clear
D)
\[\frac{2\pi Gm}{{{\ell }^{2}}}\operatorname{along}+y-axis\] done
clear
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question_answer69)
In a certain region of space, gravitational field is given by\[I=-(K/r)\]. Taking the reference point to be at \[r={{r}_{0}}\]with \[V={{V}_{0}},\] find the potential.
A)
\[K\,\log \frac{r}{{{r}_{0}}}+{{V}_{0}}\] done
clear
B)
\[K\,\log \frac{{{r}_{0}}}{r}+{{V}_{0}}\] done
clear
C)
\[K\,\log \frac{r}{{{r}_{0}}}-{{V}_{0}}\] done
clear
D)
\[\log \frac{{{r}_{0}}}{r}-{{V}_{0}}r\] done
clear
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question_answer70)
A point P lies on the axis of a fixed ring of mass M and radius R, at a distance 2R from its centre O. A small particle starts from P and reaches O under gravitational attraction only. Its speed at O will be
A)
Zero done
clear
B)
\[\sqrt{\frac{2Gm}{R}}\] done
clear
C)
\[\sqrt{\frac{2Gm}{R}\left( \sqrt{5}-1 \right)}\] done
clear
D)
\[\sqrt{\frac{2Gm}{R}\left( 1-\frac{1}{\sqrt{5}} \right)}\] done
clear
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question_answer71)
The gravitational field, due to the 'left over part* of a uniform sphere (from which a part as shown, has been 'removed out'), at a very far off point, P, located as shown, would be (nearly):
A)
\[\frac{5}{6}\frac{GM}{{{x}^{2}}}\] done
clear
B)
\[\frac{8}{9}\frac{GM}{{{x}^{2}}}\] done
clear
C)
\[\frac{7}{8}\frac{GM}{{{x}^{2}}}\] done
clear
D)
\[\frac{6}{7}\frac{GM}{{{x}^{2}}}\] done
clear
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question_answer72)
A space vehicle approaching a planet has a speed v, when it is very far from the planet. At that moment tangent of its trajectory would miss the centre of the planet by distance R. If the planet has mass M and radius r, what is the smallest value of R in order that the resulting orbit of the space vehicle will just miss the surface of the planet?
A)
\[\frac{r}{v}{{\left[ {{v}^{2}}+\frac{2GM}{r} \right]}^{\frac{1}{2}}}\] done
clear
B)
\[\operatorname{vr}\left[ 1+\frac{2GM}{r} \right]\] done
clear
C)
\[\frac{r}{v}\left[ {{v}^{2}}+\frac{2GM}{r} \right]\] done
clear
D)
\[\frac{2GMv}{r}\] done
clear
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question_answer73)
A sky lab of mass m kg is first launched from the surface of the earth in a circular orbit of radius 2R (from the centre of the earth) and then it is shifted form this circular orbit of radius 3R. The minimum energy required to place the lab in the first orbit and to shift the lab from first orbit to the second orbit are
A)
\[\frac{3}{4}\operatorname{mgR}, \frac{mgR}{6}\] done
clear
B)
\[\frac{3}{4}\operatorname{mgR}, \frac{mgR}{12}\] done
clear
C)
\[\operatorname{mgR}, mgR\] done
clear
D)
\[2\operatorname{mgR}, mgR\] done
clear
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question_answer74)
With what minimum speed should m be projected from point C in presence of two fixed masses M each at A and B as shown in the figure such that mass m should escape the gravitational attraction of A and B
A)
\[\sqrt{\frac{2GM}{R}}\] done
clear
B)
\[\sqrt{\frac{2\sqrt{2}GM}{R}}\] done
clear
C)
\[2\sqrt{\frac{GM}{R}}\] done
clear
D)
\[2\sqrt{2}\sqrt{\frac{GM}{R}}\] done
clear
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question_answer75)
An artificial satellite is first taken to a height equal to half the radius of earth. Assume that it is at rest on the earth's surface initially and that it is at rest at this height. Let \[{{E}_{1}}\]be the energy required. It is then given the appropriate orbital speed such that it goes in a circular orbit at that height. Let E be the energy required. The ratio \[\frac{{{E}_{1}}}{{{E}_{2}}}\] is
A)
4:1 done
clear
B)
3:1 done
clear
C)
1:1 done
clear
D)
1:2 done
clear
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question_answer76)
A satellite is launched in the equatorial plane in such a way that it can transmit signals up to \[60{}^\circ \] latitude on the earth. The angular velocity of the satellite is
A)
\[\sqrt{\frac{GM}{8{{R}^{3}}}}\] done
clear
B)
\[\sqrt{\frac{GM}{2{{R}^{3}}}}\] done
clear
C)
\[\sqrt{\frac{GM}{4{{R}^{3}}}}\] done
clear
D)
\[\sqrt{\frac{3\sqrt{3}GM}{8{{R}^{3}}}}\] done
clear
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question_answer77)
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 the surface of the earth, the orbital speed of the satellite is
A)
\[\frac{g{{R}^{2}}}{R+x}\] done
clear
B)
\[\frac{g{{R}^{2}}}{R-x}\] done
clear
C)
\[gx\] done
clear
D)
\[{{\left( \frac{g{{R}^{2}}}{R+x} \right)}^{1/2}}\] done
clear
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question_answer78)
For a satellite orbiting in an orbit, close to the surface of earth, to escape, what is the percentage increase in the kinetic energy required?
A)
41% done
clear
B)
61% done
clear
C)
81% done
clear
D)
98% done
clear
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question_answer79)
The radius of the earth is reduced by 4%. The mass of the earth remains unchanged. What will be the change in escape velocity?
A)
Increased by 2% done
clear
B)
Decreased by 4% done
clear
C)
Increased by 6% done
clear
D)
Decreased by 8% done
clear
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question_answer80)
A satellite of mass m is orbiting the earth in a circular orbit of radius R. It starts losing energy due to small air resistance at the rate of C J/s. Find the time taken for the satellite to reach the earth.
A)
\[\frac{GMm}{C}\left[ \frac{1}{R}-\frac{1}{r} \right]\] done
clear
B)
\[\frac{GMm}{2C}\left[ \frac{1}{R}+\frac{1}{r} \right]\] done
clear
C)
\[\frac{GMm}{2C}\left[ \frac{1}{R}-\frac{1}{r} \right]\] done
clear
D)
\[\frac{2GMm}{C}\left[ \frac{1}{R}+\frac{1}{r} \right]\] done
clear
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question_answer81)
Two spheres each of mass M are situated at a distance 2d (see figure). A particle of mass \[m(m<<M)\]is taken along the path shown in figure. The work done in the process from A to B is
A)
\[\frac{7GMm}{d}\] done
clear
B)
\[\frac{8GMm}{d}\] done
clear
C)
\[-\frac{8GMm}{d}\] done
clear
D)
zero done
clear
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question_answer82)
An asteroid of mass m is approaching earth initially at a distance of\[10{{\operatorname{R}}_{e}}\], with speed\[{{v}_{i}}\]. It hits the earth with a speed \[{{v}_{f}}({{\operatorname{R}}_{e}}\,\,and\,\,{{M}_{e}}\], are radius and mass of earth), then
A)
\[{{v}_{f}}^{2}={{v}_{i}}^{2}+\frac{2GM}{{{M}_{e}}R}\left( 1-\frac{1}{10} \right)\] done
clear
B)
\[{{v}_{f}}^{2}={{v}_{i}}^{2}+\frac{2G{{M}_{e}}}{{{\operatorname{R}}_{e}}}\left( 1+\frac{1}{10} \right)\] done
clear
C)
\[{{v}_{f}}^{2}={{v}_{i}}^{2}+\frac{2G{{M}_{e}}}{{{\operatorname{R}}_{e}}}\left( 1-\frac{1}{10} \right)\] done
clear
D)
\[{{v}_{f}}^{2}={{v}_{i}}^{2}+\frac{2GM}{{{\operatorname{R}}_{e}}}\left( 1-\frac{1}{10} \right)\] done
clear
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question_answer83)
In older times, people used to think that the earth was flat. Imagine that the earth is indeed not a sphere of radius R, but an infinite plate of thickness H. What value of is needed to allow the same gravitational acceleration to be experienced as on the surface of the actual earth? (Assume that the earth's density is uniform and equal in the two models
A)
2R/3 done
clear
B)
4R/3 done
clear
C)
8R/3 done
clear
D)
R/3 done
clear
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question_answer84)
A shell is fired vertically from the earth with speed \[\frac{{{V}_{esc}}}{N},\] where Nis some number greater than one and V is escape speed for the earth. Neglecting the nation of the earth and air resistance, the maximum altitude attained by the shell will be is radius of the earth)
A)
\[\frac{{{N}^{2}}{{\operatorname{R}}_{E}}}{{{N}^{2}}-1}\] done
clear
B)
\[\frac{N{{\operatorname{R}}_{E}}}{{{N}^{2}}-1}\] done
clear
C)
\[\frac{{{\operatorname{R}}_{E}}}{{{N}^{2}}-1}\] done
clear
D)
\[\frac{{{\operatorname{R}}_{E}}}{{{N}^{2}}}\] done
clear
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question_answer85)
A spherical uniform planet is rotating about its axis. The velocity of a point on its equator is v. Due to the rotation of planet about its axis the acceleration due to gravity g at equator is 1/2 of g at poles. The escape velocity of a particle on the pole of planet in terms of v is
A)
\[{{v}_{e}}=2v\] done
clear
B)
\[{{v}_{e}}=v\] done
clear
C)
\[{{v}_{e}}=v/2\] done
clear
D)
\[{{v}_{e}}=\sqrt{3}v\] done
clear
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question_answer86)
The acceleration due to gravity on the surface of the moon is 1/6 that on the surface of earth and the diameter of the moon is one-fourth that of earth. The ratio of escape velocities on earth and moon will be
A)
\[\frac{\sqrt{6}}{2}\] done
clear
B)
\[\sqrt{24}\] done
clear
C)
3 done
clear
D)
\[\frac{\sqrt{3}}{2}\] done
clear
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question_answer87)
Three equal masses (each m) are placed at the comers of an equilateral triangle of side 'a? Then the escape velocity of an object from the circumcentre P of triangle is:
A)
\[\sqrt{\frac{2\sqrt{3}Gm}{a}}\] done
clear
B)
\[\sqrt{\frac{\sqrt{3}Gm}{a}}\] done
clear
C)
\[\sqrt{\frac{6\sqrt{3}Gm}{a}}\] done
clear
D)
\[\sqrt{\frac{3\sqrt{3}Gm}{a}}\] done
clear
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question_answer88)
A geo-stationary satellite orbits around the earth in a circular orbit of radius 36,000 km. Then, the time period of a spy satellite orbiting a few hundred km above the earth's surface \[({{R}_{earth}} = 6,400km)\]will approximately be
A)
1/2hr done
clear
B)
1hr done
clear
C)
2hr done
clear
D)
4hr done
clear
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question_answer89)
A satellite can be in a geostationary orbit around earth at a distance r from the centre. If the angular velocity of earth about its axis doubles, a satellite can now be in a geostationary orbit around earth if its distance from the centre is
A)
\[\frac{r}{2}\] done
clear
B)
\[\frac{r}{2\sqrt{2}}\] done
clear
C)
\[\frac{r}{{{\left( 4 \right)}^{1/3}}}\] done
clear
D)
\[\frac{r}{{{\left( 2 \right)}^{1/3}}}\] done
clear
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question_answer90)
The mass M of a planet-earth is uniformly distributed over a spherical volume of radius R. Calculate the energy needed to disassemble the planet against the gravitational pull amongst its constituent particles. Given: \[\operatorname{MR}=2.5\times 1{{0}^{31}} kg-mandg= 10m/{{s}^{2}}\]
A)
\[3.0\times {{10}^{32}}\operatorname{J}\] done
clear
B)
\[2.5\times {{10}^{32}}\operatorname{J}\] done
clear
C)
\[0.5\times {{10}^{32}}\operatorname{J}\] done
clear
D)
\[1.5\times {{10}^{32}}\operatorname{J}\] done
clear
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question_answer91)
Two bodies of masses \[{{M}_{1}}\text{ }\operatorname{and}\text{ }{{M}_{2}}\]are placed at a distanced/apart. What is the potential at the position where the gravitational field due to them is zero?
A)
\[-\frac{G}{d}({{M}_{1}}+{{M}_{2}}+2\sqrt{{{M}_{1}}}\sqrt{{{M}_{2}}})\] done
clear
B)
\[-\frac{G}{d}({{M}_{1}}+{{M}_{2}}-2\sqrt{{{M}_{1}}}\sqrt{{{M}_{2}}})\] done
clear
C)
\[-\frac{G}{d}(2{{M}_{1}}+{{M}_{2}}+2\sqrt{{{M}_{1}}}\sqrt{{{M}_{2}}})\] done
clear
D)
\[-\frac{G}{2d}({{M}_{1}}+{{M}_{2}}+2\sqrt{{{M}_{1}}}\sqrt{{{M}_{2}}})\] done
clear
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question_answer92)
At what height from the surface of earth the gravitational potential and the value of g are\[-5.4\times 1{{0}^{7}}J k{{g}^{-1}} and 6.0 m{{s}^{-2}}\]respectively? Take the radius of earth as 6400 km:
A)
2600 km done
clear
B)
1600 km done
clear
C)
1400 km done
clear
D)
2000 km done
clear
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question_answer93)
Which one of the following plots represents the variation of gravitational field on a particle with distance r due to a thin spherical shell of radius R? (r is measured from the centre of the spherical shell)
A)
B)
C)
D)
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question_answer94)
The gravitational potential of two homogeneous spherical shells A and B of same surface density at their respective centres are in the ratio 3 : 4. If the two shells coalesce into single one such that surface charge density remains same, then the ratio of potential at an internal point of the view shell to shell A is equal to
A)
3:2 done
clear
B)
4:3 done
clear
C)
5:3 done
clear
D)
5:6 done
clear
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question_answer95)
The gravitational field in a region is given by\[\vec{g} =5N/kg\hat{i}+12N/kg\hat{j}\]. The change in the gravitational potential energy of a particle of mass 1 kg when it is taken from the origin to a point \[(7m,-3m)\] is:
A)
\[71\text{ }J\] done
clear
B)
\[13\sqrt{58}J\] done
clear
C)
\[-71\text{ }J\] done
clear
D)
\[1\text{ }J\] done
clear
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question_answer96)
The gravitational field due to a mass distribution is \[\operatorname{E}=K/{{x}^{3}}\]in the x-direction. (K is a constant). Taking the gravitational potential to be zero at infinity, its value at a distance x is
A)
\[K/x\] done
clear
B)
\[K/2x\] done
clear
C)
\[\operatorname{K}/{{x}^{2}}\] done
clear
D)
\[\operatorname{K}/2{{x}^{2}}\] done
clear
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question_answer97)
Infinite number of masses, each 1 kg are placed along the x-axis \[\operatorname{at} x=\pm 1 m, \pm 2m, \pm 4m, \pm \]\[8m, \pm 16M\ldots \] the magnitude of the resultant gravitational potential in terms of gravitational constant G at the or g in \[(x=0)\] is
A)
G/2 done
clear
B)
G done
clear
C)
2G done
clear
D)
4G done
clear
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question_answer98)
Two rings each of radius 'a' are coaxial and the distance between their centres is a. The masses of the rings are\[{{M}_{1}}\text{ }and\text{ }{{M}_{2}}\]. The work done in transporting a particle of a small mass m from centre \[{{\operatorname{C}}_{1}} to {{C}_{2}}\]is :
A)
\[\frac{Gm\left( {{M}_{2}}-{{M}_{1}} \right)}{a}\] done
clear
B)
\[\frac{Gm\left( {{M}_{2}}-{{M}_{1}} \right)}{a\sqrt{2}}\left( \sqrt{2}+1 \right)\] done
clear
C)
\[\frac{Gm\left( {{M}_{2}}-{{M}_{1}} \right)}{a\sqrt{2}}\left( \sqrt{2}-1 \right)\] done
clear
D)
\[\frac{Gm\left( {{M}_{2}}-{{M}_{1}} \right)}{\sqrt{2}}a\] done
clear
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question_answer99)
A system consists of two stars of equal masses that revolve in a circular orbit about a centre of mass midway between them. Orbital speed of each star is v and period is T. Find the mass M of each star (G is gravitational constant)
A)
\[\frac{2G{{v}^{3}}}{\pi T}\] done
clear
B)
\[\frac{{{v}^{3}}T}{\pi G}\] done
clear
C)
\[\frac{{{v}^{3}}T}{2\pi G}\] done
clear
D)
\[\frac{2T{{v}^{3}}}{\pi G}\] done
clear
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question_answer100)
A uniform ring of mass m and radius r is placed directly above a uniform sphere of mass M and of equal radius. The centre of the ring is directly above the centre of the sphere at a distance \[r\sqrt{3}\] as shown in the figure. The gravitational field due to the ring at a distance \[\sqrt{3}r\]is.
A)
\[\frac{Gm}{8{{r}^{2}}}\] done
clear
B)
\[\frac{Gm}{4{{r}^{2}}}\] done
clear
C)
\[\sqrt{3}\frac{Gm}{8{{r}^{2}}}\] done
clear
D)
\[\frac{Gm}{8{{r}^{2}}\sqrt{3}}\] done
clear
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