GRAVITATION

Exercise - 01
SINGLE CORRECT
1. Four similar particles of mass m are orbiting in a circle of radius r in the same
direction and same speed because of their mutual gravitational attractive force
as shown in the figure. Speed of a particle is given by
1
Gm 1+2√2 2 3 Gm
(A) [ ( )] (B) √
r 4 r

Gm
(C) √ (1 + 2√2) (D) Zero
r

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2. Three particles P, Q and R are placed as per given figure. Masses of P, Q and R
are √3m, √3m and m respectively. The gravitational force on a fourth particle
'S' of mass m is equal to
√3Gm2
(A) in ST direction only
2d2

√3Gm2 √3Gm2
(B) in SQ direction and in SU direction
2d2 2d2

√3Gm2
(C) in SQ direction only
2d2

√3Gm2 √3Gm2
(D) in SQ direction and in ST direction
2d2 2d2

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3. A mass is at the center of a square, with four masses at the corners as shown.

Rank the choices according to the magnitude of the gravitational force on

the center mass.

(A) FA = FB < FC = FD (B) FA > FB < FC < FD (C) FA = FB > FC = FD (D) None

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4. An isolated triple star system consists of two identical stars, each of mass m
and a fixed star of mass M. They revolve around the central star in the same
circular orbit of radius r. The two orbiting stars are always at opposite ends
of a diameter of the orbit. The time period of revolution of each star around
the fixed star is equal to :

4πr3⁄2 2πr3⁄2
(A) √ (B)
G(4M+m) √GM

2πr3⁄2 4πr3⁄2
(C) (D)
√G(M+m) √G(M+m)

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5. The gravitational force of attraction between a uniform sphere of mass M
and a uniform rod of length l and mass m oriented as shown is
GMm GM
(A) (B)
r(r+ℓ) r2

GMm GMm
(C) (D)
r2 r

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6. Gravitational field at the centre of a semicircle formed by a thin wire AB of
mass m and length ℓ as shown in the figure. ss
Gm Gm
(A) along + x axis (B) along + y axis
ℓ2 πℓ2

2πGm 2πGm
(C) along + x axis (D) along + y axis
ℓ2 ℓ2

–
7. Figure show a hemispherical shell having uniform mass density. The direction
of gravitational field intensity at point P will be along:

(A) a (B) b

(C) c (D) d

–
8. If Gravitational field due to uniform thin hemispherical shell at point P is I,
then the magnitude of gravitational field at Q is (Mass of hemisphere is M,
radius R) –
GM GM GM GM
(A) −I (B) +I (C) −I (D)2I −
2R2 2R2 4R 2R2

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9. A cavity of the radius R/2 is made inside a solid sphere of radius R. The centre
of the cavity is located at a distance R/2 from the centre of the sphere. The
gravitational force on a particle of mass m at a distance R/2 from the centre
of the sphere on the line joining both the centres of sphere and cavity is
(opposite to the centre of the cavity).

[Here g = GM/R2, where M is the mass of the sphere]

mg 3mg mg
(A) (B) (C) (D) None of these
2 8 16

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10. Two particles having masses m and 4m are separated by distance l. the
distance of the centre of mass from m is x1 and x2 is the distance of point at
which gravitational field intensity is zero. Find the value of x 1/x2.

(A) 1 (B) 2

(C) 2.4 (D) 3.6

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11. At what altitude will the acceleration due to gravity be 25% of that at the
earth’s surface (Given radius of earth is R)?

(A) R/4 (B) R (C) 3R/8 (D) R/2

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12. The mass and diameter of a planet are twice those of earth. What will be the
period of oscillation of a pendulum on this planet if it is seconds pendulum
on earth?
1 1
(A) √2 second (B) 2√2 second (C) second (D) second
√2 2√2

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13. A body hanging from a massless spring stretches it by 3 cm on earth’s surface.
At a place 800 km above the earth’s surface, the same body will stretch the
spring by (Radius of Earth = 6400 km)

34 64 27 27
(A) ( ) cm (B) ( ) cm (C) ( ) cm (D) ( ) cm
27 27 64 34

–
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14. A pendulum clock keeping correct time at sea level is taken to a place 1 km
above sea level. Then the clock approximately

(A) gains 13.5 seconds per day (B) gains 7 seconds per day

(C) loses 13.5 seconds per day (D) loses 7 seconds per day

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15. How many time faster than the present speed would the earth have to rotate
about its axis so that the apparent weight of the bodies on the equator
becomes zero? (The radius of the earth R = 6.4 × 106 m)

(A) 5 times (B) 9 times (C) 13 times (D) 17 times

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16. A simple pendulum is taken to 64 km above the earth’s surface. It’s time
period will:

(A) Increase by 1% (B) Decrease by 1%

(C) Increases by 2% (D) Decrease by 2%

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17. The value of gravitational acceleration g at a height h above the earth’s surface
is g/4 then (R = radius of earth)

(A) h = R (B) h = R/2 (C) h = R/3 (D) h = R/4

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18. A very large number of particles of same mass m are kept at horizontal
distances of 1m, 2m, 4m, 8m and so on from (0, 0) point. The total
gravitational potential at this point (0, 0) is :

(A) – 8G m (B) – 3G m (C) – 4G m (D) – 2G m

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19. A body starts from rest at a point, distance R 0 from the centre of the earth of
mass M, radius R. The velocity acquired by the body when it reaches the
surface of the earth will be

1 1 1 1 1 1 1 1
(A) GM ( − ) (B) 2GM ( − ) (C) √2GM ( − ) (D) 2GM√( − )
R R0 R R0 R R 0 R R 0

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20. If a tunnel is cut any orientation through earth, then a ball released from one
end will reach the other end in time (neglect earth rotation)

(A) 84.6 minutes (B) 42.3 minutes

(C) 8 minutes (D) depends on orientation

–
21. If g is the acceleration due to gravity on the earth’s surface, the gain in the
potential energy of an object of m raised from the surface of the earth to a
height equal to the radius R of the earth is
1 1
(A)( ) mgR (B) 2mgR (C) mgR (D) ( ) mgR
2 4

–
22. The escape velocity of the body from the earth is 11.2 km/sec. If the radius of
the planet be half the radius of earth and its mass is one fourth that of earth.
The escape velocity of the planet is :

(A) 8 km/sec (B) 4 km/sec (C) 16 km/sec (D) 2 km/sec

–
23. The escape velocity for a planet from the surface is ue. A particle starts from
rest at a large distance from the planet, reaches the planet only under
gravitational attraction, and passes through a smooth tunnel through its
centre. Its speed at the centre of the planet will be

(A) ue (B) 1.5ue (C) √1.5υe (D) 2ue

–
24. Let gravitation field in a space be given as E = −(k/r). If the reference point is
at distance di where potential is Vi then relation for potential is:
1 1
(A) V = Kln⁡ +0 (B) V = Kln⁡ + Vi
Vi di

1 r Vi
(C) V = ln⁡ +0 (D) V = ln⁡ +
Vi di k

–
25. Weight of a body at the equator of a planet is half of that at the poles. If
peripheral velocity of a point on the equator of this planet is v 0, what is the
escape velocity of a polar particle ?

(A) v0 (B) 2v0 (C) 3v0 (D) 4v0

–
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26. If the escape speed of a projectile on Earth’s surface is 11.2 km s–2 and a body
is projected out with thrice this speed, then determine the speed of the body
far away from the Earth

(A) 56.63 km s–1 (B) 33 km s–1 (C) 39 km s–1 (D) 31.7 km s–1

–
27. If the “escape velocity” for an object placed at the surface of the earth is v e
,then the escape velocity for the same object when ‘ejected’ from the pit of a
tunnel made upto a depth of d = 0.05R, where R, is the radius of the earth is

(A) Less than ve by more than 10 %

(B) Greater than ve by more than 10%

(C) Less than ve by less than 10 %

(D) Greater than ve by less than 10%

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28. A body is thrown the surface of the earth with velocity u ms−1. The maximum
height in meter above the surface of the earth up to which it will reach is (R =
radius of the earth, g = acceleration due to gravity)
u2 R 2u2 R u2 R2 u2 R
(A) (B) (C) (D)
2gR−u2 gR−u2 2gR2 −u2 gR−u2

–
29. A body is projected up with a velocity equal to (3/4)th of the escape velocity
from the surface of the earth. The height it reaches is, (Radius of earth = R)

(A) 10R/9 (B) 9/7 R (C) 9/8 R (D) 10R/3

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30. Two spherical massive bodies of uniform density each of mass M and radius
R are kept at a distance 4R apart. Then the minimum speed (V min) required to
project a particle from the surface of body A such that it will never return to
the surface of the same body is

12GM 2GM
(A)Vmin = √ (B) Vmin = √
5R 3R

5GM 2GM
(C)Vmin = √ (D)Vmin = √
7R R

–
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31. A planet moving around the sun sweeps area A1 in 2 days, A2 in 4 days and
A3 in 9 days. Then, relation between them

(A) A1 = A2 = A3 (B) 9A1 = 3A2 = 2A3

(C) 18A1 = 9A2 = 4A3 (D) 3A1 = 4A2 = 6A3

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32. A planet of mass m is in an elliptical orbit about the sun (m<<Msun) with an
orbital period T. If A be the area of orbit, then its angular momentum would be

(A) 2mA/T (B) mAT (C) mA/2T (D) 2mAT

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33. A small planet is revolving around a very massive star in a circular orbit of
radius r with a period of revolution T. If the gravitational force between the
planet and the star is proportional to r–5/2, then T will be proportional to

(A) r3/2 (B) r5/3 (C) r7/4 (D) r3

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34. Two identical satellites are moving around the earth in circular orbits at
heights 3R and R respectively where R is the radius of the Earth. The ratio of
their kinetic energies is

(A) 2 : 1 (B) 1 : 2 (C) 3 : 1 (D) 2 : 3

–
35. In a double star system one of mass m1 and other of mass m2 with a separation
d, rotate about their common centre of mass. Then the ratio of the areal
velocity of star of mass m1 to that of star of mass m2 about their common C.M
is

m1 m2 m12 m 22
(A) (B) (C) (D)
m2 m1 m 22 m12

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36. A satellite is seen after each 8 hours over equator at a place on the earth when
its sense of rotation is opposite to the earth. The time interval after which it
can be seen at the same place when the sense of rotation of earth & satellite
is same will be :

(A) 8 hours (B) 12 hours (C) 24 hours (D) 6 hours

–
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37. If a satellite is revolve around a planet of mass M in an elliptical orbit of semi-
major axis a, find the orbital speed of the satellite when it is at a distance r
from the focus
2 1 2 1
(A)v 2 = GM [ − ] (B) v 2 = GM [ 2 − ]
r a r a

2 1 2 1
(C) v 2 = GM [ 2 − 2 ] (D) v 2 = G [ − ]
r a r a

–
38. A satellite is revolving in a circular orbit at a height h from the earth surface,
such that h << R where R is the radius of the earth. Assuming that the effect
of the earth’s atmosphere can be neglected the minimum increase in the speed
required so that the satellite could escape from the gravitational field of earth
is :

gR
(A) √gR(√2 − 1) (B) √gR (C) √2gR (D) √
2

–
39. A planet revolves about the sun in elliptical orbit of semi major axis 2 x 10 12
m. The areal velocity of the planet when it is nearest to the sun is 4.4 x 10 16
m2/s. The least distance between planet and the sun is 1.8 x 10 12 m. Then
the minimum speed of the planet in km/s is :

(A) 10 (B) 20 (C) 30 (D) 40

–
40. A satellite can be in a geostationary orbit around earth at a distance r from
the center. 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 center
is
r r r r
(A) (B) (C) (4)1⁄3
(D) (2)1⁄3
2 2√2

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 Exercise - 02
Previous Year Questions
1. 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 from the
earth is 11 kms−1 , the velocityfrom the surface of the planet would be
[AIEEE-2008]

(A) 11 kms −1 (B) 110 kms−1 (C) 0.11 kms −1 (D) 1.1 kms −1

–
g
2. The height at which the acceleration due to gravity becomes (where g = the
9
acceleration due to gravity on the surface of the earth) in terms of R, the radius
of the earth, is- [AIEEE-2009]
R
(A) (B) R/2 (C) √2R (D) 2 R
√2

–
3. Two bodies of masses m and 4m are placed at a distance r. The gravitational
potential at a point on the line joining them where the gravitational field is
zero is [AIEEE-2011]
4Gm 6Gm 9Gm
(A) zero (B) − (C) − (D) −
r r r

–
4. 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 orbit at an altitude of
2R ? [JEE Main-2013]
2GmM GmM GmM 5GmM
(A) (B) (C) (D)
3R 2R 3R 6R

–
5. 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 attraction.
The speed of each particle is [JEE Main-2014]

GM GM 1 GM GM
(A) √2√2 (B) √ (1 + 2√2) (C) √ (1 + 2√2) (D) √
R R 2 R R

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 R
6. From a solid sphere of mass M and radius R, a spherical portion of radius is
2
removed, as shown in the figure. Taking gravitational potential V = 0 at r = ∞,
the potential at the centre of the cavity thus formed is: (G = gravitational
constant) [JEE Main-2015]
−2GM −2GM
(A) (B)
3R R

GM −GM
(C) (D)
2R R

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7. 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 increase in its orbital velocity
required, so that the satellite could escape from the earth's gravitational field,
is close to: (Neglect the effect of atmosphere.) [JEE Main-2016]

(A) √gR(√2 − 1) (B) √2gR (C) √gR (D) √gR/2

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

(A) (B)

(C) (D)

–
9. If the Earth has no rotation motion, the weight of a person on the equator is
W. Determine the speed with which the earth would have to rotate about its
3
axis so that the person at the equator will weigh W. Radius of the Earth is
4
6400 km and g = 10m/s2 . [JEE Main-2017]

(A) 0.63 × 10−3 rad/s (B) 0.83 × 10−3 rad/s

(C) 0.28 × 10−3 rad/s (D) 1.1 × 10−3 rad/s

–
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 k
10. The mass density of spherical body is given by ρ(r) = ≤ R and ρ(r) = 0 for r >
r
R, where is r is the distance from the centre. The correct graph that describes
qualitatively the acceleration, a of a test particle as a function of r is

(A) (B)

(C) (D)

–
11. A particle is moving in a circular path of radius a under the action of an
k
attractive potential U = . It total energy is [JEE Main-2018]
2r2

k k 3 k
(A) − (B) (C) Zero (D) −
4a2 2a2 2 a2

–
12. 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 period of
rotation of the particle is T, then [JEE Main-2018]
n
(A) T ∝ R3/20 for any n (B) T ∝ R 2+1

(C) T ∝ R(n+1)/2 (D) T ∝ Rn/2

–
13. 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 moves
R
in a circular orbit of radius , and the other mass, in a circular orbit of radius
2
3R
. The difference between the final initial total energies is:
2
[JEE Main Online -2018]
GMm GMm GMm GMm
(A) − (B) + (C) (D) −
2R 6R 2R 6R

–
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14. Take the mean distance of the moon and the sun from the earth to be
0.4 × 106 km and 150 × 106 km respectively. Their masses are 8 × 1022 kg and
2 × 1030 kg respectively. The radius of the earth is 6400 km. Let ΔF1 be the
difference in the forces exerted by the moon at the nearest and farthest points
on the earth and ΔF2 be the difference in the force exerted by the sun at the
ΔF1
nearest and farthest points on the earth. Then the number closest to is:
ΔF2

[JEE Main Online -2018]

(A) 2 (B) 6 (C) 10−2 (D) 0.6

–
15. Suppose that the angular velocity of rotation of earth is increased. Then, as a
consequence [JEE Main Online -2018]

(A) there will be no change in weight anywhere on the earth

(B) weight of the object, everywhere on the earth, will increase

(C) except at poles, weight of the object on the earth will decrease

(D) weight of the object, everywhere on the earth, will decrease.

–
16. Two particles of the same mass m are moving in circular orbits because of
−16
force, given by F(r) = − r 3 . The first particle is at a distance r = 1, and the
r
second at r = 4. The best estimate for the ratio of kinetic energies of the first
and the second particle is closest to [JEE Main Online -2018]

(A) 6 × 10−2 (B) 10−1 (C) 3 × 10−3 (D) 6 × 102

–
17. The energy required to take a satellite to a height 'h' above Earth surface
(radius of Earth = 6.4 × 103 km ) is E1 and kinetic energy required for the
satellite to be in a circular orbit at this height is E2 . The value of h for which
E1 and E2 are equal, is: [JEE Main-2019]

(A) 1.28 × 104 km (B) 6.4 × 103 km (C) 3.2 × 103 km (D) 1.6 × 103 km

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18. 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 just escapes from
the gravitational pull of the earth. At the time of ejection, the kinetic energy of
the object is : [JEE Main-2019]
3 1
(A) (B) mv 2 (C) 2mv 2 (D)
2mv2 2mv2

–
19. Two stars of masses 3 × 1031 kg each, and at distance 2 × 1011 m rotate in a
plane about their common centre of mass O. A meteorite passes through O
moving perpendicular to the star's rotation plane. In order to escape from the
gravitational field of this double star, the minimum speed that meteorite
should have at O is :

(Take Gravitational constant G = 6.67 × 10−11 Nm2 kg −2 ) [JEE Main-2019]

(A) 1.4 × 105 m/s (B) 24 × 104 m/s (C) 3.8 × 104 m/s (D) 2.8 ×105 m/s

–
20. A satellite is revolving in a circular orbit at a height h from the earth surface,
such that h << R where R is the radius of the earth. Assuming that the effect
of earth's atmosphere can be neglected the minimum increase in the speed
required so that the satellite could escape from the gravitational field of earth
is: [JEE Main-2019]

gR
(A) √2gR (B) √gR (C) √ (D) √gR(√2 − 1)
2

–
21. 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 length of the
rod is A + Bx 2 , is given by: [JEE Main-2019]
1 1 1 1
(A) Gm⁡ [A ( − ) − BL] (B) Gm⁡ [A ( − ) − BL]
a+L a a a−L

1 1 1 1
(D) Gm⁡ [A ( − ) + BL] (C) Gm⁡ [A ( − ) + BL]
a a+L a+L a

–
22. 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 collides with
the satellite completely inelastically. The speeds of the satellite and the
meteorite are the same, just before the collision. The subsequent motion of
the combined body will be: [JEE Main-2019]
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 (A) such that it escapes to infinity

(B) in an elliptical orbit

(C) in the same circular orbit of radius R

(D) in a circular orbit of a different radius

–
23. 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
influence of others' gravitational field in a circular orbit circumscribing the
square? [JEE Main-2019]

GM GM
(A) 1.21√ (B) 1.41√
a a

GM GM
(D) 1.16√ (D) 1.35√
a a

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24. 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 be the
minimum energy that the launcher should have if the same rocket is to be
launched from the surface of the moon? Assume that the density of the earth
and the moon are equal and that the earth's volume is 64 times the volume of
the moon. [JEE Main-2019]

(A) E/16 (B) E/32 (C) E/64 (D) E/4

–
25. 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 at distance
'3a' from the centre will be: [JEE Main-2019]

(A) 2GM/3a2 (B) GM/9a2 (C) 2GM/9a2 (D) GM/3a2

–
26. A test particle is moving in a circular orbit in the gravitational field produced
K
by a mass density ρ(r) = 2 , Identify the correct relation between the radius R
r
of the particle's orbit and its period T : [JEE Main-2019]

(A) TR is a constant (B) T/R is constant

(C) T/R2 is a constant (D) T 2 /R3 is a constant

–
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27. The value of acceleration due to gravity at Earth's surface is 9.8ms −2 . The
altitude above its surface at which the acceleration due to gravity decreases
to 4.9ms −2 , is close to (Radius of earth = 6.4 × 106 m) [JEE Main-2019]

(A) 6.4 × 106 m (B) 1.6 × 106 m (C) 2.6 × 106 m (D) 9.0 × 106 m

–
28. 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,
what will be the number of complete revolutions made by the spaceship in 24
hours around the planet?

{Given = Mass of planet = 8 × 1022 kg, radius of planet = 2 × 106 m,

Gravitational constant G = 6.67 × 10−11 Nm2 /kg 2 } [JEE Main-2019]

(A) 13 (B) 17 (C) 11 (D) 9

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29. 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 the earth),
m
it ejects a rocket of mass so that subsequently the satellite moves in a
10
circular orbit. The kinetic energy of the rocket is ( G is the gravitational
constant; M is the mass of the earth) : [JEE Main-2020]
m 113 GM 119 GM
(A) (u2 + ) (B) 5m (u2 − )
20 200 R 200 R

2 2
3m 5GM m 2GM
(C) (u + √ ) (D) (u − √ )
8 6R 20 3R

–
30. A simple pendulum is being used to determine the value of gravitational
acceleration g at a certain place. The length of the pendulum is 25.0 cm and
a stop watch with 1s resolution measures the time taken for 40 oscillations to
be 50s. The accuracy in g is : [JEE Main-2020]

(A) 4.40 % (B) 3.40% (C) 2.40% (D) 5.40%

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 Exercise - 03
JEE Advanced [Multiple Correct]
1. Assuming the earth to be a sphere of uniform density the acceleration due to
gravity
(A) At a point outside the earth is inversely proportional to the square of its
distance from the centre
(B) At a point outside the earth is inversely proportional to its distance from
the centre
(C) At any point inside is zero
(D) At a point inside is proportional to its distance from the centre
–
2. Inside a hollow isolated spherical shell
(A) Everywhere gravitational potential is zero
(B) Everywhere gravitational field is zero
(C) Everywhere gravitational potential is same
(D) Everywhere gravitational field is same
–
3. A geostationary satellite is at a height h above the surface of earth. If earth
radius is R

(A) The minimum colatitude on earth upto which the satellite can be used for
R
communication is sin−1 ( + h)
R

(B) The maximum colatitudes on earth upto which the satellite can be used
R
for communication is sin−1 ( + h)
R

(C) The area on earth escaped from this satellite is given as 2πR2 (1 + sin⁡ θ)
(D) The area on earth escaped from this satellite is given as 2πR2 (1 + cos⁡ θ)
–

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4. When a satellite in a circular orbit around the earth enters the atmospheric
region, it encounters small air resistance to its motion. Then
(A) Its kinetic energy increases
(B) Its kinetic energy decreases
(C) Its angular momentum about the earth decreases
(D) Its period of revolution around the earth increases
–
5. A communications Earth satellite
(A) Goes round the earth from east to west
(B) Can be in the equatorial plane only
(C) Can be vertically above any lace on the earth
(D) goes round the earth from west to east
–
6. An earth satellite is moved from one stable circular orbit to another larger and
stable circular orbit. The following quantities increase for the satellite as a
result of this change
(A) Gravitational potential energy (B) Angular velocity
(C) Linear orbital velocity (D) Centripetal acceleration
–
7. A satellite S is moving in an elliptical orbit around the earth. The mass of the
satellites is very small compared to the mass of the earth
(A) The acceleration of S is always directed towards the centre of the earth
(B) The angular momentum of S about the centre of earth changes in direction,
but its
(C) The total mechanical energy of s varies periodically with time
(D) The linear momentum of S remains constant in magnitude
–
8. If a satellite orbits as close to the earth’s surface as possible,
(A) Its speed is maximum
(B) Time period of its rotation is minimum
(C) The total energy of the ‘earth plus satellite’ system is minimum
(D) The total energy of the ‘earth plus satellite’ system is maximum
–
18 | P a g e
9. For a satellite to orbit around the earth, which of the following must be true?

(A) it must be above the equator at some time

(B) It cannot pass over the poles at any time

(C) Its height above the surface cannot exceed 36,000km

(D) Its period of rotation must be > 2π√R/g where R is radius of earth

–
10. For a satellite to appear stationary w.r.t. an observer on earth

(A) It must be rotating about the earth’s axis.

(B) It must be rotating in the equatorial plane.

(C) Its angular velocity must be from west to east.

(D) Its time period must be 24 hours.

–
11. tunnel is dug along a chord of the earth at a perpendicular distance R/2 from
the earth’s centre. The wall of the tunnel may be assumed to be frictionless.
A particle is released from one end of the tunnel. The pressing force by the
particle on the wall and the acceleration of the particle varies with x (distance
of the particle from the centre) according to

(A) (B)

(C) (D)

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12. A satellite close to the earth is in orbit above the equator with a period of
revolution of 1.5 hours. If it is above a point P on the equator at some time, it
will be above P again after time

(A) 1.5 hours

(B) 1.6 hours if it is rotating from west to east

(C) 24/17 hours if it is rotating from east to west

(D) 24/17 hours if it is rotating from west to east

–
13. A double star is a system of two stars of masses m and 2m, rotating about
their centre of mass only under their mutual gravitational attraction. If r is
the separation between these two stars then their time period of rotation about
their centre of mass will be proportional to

(A) r3/2 (B) r (C) m1/2 (D) m–1/2

–
14. An orbiting satellite will escape if:

(A) its speed is increased by (√2 − 1)100%

(B) its speed in the orbit is made √1.5

(C) its KE is doubled

(D) it stops moving in the orbit

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 Exercise - 04
JEE Advanced
[Subjective]
1. A small mass and a thin uniform rod each of mass ‘m’ are positioned along
the same straight line as shown. Find the force of gravitational attraction
exerted by the rod on the small mass.

–
2. A particle is fired vertically from the surface of the earth with a velocity kv e,
where ve is the escape velocity and k < 1. Neglecting air resistance and
assuming earth’s radius as Re. Calculate the height to which it will rise from
the surface of the earth.

–
3. A small body of mass is projected with a velocity just sufficient to make it
reach from the surface of a planet (of radius 2R and mass 3M) to the surface
of another planet (of radius R and mass M). The distance between the centers
of the two spherical planets is 6R. The distance of the body from the center of
bigger planet is ‘x’ at any moment. During the journey, find the distance x
where the speed of the body is (a) maximum (b) minimum. Assume motion of
body along the line joining centres of planets.

–
4. A body moving radially away from a planet of mass M, when at distance r from
planet, explodes in such a way that two of its many fragments move in
mutually perpendicular circular orbits around the planet. What will be

(a) then velocity in circular orbits.

(b) maximum distance between the two fragments before collision and

(c) magnitude of their relative velocity just before they collide

21 | P a g e
5. An astronaut in a circular orbit around earth observes a celestial body moving
in a lower circular orbit around earth in same plane as his orbit and in the
same sense. The body moves at a rate of 5 m/s relative to himself when it is
closest. What is the minimum distance (in km) between him and the body if
he is moving at a speed of 5000m/s.

Mass of earth = 6 × 1024 kg. (Round off to nearest integer)

–
6. A satellite is launched tangentially from a height h above earth's surface as
shown.
(a) Find vmin so that it just touches the earth's surface

3GM
(b) if h = R and satellite is launched tangentially with speed = √
5R

find the maximum distance of satellite from earth's center

–
7. A satellite is orbiting the Earth of mass M in equatorial plane in a circular
orbit having radius 2R and same sense of rotation as that of the Earth. Find
duration of time for which a man standing on the equator will be able to see
the satellite continuously. Assume that the man can see the satellite when it
is above horizontal. Take Earth's angular velocity = ω.

–
8. A remote sensing satellite is revolving in an orbit of radius x over the equator
of earth. Find the area on earth surface in which satellite can not send
message.

–
9. A tunnel is dug along a chord of the earth at a perpendicular distance R e/2
from the earth’s centre. The wall of the tunnel may be assumed to be
frictionless. Find the force exerted by the wall on a particle of mass m when it
is at distance x from the centre of the tunnel.

–
10. A satellite is revolving round the earth in a circular orbit of radius a with
velocity v0. A particle is projected from the satellite in forward direction with
relative velocity v = (√5⁄4 − 1) v0 . Calculate, during subsequent motion of the
particle its minimum and maximum distances from earth’ centre.

–
22 | P a g e
11. A planet of mass m moves along an ellipse around the sun so that its
maximum and minimum distances from the sun are equal to r 1 and r2
respectively. Find the angular momentum of this plane relative to the centre
of the sun.

–
12. A particle is projected from point A, that is at a distance 4R from the centre of
the Earth, with speed v1 in a direction making 30° with the line joining the
centre of the Earth and point A, as shown. Find the speed v1 of particle if
particle passes grazing the surface of the earth. Consider gravitational
interaction only between these two.

GM
(Use = 6.4 × 107 m2 /s2 )
R

23 | P a g e
 Exercise - 05
JEE Advanced
[Previous Year Questions]

1. A system of binary stars of masses mA and mB are moving in circular orbits of

radii rA and rB respectively. If TA and TB are the time periods of masses mA and
mB respectively, then [JEE-2006]
(A) TA > TB (if rA > rB ) (B) TA > TB (if mA > mB )
T 2 r 3
(C) ( A ) = ( A ) (D) TA = TB
TB rB
–
2. A geostationary satellite orbits around the earth in a circular orbit of radius
36,000km. Then, the time period of a spy satellite orbiting a few hundred km
above the earth's surface (Re = 6400km) will approximately be [JEE-2006]
1
(A) h (B) 1h (C) 2h (D) 4h
2
–
3. Column-I describes some situations in which a small object moves. Column-II
describes some characteristics of these motions. Match the situations in
Column-I with the characteristics in Column-II [JEE-2007]

Column-I Column-II

(A) The object moves on the x-axis under a (P) The object
conservative force in such a way that its executes a simple
“speed” and position” satisfy harmonic
v = c1 √c2 − x 2 c1 and c2 are positive motion.
constants.

(B) The object moves on the x axis in such a way (Q) The object does
that its velocity and its displacement from the not change its
origin satisfy v = –kx, where k is a positive Direction
constant.

(C) The object is attached to one end of a mass- (R) The kinetic
less spring of a given spring constant. The energy of the
other end of the spring is attached to the object keeps on
ceiling of an elevator. Initially everything is at decreasing
rest. The elevator starts going upwards with a
24 | P a g e
 constant acceleration a. The motion of the
object is observed from the elevator during the
period it maintains this acceleration,

(D) The object is projected from the earth’s (S) The object can
GMe change its
surface vertically upwards with a speed 2√
Re direction only
where, Me is the mass of the earth and Re is once
the radius of the earth, neglect forces from
objects other than the earth

–
4. A spherically symmetric gravitational system of particle has a mass density
ρ for r ≤ R
ρ={ 0 where ρ0 is a constant. A test mass can undergo circular
0 for r > R
motion under the influence of the gravitational field of particles. Its speed V as
a function of distance r(0 < r < ∞) from the centre of the system is represented
by: [JEE-2008]

(A) (B) (C) (D)

–
5. Statement-1: An astronaut in an orbiting space station above the Earth
experiences weightlessness. And

Statement-2: And object moving around the Earth under the influence of
Earth’s gravitational force is in a state of ‘free-fall’ [JEE-2008]

(A) Statement-1 is true, Statement -2 is true and Statement-2 is correct

explanation for Statement-1
(B) Statement-1 is true, statement-2 is true and Statement-2 is correct
explanation for Statement-1
(C) Statement-1 is true, Statement-2 is false
(D) Statement-1 is false, Statement-2 is true
–

25 | P a g e
6. A thin uniform annular disc (see figure) of mass M has outer radius 4R and
inner radius 3R. The work required to take a unit mass from point P on its axis
to infinity is [JEE-2010]
2GM 2GM GM 2GM
(A) (4√2 − 5) (B) − (4√2 − 5) (C) (D) (√2 − 1)
7R 7R 4R 5R
–
7. A binary star consists of two stars A (mass 2.2 MS) and B (mass 11 MS), where
MS is the mass of the sun. They are separated by distance d and are rotating
about their centre of mass, which is stationary. The ratio of the total angular
momentum of the binary star to the angular momentum of a star B about the
centre of mass is [JEE-2010]

–
√6
8. Gravitational acceleration on the surface of a planet is g, where g is the
11
gravitational acceleration on the surface of the earth. The average mass density
2
of the planet is times that of the earth. If the escape speed on the surface of
3
the earth is taken to be 11kms−1 , the speed on the surface of the planet in kms −1
will be [JEE-2010]

–
9. A satellite is moving with a constant speed 'V' in a circular orbit about the
earth. An object of mass 'm' is ejected from the satellite such that it just
escapes from the gravitational pull of the earth. At the time of its ejection, the
kinetic energy of the object is [JEE-2011]
1 3
(A) mV 2 (B) mV 2 (C) mV 2 (D) 2mV 2
2 2

–
10. Two spherical planets P and Q have the same uniform density ρ, masses MP
and MQ , and surface areas A and 4A, respectively. A spherical planet R also has
uniform density ρ and its mass is (MP + MQ ). The escape velocities from the
planets P, Q and R, are VP , VQ and VR , respectively. Then [JEE-2012]
1
(A) VQ > VR > VP (B) VR > VQ > VP (C) VR /VP = 3 (D) VP /VQ =
2

–
11. Two bodies each of mass M, are kept fixed with a separation 2L. A particle of
mass m is projected from the midpoint of the lien joining their centres,
perpendicular to the line. The gravitational constant is G. The correct
statement(s) is (are) [JEE-2013]

26 | P a g e
 (A) The minimum initial velocity of the mass m to escape the gravitational field
GM
of the two bodies is 4√
L

(B) The minimum initial velocity of the mass m to escape the gravitational field
GM
of the two bodies is 2√
L

(C) The minimum initial velocity of the mass m to escape the gravitational field
2GM
of the two bodies is √
L

(D) The energy of the mass m remains constant

–
1
12. A planet of radius R = × (radius of Earth) has the same mass density as
10
R
Earth. Scientists dig a well of depth on it and lower a wire of the same length
5
and of liner mass density 10 kgm into it. If the wire is not touching
−3 −1

anywhere, the force applied at the top of the wire by a person holding it in place
is 9 take the radius of Earth = 6 × 106 m and the acceleration due to gravity on
Earth is 10ms −2 ) [JEE-2014]
(A) 96 N (B) 108 N (C) 120 N (D) 150 N

–
13. A bullet is fired vertically upwards with velocity v from the surface of a
spherical planet. When it reaches its maximum height, its acceleration due to
the planet's gravity is 1/4th of its value at the surface of the planet. If the
escape velocity from the planet is vesc = v√N, then the value of N is (ignore
energy loss due to atmosphere) [JEE-2015]

–
14. A rocket is launched normal to the surface of the Earth, away from the Sun,
along the line joining the Sun and the Earth. The sun is 3 × 105 times heavier
than the Earth and is at a distance 2.5 × 104 times larger than the radius of the
Earth. The escape velocity from Earth's gravitation field is ve = 11.2kms −1 . The
minimum initial velocity (vs ) required for the rocket to be able to leave the Sun-
Earth system is closest to (Ignore the rotation and revolution of the Earth and
the presence of any the other planet) [JEE-2017]

(A) vs = 62kms −1 (B) vs = 22kms −1

(C) vs = 72kms −1 (D) vs = 42kms −1

27 | P a g e
15. A planet of mass M, has two natural satellites with masses m1 and m2 . The
radii of their circular orbits are R1 and R 2 respectively. Ignore the gravitational
force between the satellites. Define v1 , L1 , K1 and T1 to be, respectively, the
orbital speed, angular momentum, kinetic energy and time period of revolution
of satellite 1 ; and v2 , L2 , K 2 and T2 to be the corresponding quantities of satellite
2. Given m1 /m2 = 2 and R1 /R 2 = 1/4.

Match the ratios in List-I to the numbers in List-II. [JEE-2018]

List-I List-II

v1 1
(P) (1)
v2 8

L1
(Q) (2) 1
L2

K1
(R) (3) 2
K2

T1
(S) (4) 8
T2

(A) P → 4; Q → 2; R → 1; S → 3 (B) P → 3; Q → 2; R → 4; S → 1

(C) P → 2; Q → 3; R → 1; S → 4(D) P → 2; Q → 3; R → 4; S → 1

–
16. Consider a spherical gaseous cloud of mass density ρ(r) in free space where r
is the radial distance from its center. The gaseous cloud is made of particles of
equal mass m moving in circular orbits about the common center with the
same kinetic energy K. The force acting on the particles is their mutual
gravitational force. If ρ(r) is constant in time, the particle number density
n(r) = ρ(r)/m is ( G = universal gravitational constant) [JEE-2019]
K K 3K K
(A) (B) (C) (D)
6πr2 m2 G πr2 m2 G πr2 m2 G 2πr2 m2 G

28 | P a g e
 ANSWER KEY
EXERCISE-01 [Single Correct]

Ques. Ans Ques. Ans Ques. Ans Ques. Ans Ques. Ans Ques. Ans Ques. Ans
1 A 2 C 3 A 4 A 5 A 6 D 7 C
8 A 9 B 10 C 11 B 12 B 13 B 14 C
15 D 16 A 17 A 18 D 19 C 20 B 21 A
22 A 23 C 24 B 25 B 26 D 27 D 28 A
29 B 30 B 31 C 32 A 33 C 34 B 35 D
36 C 37 A 38 A 39 D 40 C

EXERCISE-02 [Previous Year Questions]

Ques. Ans Ques. Ans Ques. Ans Ques. Ans Ques. Ans Ques. Ans Ques. Ans
1 B 2 D 3 D 4 D 5 C 6 D 7 A
8 D 9 A 10 A 11 C 12 C 13 D 14 A
15 C 16 A 17 C 18 B 19 D 20 D 21 D
22 B 23 C 24 A 25 D 26 B 27 C 28 C
29 B 30 A

[JEE ADVANCED]
EXERCISE-03 [MULTIPLE CORRECT]
Ques. Ans Ques. Ans Ques. Ans Ques. Ans Ques. Ans Ques. Ans Ques. Ans
1 AD 2 BCD 3 ACD 4 AC 5 BD 6 A 7 B
8 ABC 9 AD 10 ABCD 11 C 12 BC 13 AD 14 AC

EXERCISE-04 [Subjective]
Gm2 Re k2
1. 2. 3. 2R, 3R[3 − √3]
3L2 1−k2

GM 2GM
4. (a)√ ; (b)r√2; (c)√ 5. 32
r r

2GMR 2π
6. (a) √ where r = h + R (b) 3 R 7.
r(R+r) GM
3[√ 3 −ωe ]
8R

29 | P a g e
 √x2 −R2 Gme m
8. (1 − ) 4πR2 9. [ ]
x 2R2e

10. minimum distance of the particle = a

maximum distance of the particle = 5a/3

2Gmr1 r2 8000
11. m√[ (r ] 12. m⁄s
1 +r2 ) √2

EXERCISE-04 [Previous Year Questions]

Ques. Ans Ques. Ans Ques. Ans Ques. Ans Ques. Ans Ques. Ans Ques. Ans
A-P ;
B-QR ;
1 D 2 C 3 4 C 5 A 6 A 7 6
C-P ;
D-QR
8 3 9 B 10 BD 11 BD 12 B 13 2 14 D
15 B 16 D

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