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Q. 1
Two satellites move along a circular orbit in the same direction at a small distance from each other. A container has to be thrown from the first satellite onto the second one. When will the container reach the second satellite faster: ifits is thrown in the direction of motion of the first satellite or in the opposite
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direction? The velocity of the container with respect to the satelliteu is much less than that of the satellite v.
Q.2
Because the Earth bulges near the equator, the source ofthe Mississippi River (at about 50"N latitude), although high above sea level, is about 5 km closer to the centre of the Earth than is its mouth (at about 300N latitude). How can the river flow "uphill" as it flows south?
Q.3
Use -qualitative arguments to explain why the following four periods are equal (all are 84 mill, assuming a uniform Earth density) : time of revolution of a satellite just above the Earthls surface period of oscillation of mail in a tunnel through the Earth ,period of a simple pendulum having a length equal to the Earthls radius in a uniform field 9.8 mI S2 period of an infinite simple pendulum in the Earth's real gravitational field.
(a)
(b) (c) (d)
Q.4
After Sputnik I was put into orbit, it was said that it would not return to Earth but would bum up in its descent. Considering the fact that it did not bum up in its ascent, how is this possible?
Q.5
An artificial satellite is in a circular orbit about the Earth. How will its orbit change if one of its rockets is momentarily fired. (a) towards earth, (b) away from the'Earth, (c) in a forward direction, (d) in a backward direction, and (e) at right angles to the plane of the orbit?
Q.6
A stone is dropped along the centre of a deep vertical mine shaft. Assume no air resistance but consider the Earthls rotation. WIll the stone continue along the centre of the shaft? If not, describe its motion,
Q.7
An iron cube is placed near an iron sphere at a location remote from the Earth's gravity. What can you say about the location of the centre of gravity ofthe cube? Dfthe sphere? In general, does the location ofthe centre of gravity ofan object depend on the nature of the gravitational field in which the object is ill /' placed? ,
Q.8
Figure shows a particle of mass m that is moved from an infinite distance to the
f'*-"*''''''~'/;-'
centre of a ri~ of mass M,alo~g ~he central axis of the ~ing. For the tri?, how
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does the magnitude of the gr~Vltational force on the particle due to the rmg c1lange ..
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Q.9 In figure, a particle of mass m is initially at point A, at distance d from the centre of one uniform sphere and distance 4d from the centre of another uniform sphere, both of mass M:> > m. State whether, if you moved the particle to point D, the following would be positive, negative, or zero: the change in the gravitational potential energy of the particle, the work done by the net gravitational force on the particle, the work done by your force. What are the answers if, instead, the move were from point B to point C ?
(d) (c) (b)
(a)
1-,.- d
M Q.IO
B
c
M
D
Reconsider the situation of above question. Would the work done by you be positive, negative, or zero . Ifyou moved the partic1e(a) from A to B, (b) fromAto C, (6) fromB toD? (d) Rank those moves accroding to the absolute value of the work done by your force, greatest first.
4JBansal Classes
Question Bank on Gravitation
[21
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ONLY ONE OPTION IS CORRECT. Take approx. 2 minutes for answering each question.
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A larg~ spherical plan~t ofr~dius made of a material of density d, ~asa spheri~z1 ~a:ity .of radius R/2, with center of caVity a distance R/2 from the centre of the planet. Fmd the graVitational force on a small mass 'm' at the center of the cavity.,
( ~)
A~what altitude will the (B) acceleration of that at the earth's surface (given radius of /tRGmd/3due to gravity (C)be25% 2/tRGmd (D) 4/tRGmd/3 earth2/tRGmd/3 is R)?
/
(A) R/4 Q,3~hat /
(B) R
(C) 3R/8
(D) R/2
distance from the centre of the m~on is the point at which the strength of the resultant field of earth's and moon's gravitational field is equal to zero. The. earth's mass is 81 times that of moon and the distance between centres of these planets is 60R where Ris the radius ofthe earth
(A) 6R
Q.4 /
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(C) 3R
(B) 4R
(D) 5R
Two masses m1 & ~ are initiall~ at rest and .ar~ separated ?y a very large dist~ce. If~he mas~es approach each other subsequently, due to graVitational attraction between them, therr relative velOCity of approach at a separation distance of d is :
(B) (m1+ mJ G 2d
d
(C) [ (m1 + mJ 2GJl12 (D) (m1+~)1!2,2Gd
,
Let 0) 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
/' 1/ t{Ift6
(0) ..jRg
g
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(A)GM/R2
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(B)GM/2R2·
(C)GM/8R2
0~· !.l
A spherical hole of radius R/2 is excavated from the asteroid of mass M· as shown in fig, The gravitational acceleration at a point on·the surface of the asteroid just above the excavation is
(D)7GM/8R2
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If the radius of the earth be increased by a factor of 5, by what factor its density be changed to keep the value of g the same? (A) 1/25
(B) 1/5
(C)
1/J5
(0)5
Q,8 /'- A man of mass m starts falling towards a planet of mass M and radius R. As he reaches near to the ~
surface, he realizes that he will pass through a small hole in the planet. As he enters the hole, he sees that the planet is really made oftwo pieces a spherical shell of negligible thickness of mass 2~
and a point
mass M at the centre, Change in the force of gravity experienced by the man is
3
IGMm
2GMm
(A)3'R2 4JBansal Classes
(B) 0
(C)
3R2
Question Bank on Gravitation
4GMm
(D)3~
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An infinite number of masses, each of one kg are placed on the +ve X axis at 1m, 2m, 4m, origin. The magnitude of the gravitational field at origin due to this distribution of masses is:
4G
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3G
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(A)2G
fromthe
(C)~
(D)00
With what angular velocity the earth should spin in order that a body lying at 30° latitude may become weightless[R is radius of earth and g is acceleration due to gravity on the sutface of earth]
,
(A)
V3i [2g
(C) ~3~
(D)
f3R r4i
Two concentric shells of uniform density of mass Ml and ~ are situated as shown in the figure. The forces experienced by a particle of mass m when placed at positions A, B and C respectively are (given OA = p, DB = q and DC =r)
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(A) zero, GMlm -2- an d G(Ml +M2)m q
p
A satellite of the earth is revolving in circular orbit with a uniform velocity V If the gravitational force sUddenly disappears, the satellite will (A) continlle to move with the same velocity in the same orbit. ~) move tangentially to the original orbit with velocity V (C) fall down with increasing velocity. (D) come to a stop somewhere in its original orbit. A newly discovered plan~t has adensity eight times the density of the earth and a radius twice the radius of the earth. The time taken by 2 kg mass to fall freely through a distance S near the surface of the earth is 1 second. Then the time taken for a 4 kg mass to fall freely through the same distance S near the surface of the new planet is (B) 0.5 sec. (C) 1 sec. (D) 4 sec. (A) 0.25 sec.
Q/FOU~pa:ticles "Z/
of e~ualm~ss~s.M move along a circle ~fradius R unde: th~ action of their mutual graVltatlonal attraction mamtammg a square shape. The speed of each partH:;le IS
!
(AhIR GM(2J2+1) 4 ~)VR IGM(~+ll 4)
4JBansal Classes
(C)
GM
~R(~ +1)
Question Bank on Gravitation
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earth's At whatsurface? height above the earth's surface does the a.coelerationdue to gravity fall to 1% ofits . value at the
/ Q. ~
(A) 9R
? (C) 99R
(B) lOR
(0) IOOR
Q.l6~ind distanceto between ~~n~ centre massmass. ofa Length two particle <'" systemtheattached the ends of a lightofgravi~(and rod. Each particle hasofsame of the. rod is R, where R is the radius of earth
(A)R
(B)R/2
(O)R/4
,,~/(C) z~ro . ~~
r/
Q. l7/"The radius of.a pla~et is R. A sat~llite revolves around ~tin a circle of radius r with angular velocity The acceleratiOn due to the graVIty on planet's surface IS r3ro3
3
b
\
(A) r :0
3 2
(B) R20
roo'
r3ro2
(C) r :0
(D) R20
A solid sphere of uniform density and radius R applies a gravitational force of attraction equal to F 1 on a particle placed at a distance 3R from the centre of the sphere. A spherical cavity of radius R/2 is now made in the sphere as shown in the figure. The sphere with cavity now applies a
A
gravitational force F 2 on the same particle. The ratio F iF 1 is:
9 (A) 50
41 .(B) 50
3 (C) 25
22 (D) 25
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 ifit is a seconds pendulum on earth?
1
1 (A)
IQ./
..fi second
(B)
2J2
seconds
(C)
J2
second
(D~
2/2
second
M
shell of equal mass and having radius a .... (A) Gravitational field and potential both are zero at centre of the shell ..• A of mass M is atisazero distance frominside surface a thin spherical' (B)particle Gravitational field not aonly theofshell.but at a point outside the shell also.
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(C) Inside the shell, gravitational field alone is zero. (D) Neither gravitational field nor gravitational potential is zero inside the shell. /" :k:~~hree
.
point masses, M each, are moving in a circle, each with a speed v, under their mutual gravitational attractive force. The distance between any two masses must be: ".
(A)2GM/v2
voi..,/0v --.f
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(C) GM.J3 Iv2
.
(D)GM/v2
A cavity R/2 of radius madeofinside a solid Find spheretheofr.adius R. Theforce centre the cavity loc.ated distance from R/2 the is centre the sphere. gravitational on of a particle of is mass 'm' at a distance R/2 from the centre of the sphere on the line joining both the centres of sphere and cavity (opposite to the centre of cavity), [Here g = GM!R2, where M is the mass of the sphere] '" (A) mg
L.
-.
(B)3GM/v2
(B) 3m~ 8
mg (C)
'16
(D) none of these
---------------------------------
~Bansal Classes
Question Bank on Gravitation
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How much deep inside the earth should a man go ~o that his weight becomes one fourth of that at a point which is at a height R above the surface of earth.
?4
.'
~? :enti~. spberical (B) b~1s 15R/16 ,:""h of~ m are (C) placed 3R/4":' shown in figure (D) R/2 y!I Plot the vanation of g(graVltation mtenslty) along the X-axIS. ···0 ...·..·..···"1"...·.....··0··· x m
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m
~
••••.•
:.
x
(A)
x
(B)
x
(C)
x
(D)
A satellite revolves in the geostationary orbit but ina direction east to west. The time intelVal between its successive passing about a point on the equator is : (A) 48 hrs (B) 24 hrs (C) 12 hrs
V2a/
, Q.
(D) never
Aparticle starts' fromR rest at a distance R fromatthe theis:axis of a fixed ring of radius & mass M. Its velocity thecentre centreand of along the ring
R m«M
(A)~~~ R (C) "\/(1__.J2 1 )GM Q.27
A spherical uniform planet is rotating about its axis. The velocity of a point on its equator is VDue 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 t~ planet in terms of V
(A)Ve=2V ~
(B)Ve=V
;r:Q.29
the escape velocity
(B) 1 : 5
....b-.
is-
, T,WOPlanetsA~ B~ethe VB
.
(A) 2 ~ ,~Q.
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(D)Ve=
fjv
Two point masses of mass 4m and m respectively separated by d distance are revolving under mutual force of attfliction. Ratio of their kinetic energies will be :
/
(A) 1 : 4
"
(C)Ve= V/2
~ 1
same~
(B) .J2
(C) 1 : 1
(D) 1 : 2
density. If the radius ofAis twice that ofB, then the ratio of
(C) 1/.J2
(D) 1/2
The respectively, esca,pe,velocity 'then theonescape the surface velocity of the on earth another is Yo' planet JfMofand mass R are 2Mthe andmass radius andR/2 the will radius be of : the earth
. 1 Aball'A' of mass m falls to the surface of the earth from infinity, Another ball 'BI of mass 2m falls to the , , , earth A)4 from Vo the height equal (B)2Vo to six times radius (C)Vo of the earth then ratio (D)Vo/2 of velocities oflAI and IB' on reaching the earth is
(A)
...J(6/5)
4JBansal Classe$
(B)
...J(5/6)
(C) 1
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Question Ba:nk on Gravitation
(D)
--J(7/6)
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an artificial satellite revolves in circular orbit around the earth with a speed equal to halfits escape velocity from the earth. Then its height above the surface of earth will be: [Radius of earth=6400 km] .
Y1)
Q.' .
The ratio of gravitational acceleration at height 3R to that at height 4R from the surface of the earth is : 3200Riskmthe radius ofthe (B) 6400 (C) 12800 km (0)24000 km (where earth)km (A) 9/16 (B) 25/16 (C) 16125 (D) 16/9
, Q.3 4 /'"
A small body of superdense material, whose mass is twice the mass of the earth but whose size is very small compared to the size ofthe earth, starts from rest at a height H« R above the earth's surface, and reaches the earth's surface in time 1. Then t is equal to
(B) ~HI g
(A)~2H/g
, s
Q.35
\.d
(C) ..j2H 1 3g
(D) ..j4H/3g.
A rocket is launched straight up from the surface ofthe earth. When its altitude is one fourth of the radius of the earth, its fuel runs dut and therefore it coasts. The minimum velocity which the rocket must have when it starts to coast if it is to escape from the gravitational pull of the earth is [escape velocity on surface ofeartlJ..is 11.2km/s] . (A) lkm/s (B)5km/s (C) lOkm!s (D) i5km/s Gravitational potential difference between a point on surface of planet and another point 10m above is 4J/kg. Considering gravitational field to be uniform, how much work is done in moving a mass of2.0 kg from the surface to a point 5 .Om above the, surface ? (B) 2.5 J
(A) 0.40 J
Q.37
(C) 4.0 J
(D) 8.0
J
Referring to previous problem, what is the gravitational field strength in this region ? (A) 0.025 N kg-l (B) 0.40 N kg-1 (C) 2.5 Nkg-I (D) 4.0 N kg~l
~>/
?) .)YS9
Earth's satellite must be in equatorial plane .
The orbital :elocity ~f an artificial satelli.te in a .c~rcular orbi~ just above the earth's s~rfa~e is Vo. The value of orbItal velOCIty for another satelhte orbltmg at an altitude ofhalf of earth's radius IS
>:-~4lY/Aparticle /'
is projected with a velocity ~
4~
vertically upward from the surfuce of the earth, R being
the radius ofthe earth & g being the acceleration due to gravity on the surface of the earth. The velocity of the particle when it is at half the maximum height reached by it is
(A)
ff
tBansal Classes
,',I'
(C) .JgR
Question Bank on Gravitation
[7}
·J.. Q A 1
A satellite is in a circular orbit very close to the surface of a planet. At some point it is given an impulse along its direction of motion, causing its velocity to increase 11 times. It now goes into an elliptical orbit. The maximum possible value of
11
for this to occur is 1
Q. 42
\ \
QA3
(D) -
A satellite of mass In, initially at rest on the earth, is launched into a circular orbit at a height equal to the radius of the earth. The minimum energy required is (A)
(
(C) .J2 + 1
(B) .J2
(A) 2
.J3 mgR 4
I
1
3
(B) "2mgR
(C) "4mgR
(D) 4mgR
The figure shows the variation of energy with the orbit radius of a body in circular planetary motion. Find the correct statement about the curves A, B and C (A) A shows the kinetic energy, B the total energy and C the potential energy of ~ the system. &'j (B) C shows the total energy, B the kinetic energy and A the potential energy of the system. (C) C and A are kinetic and potential energies respectively and B is the total energy of the system. (D) A and B are kinetic and potential energies and C is the total energy of the system.
QA4
The ratio of the radius of the earth to that of the moon is 10. The ratio of the acceleration due to gravity on the earth to that on the moon is 6. The ratio of the escape velocity from the earth's sUlface to that from the moon is
(B) 1.5
(A) 6
(C) 12
(D)f60
Q. 45
An artificial satellite of the earth releases a package. If air resistance is neglected, the point where the package will hit (with respect to the position at the time of release) will be (A) ahead (B) exactly below (C) behind (D) it will never reach the earth
QA6
A space ship of mass m is in circular orbit of radius 2~ about the earth of mass M and radius Re' Energy required to transfer the space ship to circular orbit of radius 3Re is GMm
(A) ~ .Q.47
e
GMm
GMm
(C) 24R e
(D) 12Re
GMm
(B) ~
e
A particle is projected from the mid-point of the line joining two fixed particles each of mass m. Ifthe distance of separation between the fixed particles is I, the minimum velocity ofprojection of the particle so as to escape is equal to
1 . (A) ~GM
21 (B) ~GM
(C) ~2~M
(D)2~?-~~
'\{.' t~
048
The escape velocity for a planet is ve' A tunnel is dug along a diameter ofthe planet and a small body is dropped into it at the surface: When the body reaches the centre of the planet, its speed will be ve
(C)
4JBansal' Classes
2"
Question Bank on Gravitation
(D) zero
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Q.49
A per son brings a mass of 1 kg from infinity to a point A. Initially the mass was at rest but it moves at a speed of 2m1s as itreachesA.Theworkdonebythepersononthemassis~3J.ThepotentialatAis: (A) -3 J/kg (B) -2 J/kg (C) -5 J/kg (D) - 7 J/kg
Q. 50
A small ball of mass 'm' is released at a height 'R' above the earth surface, as shown in the :figure above. If the maximum depth of the ball to which it goes is R/2 inside the earth through a narrpwgrove before coming to rest momentarily. The grove, contain an ideal spring of spring constant K and natural length R, find the value ofK ifR isradius of earth and M mass of earth
b.e
3GMm R3
Q.51
§.
6GMm 7GMin R3 (D)R3
The magnitude ofthe potential energy per unit mass of the object at the surrace of earth is E. Then the escape velocity of the object is: (A) -fiE"
t ~ Q.52
(B)
(C) -JE
(B) 4E2
(D) 2E
Suppose a smooth tunnel is dug along a straight linejoining two points on the surface of the earth and a particle is dropped from rest at its one end. Assume that mass of earth is uniformly distributed over its
:y
n Volume. (A) the particle Then will emerge from the other end with velocity and radius respectively,
V~~
2Re
where Me and Re are earth's mass .
(B) the particle will come to rest at centre of the tunnel because at this position, particle is closest to earth centre. (C) potential energy of the particle will be equal to zero at centre of tunnel ifit is along a diameter.
e
(D )
~. 53
Q.54
acceleration of the particle will be proportional to its distance from midpoint of the tunneL
A hollow spherical shell is compressed to halfits radius. The gravitational potential at the centre (A) increases (B) decreases (C) remains same (D) during the compression increases then returns at the previous value. A body is projection horizontally from the surface of the Earth (radius = R) with a velocity equal to 'n' times the escape velocity. Neglect rotational effects of the earth. The maXhnumheight att~edbythe body from the Earth's surrace is R/2. Then, 'n' must be " (A) -JO:6
Q.55
(B) (.J3)/2
(C) .JO.4
(D) None
Consider two configurations of a system ofthree particles of masses ill, 2m and 3m. The work done by external agent in changing the configuration of the system from :figure (i) to :figure (ii) is
(A) zero
.Bansal
Classes
(B)
a 6Gm2
(
1 + .J2" 1)
(D)
a 60m2
(
2-121)
Question Bank on Gravitation
[9}
C/
6571. Two of mass m1 & ~ are ininelastic same circular orbit earth sense.satellites When they undergo completely collision, thearound combination
?
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(A) continues in same orbit (B) goes to a circular orbit oflesser radius (C) goes in an elliptical orbit within the original circle (D) goes in an elliptical orbit outside the original circle
rA uniform spherical planet (Radius R) has acceleration due to gravity at its surface g. Points P and Q
. I /
"
but are revolving in opposite
\
located inside and ?utside the planet have acceleration due to gravity ~ . Maximum possible separation between P and Q is
w7R 4
Q.58
9R
~-. 4
~~
A particle is dropped on Earth from height R (radius of Earth) and it bounces back to a height Rl2 the coefficient of restitution for collision is (ignore air resistance and rotation of Earth)
(A) -3 2 Q. 59
3R
002
(B)
H-3
(C)
If-3
(D)
~-2
A body of mass m is lifted up from the surface of the earth to a height three times the radius of the earth. The change in potential energy ofthe body is (A) 3mgR (B) 3/4 mgR (C) 1/3 mgR (D) 2/3 mgR where g is acceleration due to gravity at the surface of earth.
/' ~TWO ,
satellites S1 and S2 describe circular orbits of radius r and 2r respectively orbital angular velocity ofS1 is ro, that ofS2 is :
(A) ~en . Q.62
ro/(2..J2)
(B)
(ro..J2)/3
(C)
ro/2
(D)
around a planet. If the
w/..J2
a satellite ~oves aroun.d t~e earth in a certain orb~t, the q~alltity which remai~s constant is : (A) angularveloClty (B) kinet:1cenergy (C) aenal velOCity (D) potential energy Asatellite is launched into a circular orbit of radius R around the earth. A second satellite is launched into an orbit o~dius
~5%
1.02R The period of second satellite is larger than the first one by approximately
. (B) 3%
(C) 1%
(D) 2%
Asatellite ofmass 5M orbits the earth in a circular orbit. At one point in its orbit, the satellite explodes into twe pieces, one of mass M and the other of mass 4M. After the explosion the mass M ends up travelling in thf. same circular orbit, but in opposite direction. After explosion the mass 4M is in (A) bound orbit (B) unbound orbit (C) partially bound orbit (D) data is insufficient to determine the nature of the orbit.
Q.64 ~
V
ft-sitellitecan be ina ofgeostationary orbit itsaround a planet aatsatellite a distance from be theincentre of the planet. the angular velocity the planet about axis doubles, canr now a geostationary orbj1 around the planet ifits distance from the centre of the planet is ..
r (A) "2
.Bansal Classes
r
r (B) 2.../2
(C)
(4)113
Question Bank on Gravitation
r (D) (2)1/3
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A planet of mass m is in an elliptical orbit about the sun (m < < Msun) with an orbital period T. !fA be the area of orbit, then its angular momentum would be:
2mA
(A)-T
mA (B) mAT
(C)
2T'
(D)2mAT
9-66/ The planets
with radii Rl, ~ have densities Pl' P2 respectively. Their atmospheric pressures are Pp P2 respectively. Therefore, the ratio of masses of their atmospheres, neglecting variation of g within the limits of atmosphere, is
(A) Pl~P1 / P2R1P2 Q, 67
/
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/ /Q, 68
l.
(B) Pl~P2
/ P2RlPl'
(C) P1R1PI
I P2~P2
(D)P1R1P2/
P2~PI
Suppose the gravitational force varies inversely as the nth power of distance. Then the time period of a planet in circular orbit of radius R around the sun will be proportional to
(A) R (n+1) 2
(B)R(n-l) 2
(0)R(n-2) 2
A satellite is seen after every 6 hours over the equator. It is known that it rotates opposite to that of earth's direction, Then the angular velocity of the satellite about the centre of earth will be : .. (A)
Te/2
rad/hr
(B) 1[/3 rad/hr
(C)
Te/4
rad/hr
(D) n/8 rad/hr
A satellite is orbiting round the earth. In a particular orbit its time period is T and orbital speed is V. In another orbit the orbital speed is 2V, then time period will be
WIT Q .~In
;r
~m
~n
~m
a double star system, the ~asses of the two stars are M and 3M. The orbit radius ofthe lighter star is R. The time period of each star is .. (B) 16rc[R3/GM]1/2 (C)4~[2R3/GM]1/2 (D) None (A) 811: [R3/GM]1/2
Question No. 71 to 72
,an
Figure shows the orbit of a planet P round the sun S. AB and CD are the minor and major axes of the ellipse. Q ,11" ../ I .~
Ift1 is t4e time taken by the planet to travel along ACB and ~ the time along BDA, then (B)tl >t2 (C)tl <~ (D) nothing can be concluded (A)t1 =~
Q.?Y' IfU
is the potential energy and K kinetic energy then IUI > IKI at (A)OnlyD (B)OnlyC (C) both D & C (0) neither D nor C
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-
Y'
end in time(neglect earthorientation rotation) through earth, then a ball released .'. 73 / If a tunnel is cut at any from one end will reac,h th,e other (A) 84.6 minutes (B) 42.3 minutes (C) 8 minutes (D) depends on orientation
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Question Bank on Gravitation
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Questions 74 to 79 (6 questions) Two stars bound together by gravity orbit each other because of their mutual attraction. Such a pair of stars is referred to as a binary star system. One type ofbinary system is that of a black hole and a companion star, The black hole is a star that has collapsed on itself and is so massive that not even light rays can escape it~ gravitational pull. Therefore, when describing the relative motion of a black hole and a companion star, the motion of the black hole can be assumed negligible compared to that of the companion. The orbit of the companion star is either elliptical with the black hole at one of the foci or circular with the black hole at the centre. The gravitational potential energy is given by U = - GmMir, where G is the universal gravitational constant, m is the mass of the companion star, M is the mass of the black hole, ane r is the distance between the centre of the companion star and the centre of the black hole. Since th{ gravitational force is conservative, the companion star's total mechanical energy is a constant oftht motion. Because of the periodic nature of the orbit, there is a simple relation betweenthe average kinetic energy ofthe companion star and its average potential energy < U>. In particular, = -
At which point in the elliptical orbit does the companion star attain its maximum kinetic energy? (B) Perigee (C) The point midway from apogee to perigee (A) Apogee (D) All points in the orbit, since the kinetic energy is a constant of the motion.
Q.75
For circular orbits, the potential energy of the companion star is constant throughout the orbit. Ifth, radius of the orbit doubles, what is the new value of the velocity ofthe companion star? (A) It is
1/2of the
old value
(B) It is
(C) It is the, same as the old value.
1/.Ji
ofthe old value
(D) It is double the old value
Q. 76
Which of the following prevents the companion star from leaving its orbit and falling into the black hole' (A) The centripetal force (B) The gravitational force (C) The companion star's potential energy (D) the companion star's kinetic energy
Q.77
The work done on the companion star in one complete orbit by the gravitational force ofthe black hole equal (A) the difference hi the kinetic energy of the companion star between apogee and perigee. (B) the total mechanical energy of the companion star (C) zero (D) the gravitational force on the companion star times the distance that it travels in one orbit.
Q.78
For a circular orbit, which of the following gives the correct expression for the total energy? (A)-(1/2)ri1v2 (B)mv2 (C)-(GmM)/r (D)(GmM)/2r
Q.79
What is the ratio of the acceleration of the black hole to that of the companion star? (A)M/m (B)m/M (C)mM/r (D) 111
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Question Bank on Gravitation
{12
s r.
ONE OR MORE THAN ONE OPTION AlAr BE CORRECT
s e
Take approx. 3 minutes for answering each question. Q. 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 a point inside is zero (D) at a point inside is proportional to its distance from the centre.
Q.2
Mark the correct statement/s (A) Gravitational potential at curvature centre of a thin hemispherical
shell of radius R and mass M is
GM
)n equalto
R'
(B) Gravitational field strength at a point lying on the axis of a thin, uniform circular ring of radius R and
GMx mass M is equal to (R 2 + x2 )3/ 2 where x is distance of that point from centre of the ring. (C) Newton's law of gravitation for gravitational force between two bodies is applicable only when bodies have spherically symmetric distribution of mass . (D) None ofthe~e.
:he
Q.3
Three particles are projected vertically upward from a point on the surface of the earth with velocities ~(2gR13), ~(gR), ~(4gR13) respectively where Ris the radius of the earth and g is the acceleration due to gravity on the surface of the earth. The maximum heights attained are respectively hI' ~, ~. (B) ~: h3 = 3 : 4 (C) hI : h3 = 1 : 4 (D) ~ = R (A) hI : ~:= 2 : 3
Q.4
A geostationary satellite is at a height h above the surface of earth. If earth radius is R (A) The minimum colatitude q on earth upto which the satellite can be used for communication sin-I (R/R
is
+ h).
(B) The maximum colatitudes q on earth upto which the satellite can be used· for communication is sin-I (R/R
+ h) ..
(C) The area on earth escaped fromthis satellite is given as 2pR2 (1 + sinq) (D) The area on earth escaped from this satellite is given as 2pR2 (1 + cosq) Q. 5
•
Gravitational potential at the centre of curvature of a hemispherical bowl of radius R and mass Mis V (A) gravitational potential at the centre of curvature of a.thinunifQrmwire ofmassM, bent into a semicircle of radius R, is also equal to V (B) In part (A) if the same wire is bent into a quarter of a circle then. also the gravitational potential at the centre of curvature will be V (C) In part (A) if the same wire mass is nonuniformly distributed along its length and it is bent into a semicircle of radius R, gravitational potential at the centre is V (D) none ofthese
Q.6
[12
In a solid sphere two small symmetrical cavities are created whose centres lie on a diameter AB of sphere on opposite sides ofthe centre. (A) The gravitational field at the centre ofthe sphere is zero. (B) The gravitational potential at the centre remains unaffected ifcavitiesare not present' (C) A circle at which all points have same potential is in the plane of diameter AB. (D) A circle at which all points have same potential is in the plane perpendicular to the diameter AB .
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Question Bar'k on Gravitation
{I3}
I
Q.7
Q .14
The spherical planets have the same mass but densities in the ratio 1 : 8. For these planets, the (A) acceleration due to gravity will be in the ratio 4: 1 (B) acceleration due to gravity will be in the ratio 1 : 4
.fi : 1 escape velocities from their surfaces will be in the ratio 1 : .fi
(C) escape velocities from their surfaces will be in the ratio (0)
Q.1 Q.8
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 (0) its period of revolution around the earth increases
Q.9
A communications Earth satdlite (A) goes round the earth from east to west (B) can be in the equatorial plane only (C) can be vertically above any place on the earth (D) goes round the earth from west to east
Q. lOAn earth satellite is moved from one stable circular orbit to another larger and stable circular orbit. The following quantities increase for the satellite asa result of this change (A) gravitational potential energy (B) angularvleocity (C) linear orbital velocity (0) centripetal acceleration Q.l1
Two satellites of same mass of a planet in circular orbits have periods of revolution 32 days and 256 days. If the radius ofthe orbit of the first is x, then the . (A) radius ofthe orbit of the second is 8x (B) radius of the orbit of the second is 4x (C) total mechanical energy of the second is greater than that of the first (0) kinetic energy of the second is greater than that of the fIrst.
Q .12
Two satellites S1'& S2of equal masses revolve in the same sense around a heavy planet in coplanar circular orbit of radii R & 4R (A) the ratio ofperiod of revolution s1 & s2is 1 : 8. (B) their velocities are in the ratio 2 : 1 (C) their angular momentum aboutthe planet are in the ratio 2 : 1 (D) the ratio of angular velocities of S2w.r.t. S1when all three are in the same line is 9 : 5.
Q.13
A satellite S is moving in an elliptical orbit around the earth. The mass of the satellite is very small compared to the mass of the earth (A) the acceleration ofS is always directed towards the centre ofthe earth (B) the angular momentum ofS about the centre of the earth changes in direction, but its magnitude remains constant (C).the total mechanical energy of S varies periodically with time (D) the linear momentum ofS remains constant in magnitude
4JBansal Classes
Question Bank on Gravitation
[14J
,
Q .14
If a satellite orbits as close to the earth's surface as possible, (i\) its speed is O1axllnUnl (B) time period ofits rotation is Il1inimum (C) the total energy of the 'earth plus satellite' system is minimum (D) the total energy of the 'earth plus satellite'system is maximum
Q. 15
For a satellite to orbit around the earth, which of the following O1Ustbe true? (i\) 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 36,000 km
.15
T
(D) Its period of rotation must be
4JBansal Classes
> 21t~R I g where Ris radius of earth
Question Bank on Gravitation
[15J
ex
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if/I
C A D B A .12 B Q.4 .13 .7 .14 A Q .9 C,D A,D A,C Q .6 Q.28 Q.12 A,B,D .42 Q.40 Q.19 D Q.3 Q.41 Q.20 Q.21 Q.2 B .5 .4 Q.10 Q.8 .35 Q.27 .I0 .l1 Q.3 Q.16 Q.17 C B,C Q.75 Q.39 B,D A,C .34. .:::Q.44 Q.33 Q.26 CQ.38 Q.47 Q.48 Q.49 Q.72 Q.15 Q.18 Q.ll .Q.37 Q.23 Q.79 Q.74 Q.73 Q.76 B CD' A,D Q.7 Q.30 Q.32 .56 Q.51 Q.54 C Q.55 .62 Q.65 Q.68 .63 ·B Q.14 A,B,C Q.24 Q.25 A Q.6 Q.46 Q.31 D Q.52 Q .53 Q.58 Q.61 Q.60 Q.59 Q.67 Q.66 Q~69 Q.70 A D Q.77 C Q.2 Q.45A A,D D'
Q.l . Q:l ONE OR MORE THAN ONE OPTION MAYBE CORRECT ON IS CORRECT
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Question Bank on Gravitation
[16J
