Master Resource Book in JEE Main Physics

D. B. Singh

Chapter 10

Gravitation - all with Video Answers

Educators

Section 1

Round 1

Problem 1

Two spheres of radius $r$ and $2 r$ are touching each other. The force of attraction between them is proportional to
(a) $r^{6}$
(b) $r^{4}$
(c) $r^{2}$
(d) $r^{-2}$

Subash Charan

Subash Charan

Numerade Educator

Problem 2

A solid sphere of uniform density and radius $R$ applies gravitational force of attraction equal to $F_{1}$ on a particle placed at
$P$, distance $2 R$ from the centre $O$ of the sphere. A spherical cavity of radius $R / 2$ is now made in the sphere as shown in figure. The sphere with cavity now applies an gravitational force $F_{2}$ on same particle placed at $P$. The ratio $F_{2} / F_{1}$ will be
(a) $1 / 2$
(b) $7 / 9$
(c) 3
(d) 7

Subash Charan

Numerade Educator

Problem 3

A uniform ring of mass $M$ and radius $r$ is placed directly above a uniform sphere of mass $8 M$ and of same radius $R$. The centre of the ring is at a distance of $d=\sqrt{3} R$ from the centre of the sphere. The gravitational attraction between the sphere and the ring is
(a) $\frac{G M^{2}}{R^{2}}$
(b) $\frac{3 G M^{2}}{2 R^{2}}$
(c) $\frac{2 G M^{2}}{\sqrt{2} R^{2}}$
(d) $\frac{\sqrt{3} G M^{2}}{R^{2}}$

Subash Charan

Numerade Educator

Problem 4

Imagine a light planet revolving around a very massive star in a circular orbit of radius $r$ with a period of revolution $T$. If the gravitational force of attraction between the planet and the star is proportional to $R^{-3 / 2}$, then $T_{2}$ is proportional to
(a) $R^{3}$
(b) $R^{5 / 2}$
(c) $R^{3 / 2}$
(d) $R^{7 / 2}$

Subash Charan

Numerade Educator

Problem 5

If a planet of given density were made larger its force of attraction for an object on its surface would increase because of planet's greater mass but would decrease because of the greater distance from the object to the centre of the planet. Which effect predominate?
(a) Increase in mass
(b) Increase in radius
(c) Both affect attraction equally
(d) None of the above

Subash Charan

Numerade Educator

Problem 6

Both earth and moon are subject to the gravitational force of the sun. As observed from the sun, the orbit of the moon $\quad$ [NCERT Exemplar]
(a) will be elliptical
(b) will not be strictly elliptical because the total gravitational force on it is not central
(c) is not elliptical but will necessarily be a closed curve
(d) deviates considerably from being elliptical due to influence of planets other than earth

Subash Charan

Numerade Educator

Problem 7

Different points in earth are at slightly different distances from the sun and hence experience different forces due to gravitation. For a rigid body, we know that if various forces act at various points in it, the resultant motion is as if a net force acts on the CM (centre of mass) causing translation and a net torque at the CM causing rotation around an axis through the CM For the earth-sun system (approximating the earth as a uniform density sphere) $\quad$ [NCERT Exemplar]
(a) the torque is zero
(b) the torque causes the earth to spin
(c) the rigid body result is not applicable since the earth is not even approximately a rigid body
(d) the torque causes the earth to move around the sun

Subash Charan

Numerade Educator

Problem 8

Two astronauts have deserted their space ships in a region of space far from the gravitational attraction of any other body. Each has a mass of $100 \mathrm{~kg}$ and they are $100 \mathrm{~m}$ apart. They are initially at rest relative to one another. How long will it be before the gravitational attraction brings them $1 \mathrm{~cm}$ closer together?
(a) $2.52$ days
(b) $1.41$ days
(c) $0.70$ days
(d) $0.41$ days

Subash Charan

Numerade Educator

Problem 9

If three particles each of mass $M$ are placed at the three corners of an equilateral triangle of side $a$, the forces exerted by this system on another particle of mass $M$ placed (i) at the mid point of a side and (ii) at the centre of the triangle are respectively
(a) 0,0
(b) $\frac{4 G M^{2}}{3 a^{2}}, 0$
(c) $0, \frac{4 G M^{2}}{3 a^{2}}$
(d) $\frac{3 G M^{2}}{a^{2}}, \frac{G M^{2}}{a^{2}}$

Subash Charan

Numerade Educator

Problem 10

The gravitational attraction between the two bodies increases when their masses are
(a) reduced and distance is reduced
(b) increased and distance is reduced
(c) reduced and distance is increased
(d) increased and distance is increased

Subash Charan

Numerade Educator

Problem 11

A spherical hollow is made in a lead sphere of radius $R$ such that its surface touches the outside surface of the lead sphere and passes through the centre. The mass of the lead sphere before hollowing was $M$. The force of attraction that this sphere would exert on a particle of mass $m$ which lies at a distance $d(>R)$ from the centre of the lead sphere on the straight line joining the centres of the sphere and the hollow is
(a) $\frac{G M m}{d^{2}}$
(b) $\frac{G M m}{8 d^{2}}$
(c) $\frac{G M m}{d^{2}}\left[1+\frac{1}{8\left(1+\frac{R}{2 d}\right)}\right]$
(d) $\frac{G M m}{d^{2}}\left[1-\frac{1}{8\left(1-\frac{R}{2 d}\right)^{2}}\right]$

Subash Charan

Numerade Educator

Problem 12

A spherical hollow is made in a lead sphere of radius $R$ such that its surface touches the outside surface of the lead sphere and passes through the centre. The mass of the lead sphere before hollowing was $M$. The force of attraction that this sphere would exert on a particle of mass $m$ which lies at a distance $d(>R)$ from the centre of the lead sphere on the straight line joining the centres of the sphere and the hollow is
(a) $\frac{G M m}{d^{2}}$
(b) $\frac{G M m}{8 d^{2}}$
(c) $\frac{G M m}{d^{2}}\left[1+\frac{1}{8\left(1+\frac{R}{2 d}\right)}\right]$
(d) $\frac{G M m}{d^{2}}\left[1-\frac{1}{8\left(1-\frac{R}{2 d}\right)^{2}}\right]$

Prem Bijarniya

Numerade Educator

Problem 13

If suppose moon is suddenly stopped and then released (given radius of moon is one-fourth the radius of earth) and the acceleration of moon with respect to earth is $0.0027 \mathrm{~ms}^{-2}$ ), then the acceleration of the moon just before striking the earth's surface is (Take $g=10 \mathrm{~ms}^{-2}$ )
(a) $0.0027 \mathrm{~ms}^{-2}$
(b) $5.0 \mathrm{~ms}^{-2}$
(c) $6.4 \mathrm{~ms}^{-2}$
(d) $10 \mathrm{~ms}^{-2}$

Subash Charan

Numerade Educator

Problem 14

If suppose moon is suddenly stopped and then released (given radius of moon is one-fourth the radius of earth) and the acceleration of moon with respect to earth is $0.0027 \mathrm{~ms}^{-2}$ ), then the acceleration of the moon just before striking the earth's surface is (Take $g=10 \mathrm{~ms}^{-2}$ )
(a) $0.0027 \mathrm{~ms}^{-2}$
(b) $5.0 \mathrm{~ms}^{-2}$
(c) $6.4 \mathrm{~ms}^{-2}$
(d) $10 \mathrm{~ms}^{-2}$

Subash Charan

Numerade Educator

Problem 15

If different planets have the same density but different radii, then the acceleration due to gravity on the surface of the planet is related to the radius $(R)$ of the planet as
(a) $g \propto R^{2}$
(b) $g \propto R$
(c) $g \propto \frac{1}{R^{2}}$
(d) $g \propto \frac{1}{R}$

Subash Charan

Numerade Educator

Problem 16

A thief stole a box full of valuable articles of weight $w$ and while carrying it on his head jumped down from a wall of height $h$ from the ground. Before he reaches the ground, he experienced a load
(a) zero
(b) $w / 2$
(c) $w$
(d) $2 w$

Subash Charan

Numerade Educator

Problem 17

Assuming the earth to be a sphere of uniform mass density, how much would body weigh half way down to the centre of earth if it weighed $250 \mathrm{~N}$ on the surface?
(a) $225 \mathrm{~N}$
(b) $325 \mathrm{~N}$
(c) $100 \mathrm{~N}$
(d) $125 \mathrm{~N}$

Subash Charan

Numerade Educator

Problem 18

The maximum vertical distance through which a full dressed astronaut can jump on the earth is $0.5 \mathrm{~m}$. Estimate the maximum vertical distance through which he can jump on the moon, which has a mean density $2 / 3 \mathrm{rd}$ that of earth and radius one quarter that of the earth
(a) $1.5 \mathrm{~m}$
(b) $3 \mathrm{~m}$
(c) $6 \mathrm{~m}$
(d) $7.5 \mathrm{~m}$

Subash Charan

Numerade Educator

Problem 19

In the above problem, the ratio of the time duration of his jump on the moon to that of his jump on the earth is
(a) $1: 6$
(b) $6: 1$
(c) $\sqrt{6}: 1$
(d) $1: \sqrt{6}$

Subash Charan

Numerade Educator

Problem 20

Particles of masses $2 M, m$ and $M$ are respectively at points $A, B$ and $C$ with $A B=\frac{1}{2}(B C) . m$ is much-much smaller than $M$ and at time $t=0$, they are all at rest
(a) $m$ will remain at rest
(b) $m$ will move towards $M$
(c) $m$ will move towards $2 M$
(d) $m$ will have oscillatory motion

Subash Charan

Numerade Educator

Problem 21

The earth is an approximate sphere. If the interior contained matter which is not of the same density everywhere, then on the surface of the earth, the acceleration due to gravity $\quad$ [NCERT Exemplar]
(a) will be directed towards the centre but not the same everywhere
(b) will have the same value everywhere but not directed towards the centre
(c) will be same everywhere in magnitude directed towards the centre
(d) cannot be zero at any point

Subash Charan

Numerade Educator

Problem 22

The masses and radii of the earth and moon are $M_{1}, R_{1}$ and $M_{2}, R_{2}$ respectively. Then centres are distance $d$ apart. The minimum velocity with which a particle of mass $M$ should be projected from a point midway between their centres so that it escapes to infinity is
(a) $2 \sqrt{\frac{G}{d}\left(M_{1}+M_{2}\right)}$
(b) $2 \sqrt{\frac{2 G}{d}\left(M_{1}+M_{2}\right)}$
(c) $2 \sqrt{\frac{G M}{d}\left(M_{1}+M_{2}\right)}$
(d) $2 \sqrt{\frac{G M\left(M_{1}+M_{2}\right)}{d\left(R_{1}+R_{2}\right)}}$

Subash Charan

Numerade Educator

Problem 23

At a given place where, acceleration due to gravity is $g \mathrm{~ms}^{-2}$, a sphere of lead of density $d \mathrm{kgm}^{-3}$ is gently released in a column of liquid of density $\rho \mathrm{kgm}^{-3}$. If $d>\rho$, the sphere will
(a) fall vertically with an acceleration of $\mathrm{g} \mathrm{ms}^{-2}$
(b) fall vertically with no acceleration
(c) fall vertically with an acceleration $g\left(\frac{d-\rho}{d}\right)$
(d) fall vertically with an acceleration $\rho / d$

Subash Charan

Numerade Educator

Problem 24

What is the height, the weight of body will be the same as at the same depth from the surface of the earth? Radius of earth is $R ?$
(a) $\frac{R}{2}$
(b) $\sqrt{5} R-R$
(c) $\frac{\sqrt{5} R-R}{2}$
(d) $\frac{\sqrt{3} R-R}{2}$

Subash Charan

Numerade Educator

Problem 25

There is a mine of depth about $2.0 \mathrm{~km}$. In this mine the conditions as compared to those at the surface are
(a) lower air pressure, higher acceleration due to gravity
(b) higher air pressure, lower acceleration due to gravity
(c) higher air pressure, higher acceleration due to gravity
(d) lower air pressure, lower acceleration due to gravity

Subash Charan

Numerade Educator

Problem 26

A clock $S$ is based on oscillation of a spring and a clock $P$ is based on pendulum motion. Both clock run at the same rate on earth. On a planet having the same density as earth but twice the radius,
(a) $S$ will run faster than $P$
(b) $P$ will run faster than $S$
(c) both will run at the same rate as on the earth
(d) both will run at the same rate which will be different from that on the earth

Subash Charan

Numerade Educator

Problem 27

If the radius of the earth were to shrink by $1 \%$ its mass remaining same, the acceleration due to gravity on the earth's surface would
(a) decrease by $2 \%$
(b) remain unchanged
(c) increase by $2 \%$
(d) become zero

Subash Charan

Numerade Educator

Problem 28

Two spherical planets $A$ and $B$ have same mass but densities in the ratio $8: 1$. For these planets, the ratio of acceleration due to gravity at the surface of $A$ to its value at the surface of $B$ is
(a) $1: 4$
(b) $1: 2$
(c) $4: 1$
(d) $8: 1$

Subash Charan

Numerade Educator

Problem 29

The height at which the acceleration due to gravity decreases by $36 \%$ of its value on the surface of the earth. (The radius of the earth is $R$ ).
(a) $\frac{R}{6}$
(b) $\frac{R}{4}$
(c) $\frac{R}{2}$
(d) $\frac{2}{3} R$

Subash Charan

Numerade Educator

Problem 30

If the value of $g$ acceleration due to gravity at earth surface is $10 \mathrm{~ms}^{-2}$, its value in $\mathrm{ms}^{-2}$ at the centre of the earth, which is assumed to be a sphere of radius $R$ metre and uniform mass density is
(a) 5
(b) $10 / R$
(c) $10 / 2 R$
(d) zero

Subash Charan

Numerade Educator

Problem 31

When of the following graphs correctly represents the variation of $g$ on earth?

Subash Charan

Numerade Educator

Problem 32

If the force inside the earth surface varies as $x^{n}$, where $r$ is the distance of body from the centre of earth, then the value of $n$ will be
(a) $-1$
(b) $-2$
(c) $]$
(d) 2

Subash Charan

Numerade Educator

Problem 33

$320 \mathrm{~km}$ above the surface of earth, the value of acceleration due to gravity is nearly $90 \%$ of its value on the surface of the earth. Its value will be $95 \%$ of the value on the earth's surface
(a) nearly $160 \mathrm{~km}$ below the earth's surface
(b) nearly $80 \mathrm{~km}$ below the earth's surface
(c) nearly $640 \mathrm{~km}$ below the earth's surface
(d) nearly $320 \mathrm{~km}$ below the earth's surface

Subash Charan

Numerade Educator

Problem 34

The acceleration due to gravity at a height $1 / 20$ th of the radius of the earth above the earth surface is $9 \mathrm{~ms}^{-2}$. Its value at a point at an equal distance below the surface of the earth in $\mathrm{ms}^{-2}$ is about
(a) $8.5$
(b) $9.5$
(c) $9.8$
(d) $11.5$

Subash Charan

Numerade Educator

Problem 35

At a distance $320 \mathrm{~km}$ above the surface of earth, the value of acceleration due to gravity will be lower than its value on the surface of the earth by nearly (radius of earth $=6400 \mathrm{~km}$ )
(a) $2 \%$
(b) $6 \%$
(c) $10 \%$
(d) 1496

Subash Charan

Numerade Educator

Problem 36

The depth from the surface of the earth of radius $R$ at which the acceleration due to gravity will be $75 \%$ of the value on the surface of the earth is
(a) $R / 4$
(b) $R / 2$
(c) $3 \mathrm{R} / 4$
(d) $R / 8$

Subash Charan

Numerade Educator

Problem 37

Two equal masses $m$ and $m$ are hung from a balance whose scale pans differ in height by $h$. If $\rho$ is the mean density of earth, then the error in weighing machine is
(a) zero
(b) $4 \pi \mathrm{Gpmh} / 3$
(c) $8 \pi \mathrm{Gpmh} / 3$
(d) $2 \pi G \rho m h / 3$

Subash Charan

Numerade Educator

Problem 38

One goes from the centre of the earth to a distance two-third the radius of the earth, where will the acceleration due to gravity be the greatest?
(a) At the centre of the earth
(b) At a height half the radius of the earth
(c) At a height one-third the radius of the earth
(d) At a height two-third the radius of the earth

Subash Charan

Numerade Educator

Problem 39

Mass of moon is $7.34 \times 10^{22} \mathrm{~kg}$. If the acceleration due to gravity on the moon is $1.4 \mathrm{~ms}^{-2}$, the radius of the moon is $\left(G=6.667 \times 10^{-11} \mathrm{Nm}^{2} \mathrm{~kg}^{-2}\right)$
(a) $0.56 \times 10^{4} \mathrm{~m}$
(b) $1.87 \times 10^{6} \mathrm{~m}$
(c) $1.92 \times 10^{6} \mathrm{~m}$
(d) $1.01 \times 10^{8} \mathrm{~m}$

Subash Charan

Numerade Educator

Problem 40

The ratio of acceleration due to gravity at a height $h$ above the surface of the earth and at a depth $h$ below the surface of the earth for $h<$ radius of earth
(a) is constant
(b) increases linearly with $h$
(c) decreases linearly with $h$
(d) decreases parabolically with $h$

Subash Charan

Numerade Educator

Problem 41

At what height in $\mathrm{km}$ over the earth's pole the free fall acceleration decreases by one percent? (Assume the radius of the earth to be $6400 \mathrm{~km}$ )
(a) 32
(b) 64
(c) 80
(d) $1.253$

Subash Charan

Numerade Educator

Problem 42

If a man weighs $90 \mathrm{~kg}$ on the surface of earth, the height above the surface of the earth of radius $R$, where the weight is $30 \mathrm{~kg}$, is
(a) $0.73 R$
(b) $R[\sqrt{3}$
(c) $R / 3$
(d) $\sqrt{3} R$

Subash Charan

Numerade Educator

Problem 43

Two equal masses $m$ and $m$ are hung from a balance whose scale pan differs in vertical height by $h / 2$. 'The error in weighing in terms of density of the earth $\rho$ is
(a) $\frac{1}{3} \pi G \rho m h$
(b) $\pi G \pi m h$
(c) $\frac{4}{3} \pi G \rho m h$
(d) $\frac{8}{3} G \rho m h$

Subash Charan

Numerade Educator

Problem 44

The radius of the earth is $6400 \mathrm{~km}$ and $g=10 \mathrm{~m} / \mathrm{s}^{2}$ in order that a body of $5 \mathrm{~kg}$ weight zero at the equator the angular speed of the earth is
(a) $1 / 80 \mathrm{rad} / \mathrm{s}$
(b) $1 / 400 \mathrm{rad} / \mathrm{s}$
(c) $1 / 800 \mathrm{rad} / \mathrm{s}$
(d) $1 / 1600 \mathrm{rad} / \mathrm{s}$

Subash Charan

Numerade Educator

Problem 45

What should be the angular speed of earth in $\mathrm{rads}^{-1}$ so that a body of $5 \mathrm{~kg}$, weighs zero at the equator? (Take $g=10 \mathrm{~ms}^{-2}$ and radius of earth $=6400 \mathrm{~km}$ ).
(a) $1 / 1600$
(b) $1 / 800$
(c) $1 / 400$
(d) $1 / 80$

Subash Charan

Numerade Educator

Problem 46

The bodies situated on the surface of earth at its equator, becomes weightless, when the earth has KE about it axis
(a) $m g R$
(b) $2 m g R / 5$
(c) $M g R / 5$
(d) $5 \mathrm{Mg} R / 2$

Subash Charan

Numerade Educator

Problem 47

At what height above the earth's surface, does the force of gravity decrease by $10 \% ?$ The radius of the earth is $6400 \mathrm{~km} ?$
(a) $345.60 \mathrm{~km}$
(b) $687.20 \mathrm{~km}$
(c) $1031.8 \mathrm{~km}$
(d) $12836.80 \mathrm{~km}$

Subash Charan

Numerade Educator

Problem 48

The value of $g$ on the earth's surface is $980 \mathrm{cms}^{-2}$. Its value at a height of $64 \mathrm{~km}$ from the earth's surface is (Radius of the earth $R=6400 \mathrm{~km}$ )
(a) $960.40 \mathrm{cms}^{-2}$
(b) $984.90 \mathrm{cms}^{-2}$
(c) $982.45 \mathrm{cms}^{-2}$
(d) $977.55 \mathrm{cms}^{-2}$

Subash Charan

Numerade Educator

Problem 49

The speed of earth's rotation about its axis is $\omega$. Its speed is increased to $x$ times to make the effective acceleration due to gravity equal to zero at the equator, then value of $x$ is around $\left(g=10 \mathrm{~ms}^{-2} ; R=6400 \mathrm{~km}\right)$
(a) 1
(b) $8.5$
(c) 17
(d) 34

Subash Charan

Numerade Educator

Problem 50

For a body lying on the equator to appear weightless, what should be the angular speed of the earth? (Take $g=10 \mathrm{~ms}^{-2}$; radius of earth $=6400 \mathrm{~km}$ )
(a) $0.125 \mathrm{rads}^{-1}$
(b) $1.25 \mathrm{rads}^{-1}$
(c) $1.25 \times 10^{-3} \mathrm{rads}^{-1}$
(d) $1.25 \times 10^{-2} \mathrm{rads}^{-1}$

Subash Charan

Numerade Educator

Problem 51

$P$ is a point at a distance $r$ from the centre of a solid sphere of radius $r$. The variation of gravitational potential at $P$ (i.e., $V$ ) and distance $r$ from the centre of sphere is represented by the curve.

Subash Charan

Numerade Educator

Problem 52

The gravitational field due to a mass distribution is $I=k / x^{3}$ in the $x$-direction ( $k$ is a constant). Taking the gravitational potential to be zero at infinity, its value at a distance $x / \sqrt{2}$ is
(a) $k / x$
(b) $k / 2 x$
(c) $k / x^{2}$
(d) $k / 2 x^{2}$

Prem Bijarniya

Numerade Educator

Problem 53

Select the proper graph between the gravitational potential $\left(V_{g}\right)$ due to hollow sphere and distance $(r)$ from its centre.

Subash Charan

Numerade Educator

Problem 54

Which of the following graphs represents correctly the variation of the intensity of gravitational field $(I)$ with the distance $(r)$ from the centre of a spherical shell of mass $M$ and radius $a$ ?

Subash Charan

Numerade Educator

Problem 55

A particle of mass $m$ is placed inside a spherical shell, away from its centre. The mass of the shell is $M$.
(a) The particle will move towards the centre if $m<M$, and away from the centre if $m>M$
(b) The particle will move towards the centre
(c) The particle will oscillate about the centre of shell
(d) The particle will remain stationary

Subash Charan

Numerade Educator

Problem 56

The distance between the earth and the moon is $3.85 \times 10^{\mathrm{B}} \mathrm{m}$. At what distance from the earth's centre, the intensity of gravitational field will be zero? The masses of earth and moon are $5.98 \times 10^{24} \mathrm{~kg}$ and $7.35 \times 10^{22} \mathrm{~kg}$ respectively.
(a) $3.47 \times 10^{8} \mathrm{~m}$
(b) $0.39 \times 10^{8} \mathrm{~m}$
(c) $1.82 \times 10^{8} \mathrm{~m}$
(d) None of these

Subash Charan

Numerade Educator

Problem 57

Two bodies of masses $100 \mathrm{~kg}$ and $1000 \mathrm{~kg}$ are separated by a distance of $1 \mathrm{~m}$. What is the intensity of gravitational field at the mid point of the line joining them?
(a) $6.6 \times 10^{-11} \mathrm{~N} \mathrm{~m}^{2} \mathrm{~kg}^{-2}$
(b) $2.4 \times 10^{-8} \mathrm{Nkg}^{-1}$
(c) $2.4 \times 10^{-7} \mathrm{Nkg}^{-1}$
(d) $2.4 \times 10^{-6} \mathrm{Nkg}^{-1}$

Subash Charan

Numerade Educator

Problem 58

There are two bodies of masses $100000 \mathrm{~kg}$ and $1000 \mathrm{~kg}$ separated by a distance of $1 \mathrm{~m} .$ At what distance (in metre) from the smaller body, the intensity of gravitational field will be zero?
(a) $1 / 9$
(b) $1 / 10$
(c) $1 / 11$
(d) $10 / 11$

Subash Charan

Numerade Educator

Problem 59

In a certain region of space, the gravitational field is given by $-k / r$, where $r$ is the distance and $k$ is a constant. If the gravitational potential at $r=r_{0}$ be $V_{0}$, then what is the expression for the gravitational potential $V$ ?
(a) $k \log \left(\frac{r}{r_{0}}\right)$
(b) $k \log \left(\frac{r_{0}}{r}\right)$
(c) $V_{0}+k \log \left(\frac{r}{r_{0}}\right)$
(d) $V_{0}+k \log \left(\frac{r_{0}}{r}\right)$

Subash Charan

Numerade Educator

Problem 60

The depth $d$ at which the value of acceleration due to gravity becomes $\frac{1}{n}$ times, the value at the surface is $(R=$ radius of the earth)
(a) $\frac{R}{n}$
(b) $R\left(\frac{n-1}{n}\right)$
(c) $\frac{R}{n^{2}}$
(d) $R\left(\frac{n}{n+1}\right)$

Subash Charan

Numerade Educator

Problem 61

A solid sphere is of density $\rho$ and radius $R$. The gravitational field at a distance $r$ from the centre of the sphere, where $r<R$, is
(a) $\frac{\rho \pi G R^{3}}{r}$
(b) $\frac{4 \pi G \rho r^{2}}{3}$
(c) $\frac{4 \pi G \rho R^{3}}{3 r^{2}}$
(d) $\frac{4 \pi G \rho r}{3}$

Subash Charan

Numerade Educator

Problem 62

Two bodies of masses $2 \mathrm{~kg}$ and $8 \mathrm{~kg}$ are separated by a distance of $9 \mathrm{~m}$. The point where the resultant gravitational field intensity is zero is at a distance of
(a) $4.5 \mathrm{~m}$ from each mass
(b) $6 \mathrm{~m}$ from $2 \mathrm{~kg}$
(c) $6 \mathrm{~m}$ from $8 \mathrm{~kg}$
(d) $2.5 \mathrm{~m}$ from $2 \mathrm{~kg}$

Subash Charan

Numerade Educator

Problem 63

Gravitational potential on the surface of earth is $(m=$ mass of the earth, $R=$ radius of earth)
(a) - GM / $2 R$
(b) $-g R$
(c) $g R$
(d) $G M / R$

Subash Charan

Numerade Educator

Problem 64

A particle of mass $m$ is placed at the centre of a uniform spherical shell of mass $3 \mathrm{~m}$ and radius $R$. The gravitational potential on the surface of the shell is
(a) $-\frac{G m}{R}$
(b) $-\frac{3 \mathrm{Gm}}{R}$
(c) $-\frac{4 G m}{R}$
(d) $-\frac{2 G m}{R}$

Subash Charan

Numerade Educator

Problem 65

The gravitational field due to a mass distribution is $1=\frac{C}{x^{2}}$ in $x$ direction. Here $C$ is constant. Taking the gravitational potential to be zero at infinity, potential at $x$ is
(a) $\frac{2 C}{x}$
(b) $\frac{C}{x}$
(c) $\frac{2 C}{x^{2}}$
(d) $\frac{C}{2 x^{2}}$

Subash Charan

Numerade Educator

Problem 66

A space ship moves from earth to moon and back. Th. greatest energy required for the space ship is $t$ overcome the difficulty in
(a) entering the earth's gravitational field
(b) take off from earth's field
(c) take off from lunar surface
(d) entering the moon's lunar surface

Subash Charan

Numerade Educator

Problem 67

The mass of the earth is $6.00 \times 10^{22} \mathrm{~kg}$. The constant of gravitation $G=6.67 \times 10^{-11} \mathrm{Nm}^{2} \mathrm{~kg}^{-2}$. The
potential energy of the system is $-7.79 \times 10^{28} \mathrm{~J}$. The mean distance between earth and moon is
(a) $3.80 \times 10^{8} \mathrm{~m}$
(b) $3.37 \times 10^{6} \mathrm{~m}$
(c) $7.60 \times 10^{4} \mathrm{~m}$
(d) $1.90 \times 10^{2} \mathrm{~m}$

Subash Charan

Numerade Educator

Problem 68

The change in potential energy when a body of mass $m$ is raised to a height $n R$ from the centre of earth ( $R=$ radius of earth)
(a) $m g R \frac{(n-1)}{n}$
(b) $n m g R$
(c) $m g R \frac{n^{2}}{n^{2}+1}$
(d) $m g R \frac{n}{n+1}$

Subash Charan

Numerade Educator

Problem 69

A mass $m$ is placed at a point $B$ in the gravitational field of mass $M$. When the mass $m$ is brought from $B$ to near point $A$, its gravitational potential energy will
(a) remain unchanged
(b) increase
(c) decrease
(d) become zero

Subash Charan

Numerade Educator

Problem 70

The gravitational field in a region is given by $\mathbf{I}=(4 \hat{\mathbf{i}}+\hat{\mathbf{j}}) \mathbf{N k g}^{-1}$. Work done by this field is zero when a particle is moved along the line
(a) $x+y=6$
(b) $x+4 y=6$
(c) $y+4 x=6$
(d) $x-y=6$

Subash Charan

Numerade Educator

Problem 71

A satellite orbits the earth at a height of $400 \mathrm{~km}$ above the surface. How much energy must be expended to rocket the satellite out of the earth's gravitational influence? Mass of the satellite $=200 \mathrm{~kg}$, mass of the earth $=6.0 \times 10^{24} \mathrm{~kg}$, radius of the earth $=6.4 \times 10^{6} \mathrm{~m}, G=6.67 \times 10^{-11} \mathrm{~N}-\mathrm{m}^{2} / \mathrm{kg}^{2}$.
(a) $5.2 \times 10^{10} \mathrm{~J}$
(b) $3 \times 10^{6} \mathrm{~J}$
(c) $4 \times 10^{6} \mathrm{~J}$
(d) $6 \times 10^{9} \mathrm{~J}$

Subash Charan

Numerade Educator

Problem 72

A body of mass $m$ rises to a height $h=R / 5$ from the surface of earth, where $R$ is the radius of earth. If $g$ is the acceleration due to gravity at the surface of earth, the increase in potential energy is
(a) $(4 / 5) m g h$
(b) $(5 / 6) m g h$
(c) $(6 / 7) m g h$
(d) $m g h$

Subash Charan

Numerade Educator

Problem 73

The gravitational potential difference between the surface of a planet and a point $20 \mathrm{~m}$ above it is $14 \mathrm{~J} \mathrm{~kg}^{-1}$. The work done in moving a $2.0 \mathrm{~kg}$ mass by $8.0 \mathrm{~m}$ on a slope of $60^{\circ}$ from the horizontal, is equal to
(a) 7$]$
(b) $9.6 \mathrm{~J}$
(c) 16$]$
(d) $32 \mathrm{~J}$

Subash Charan

Numerade Educator

Problem 74

If $W_{1}, W_{2}$ and $W_{3}$ represent the work done in moving a particle from $A$ to $B$ along three different paths 1,2 and 3 respectively (as shown) in a gravitional field of point mass $m$, then
(a) $W_{1}=W_{2}=W_{3}$
(b) $W_{1}>W_{2}>W_{3}$
(c) $W_{1}>W_{2}<W_{3}$
(d) $W_{1}<W_{2}<W_{3}$

Subash Charan

Numerade Educator

Problem 75

Out of the following, the only correct statement about satellites is
(a) A satellite cannot move in a stable orbit in a plane passing through the earth's centre
(b) Geostationary satellites are launched in the equatorial plane
(c) We can use just one geostationary satellite for global communication around the globe
(d) The speed of satellite increases with an increase in the radius of its orbit

Subash Charan

Numerade Educator

Problem 76

A satellite $S$ is moving in an elliptical orbit around earth. The mass of the satellite 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 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 of $S$ remains constant in magnitude

Ajay Singhal

Numerade Educator

Problem 77

A satellite is placed in a circular orbit around earth at such a height that it always remains stationary with respect to earth surface. In such case, its height from the earth surface is
(a) $32000 \mathrm{~km}$
(b) $36000 \mathrm{~km}$
(c) $6400 \mathrm{~km}$
(d) $4800 \mathrm{~km}$

Subash Charan

Numerade Educator

Problem 78

Satellites orbiting the earth have finite life and sometimes debris of satellites fall to the earth. This is because,
(a) the solar cells and batteries in satellites run out
(b) the laws of gravitation predict a trajectory spiralling inwards
(c) of viscous forces causing the speed of satellite and hence height to gradually decrease
(d) of collisions with other satellites

Ashok Prajapati

Ashok Prajapati

Numerade Educator

Problem 79

The orbital velocity of an artificial satellite in a circular orbit just above the earth's surface is $v$. For a satellite orbiting at an altitude of half of the earth's radius, the orbital velocity is
(a) $\frac{3}{2} v$
(b) $\sqrt{\frac{3}{2}} v$
(c) $\sqrt{\frac{2}{3}} v$
(d) $\frac{2}{3} v$

Subash Charan

Numerade Educator

Problem 80

For a body to escape from earth, angle at which it should be fired is?
(a) $45^{\circ}$
(b) $>45^{\circ}$
(c) $<45^{\circ}$
(d) any angle

Subash Charan

Numerade Educator

Problem 81

If the moon is to escape from the gravitational field of the earth forever, it will require a velocity
(a) $11.2 \mathrm{kms}^{-1}$
(b) less than $11.2 \mathrm{kms}^{-1}$
(c) slightly more than $11.2 \mathrm{kms}^{-1}$
(d) $22.4 \mathrm{kms}^{-1}$

Subash Charan

Numerade Educator

Problem 82

The escape velocity from the earth is $11 \mathrm{kms}^{-1}$. The escape velocity from a planet having twice the radius and the same mean density as the earth would be
(a) $5.5 \mathrm{kms}^{-1}$
(b) $11 \mathrm{kms}^{-1}$
(c) $15.5 \mathrm{kms}^{-1}$
(d) $22 \mathrm{kms}^{-1}$

Subash Charan

Numerade Educator

Problem 83

The escape velocity for a body projected vertically| upwards from the surface of the earth is $11.2 \mathrm{kms}^{-1}$ If the body is projected in a direction making an angle of $45^{\circ}$ with the vertical, the escape velocity will be
(a) $11.2 \mathrm{kms}^{-1}$
(b) $11.2 \times \sqrt{2} \mathrm{kms}^{-1}$
(c) $11.2 \times 2 \mathrm{kms}^{-1}$
[d) $11.2 / \sqrt{2} \mathrm{kms}^{-1}$

Subash Charan

Numerade Educator

Problem 84

The ratio of the radii of the planets $P_{1}$ and $P_{2}$ is $a$. The ratio of their acceleration due to gravity is $b .$ The ratio of the escape velocities from them will be
(a) $a b$
(b) $\sqrt{a b}$
(c) $\sqrt{a / b}$
(d) $\sqrt{b / a}$

Subash Charan

Numerade Educator

Problem 85

The mass of the moon is $1 / 81$ of earth's mass and its radius $1 / 4$ th that of the earth. If the escape velocity from the earth's surface is $11.2 \mathrm{kms}^{-1}$, its value for the moon will be
(a) $0.15 \mathrm{kms}^{-1}$
(b) $5 \mathrm{kms}^{-1}$
(c) $2.5 \mathrm{kms}^{-1}$
(d) $0.5 \mathrm{kms}^{-1}$

Subash Charan

Numerade Educator

Problem 86

If the radius of earth's orbit is made $1 / 4$ th, then duration of an year will become
(a) 8 times
(b) 4 times
(c) $1 / 8$ times
(d) $1 / 4$ times

Subash Charan

Numerade Educator

Problem 87

The period of revolution of planet $A$ around the sun is 8 times that $B$. The distance of a from the sun is how many times greater than that of $B$ from the sun?
(a) 2
(b) 3
(c) 4
(d) 5

Narayan Hari

Numerade Educator

Problem 88

The largest and the shortest distance of the earth from the sun are $r_{1}$ and $r_{2}$, its distance from the sun when it is perpendicular to the major axis of the orbit drawn from the sun, is
(a) $\frac{r_{1}+r_{2}}{4}$
(b) $\frac{r_{1} r_{2}}{r_{1}+r_{2}}$
(c) $\frac{2 r_{1} r_{2}}{r_{1}+\hbar_{2}}$
(d) $\frac{r_{1}+r_{2}}{3}$

Subash Charan

Numerade Educator

Problem 89

In our solar system, the inter-planetary region has chunks of matter (much smaller in size compared to planets) called asteroids. They [NCERT Exemplar]
(a) will not move around the sun since they have very small masses compared to sun
(b) will move in an irregular way because of their small masses and will drift away into outer space
(c) will move around the sun in closed orbits but not obey Kepler's laws
(d) will move in orbits like planets and obey Kepler's laws

Subash Charan

Numerade Educator

Problem 90

A comet of mass $m$ moves in a highly elliptical orbit around the sun of mass $M$. The maximum and minimum distances of the comet from the centre of the sun are $r_{1}$ and $r_{2}$ respectively. The magnitude of angular momentum of the comet with respect to the centre of sun is
(a) $\left[\frac{G M r_{1}}{\left(r_{1}+r_{2}\right)}\right]^{1 / 2}$
(b) $\left[\frac{G M m r_{1}}{\left(r_{1}+r_{2}\right)}\right]^{1 / 2}$
(c) $\left(\frac{2 G m^{2} r_{12}}{r_{1}+r_{2}}\right)^{1 / 2}$
(d) $\left(\frac{2 G M m^{2} r_{12}}{\left(r_{1}+r_{2}\right)}\right)^{1 / 2}$

Prem Bijarniya

Numerade Educator