Gravitational Potential and Potential Energy

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Two bodies of masses $m$ and $4 m$ are placed at a distance $r$ . The gravitational potential at a point on the line joining them, where the gravitational field is zero, is

- \frac{4 G m}{r}

(B)

- \frac{6 G m}{r}

(C)

- \frac{9 G m}{r}

The initial velocity $v_{i}$ required to project a body vertically upward from the surface of the Earth to reach a height of $10 R$ , where $R$ is the radius of the Earth, may be described in terms of escape velocity $v_{e}$ such that $v_{i} = \frac{x}{y} \times v_{e}$ . The value of $x$ will be ............. . [2021, 25 Feb Shift-II]

JEE Main 2021, 27 Aug Shift-II

LEVELJEE Main

A mass of $50 kg$ is placed at the centre of a uniform spherical shell of mass $100 kg$ and radius $50 m$ . If the gravitational potential at a point, $25 m$ from the centre is $V kg/m$ . The value of $V$ is

(A)

- 60 G

(B)

+ 2 G

(C)

- 20 G

(D)

- 4 G

JEE Main 2015

LEVELJEE Advanced

From a solid sphere of mass $M$ and radius $R$ , a spherical portion of radius $(R /2)$ is removed as shown in the figure. Taking gravitational potential $V = 0$ at $r = \infty$ , the potential at the centre of the cavity thus formed is ( $G =$ gravitational constant)

(A)

\frac{- GM}{2 R}

(B)

\frac{- GM}{R}

(C)

\frac{- 2 GM}{3 R}

(D)

\frac{- 2 GM}{R}

LEVELBoard

The escape velocity of a body depends upon mass as

(A)

m^{0}

(B)

m^{1}

(C)

m^{2}

(D)

m^{3}

JEE Main 2021, 27 July Shift-I

LEVELJEE Advanced

Suppose two planets (spherical in shape) of radii $R$ and $2 R$ , but mass $M$ and $9 M$ respectively have a centre to centre separation $8 R$ as shown in the figure. A satellite of mass $m$ is projected from the surface of the planet of mass $M$ directly towards the centre of the second planet. The minimum speed $v$ required for the satellite to reach the surface of the second planet is $\frac{a GM}{7 R}$ , then the value of $a$ is …………… . [Take, the two planets are fixed in their position]

LEVELBoard

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 escape velocity from the surface of the planet would be

(A)

1.1 kms^{- 1}

(B)

11 kms^{- 1}

(C)

110 kms^{- 1}

(D)

0.11 kms^{- 1}

LEVELJEE Main

Energy required to move a body of mass $m$ from an orbit of radius $2 R$ to $3 R$ is

(A)

\frac{GM m}{12 R ^{2}}

(B)

\frac{GM m}{3 R ^{2}}

(C)

\frac{GM m}{8 R}

(D)

\frac{GM m}{6 R}

JEE Main 2021, 27 Aug Shift-I

LEVELJEE Advanced

A body of mass $2 M$ splits into four masses ${m, M - m, m, M - m}$ , which are rearranged to form a square as shown in the figure. The ratio of $\frac{M}{m}$ for which, the gravitational potential energy of the system becomes maximum is $x : 1$ . The value of $x$ is……… .

JEE Main 2019, 10 Jan Shift-II

LEVELJEE Advanced

Two stars of masses $3 \times 1 0^{31} kg$ each and at distance $2 \times 1 0^{11} m$ 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, $G = 6.67 \times 1 0^{- 11} N-m^{2} kg^{- 2}$ )

(A)

2.8 \times 1 0^{5} m/s

(B)

3.8 \times 1 0^{4} m/s

(C)

2.4 \times 1 0^{4} m/s

(D)

1.4 \times 1 0^{5} m/s

JEE Main 2020, 6 Sep Shift-II

LEVELJEE Advanced

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

(A)

2 \frac{GM}{a}

(B)

4 \frac{GM}{a}

(C)

\frac{G M ^{2}}{ma}

(D)

\frac{3}{2} \frac{5 GM}{a}

LEVELJEE Main

The mass of a spaceship is $1000 kg$ . It is to be launched from the earth's surface out into free space. The value of $g$ and $R$ (radius of earth) are $10 m/s^{2}$ and $6400 km$ , respectively. The required energy for this work will be

(A)

6.4 \times 1 0^{11} J

(B)

6.4 \times 1 0^{8} J

(C)

6.4 \times 1 0^{9} J

(D)

6.4 \times 1 0^{10} J

LEVELJEE Main

A particle of mass $10 g$ is kept on the surface of a uniform sphere of mass $100 kg$ and radius $10 cm$ . Find the work to be done against the gravitational force between them, to take the particle far away from the sphere, (you may take $G = 6.67 \times 1 0^{- 11} Nm^{2} / kg^{2}$ )

(A)

13.34 \times 1 0^{- 10} J

(B)

3.33 \times 1 0^{- 10} J

(C)

6.67 \times 1 0^{- 9} J

(D)

6.67 \times 1 0^{- 10} J

JEE Main 2021, 16 March Shift-II

LEVELJEE Main

If one wants to remove all the mass of the earth to infinity in order to break it up completely. The amount of energy that needs to be supplied will be $\frac{x G M ^{2}}{5 R}$ , where $x$ is ………. (Round off to the nearest integer) ( $M$ is the mass of earth, $R$ is the radius of earth and $G$ is the gravitational constant.)

LEVELJEE Main

If $g$ is the acceleration due to gravity on the earth's surface, the gain in the potential energy of an object of mass $m$ raised from the surface of the earth to a height equal to the radius $R$ of the earth, is

(A)

2 m g R

(B)

\frac{1}{2} m g R

(C)

\frac{1}{4} m g R

(D)

m g R

JEE Main 2021, 26 Aug Shift I

LEVELJEE Main

Inside a uniform spherical shell I. the gravitational field is zero. II. the gravitational potential is zero. III. the gravitational field is same everywhere. IV. the gravitation potential is same everywhere. V. All of the above Choose the most appropriate answer from the options given below .

JEE Main 2020, 4 Sep Shift-I

LEVELJEE Main

On the X-axis and at a distance $x$ from the origin, the gravitational field due to a mass distribution is given by $\frac{A x}{( x ^{2} + a ^{2} ) ^{3/2}}$ in the $x$ -direction. The magnitude of gravitational potential on the X-axis at a distance $x$ , taking its value to be zero at infinity, is

(A)

\frac{A}{( x ^{2} + a ^{2} ) ^{1/2}}

(B)

A (x^{2} + a^{2})^{3/2}

(C)

\frac{A}{( x ^{2} + a ^{2} ) ^{3/2}}

(D)

A (x^{2} + a^{2})^{1/2}

JEE Advanced 2012

LEVELJEE Advanced

Two spherical planets $P$ and $Q$ have the same uniform density $ρ$ , masses $M_{P}$ and $M_{Q}$ , and surface areas $A$ and $4 A$ , respectively. A spherical planet $R$ also has uniform density $ρ$ and its mass is $(M_{P} + M_{Q})$ . The escape velocities from the planets $P, Q$ , and $R$ , are $v_{P}, v_{Q}$ , and $v_{R}$ , respectively. Then

* Multiple Correct Options

(A)

v_{Q} > v_{R} > v_{P}

(B)

v_{R} > v_{Q} > v_{P}

(C)

v_{R} / v_{P} = 3

(D)

v_{P} / v_{Q} = 1/2

LEVELJEE Main

Gravitational acceleration on the surface of a planet is $\frac{6}{11} g$ , where $g$ is the gravitational acceleration on the surface of the earth. The average mass density of the planet is $\frac{2}{3}$ times that of the earth. If the escape speed on the surface of the earth is taken to be $11 km s^{- 1}$ , the escape speed on the surface of the planet in $km s^{- 1}$ will be

JEE Advanced 1983

LEVELJEE Main

If $g$ is the acceleration due to gravity on the earth's surface, the gain in the potential energy of an object of mass $m$ raised from the surface of the earth to a height equal to the radius $R$ of the Earth, is

(A)

\frac{1}{2} m g R

(B)

2 m g R

(C)

m g R

(D)

\frac{1}{4} m g R

JEE Advanced 2010

LEVELJEE Advanced

A thin uniform annular disc (see figure) of mass $M$ has outer radius $4 R$ and inner radius $3 R$ . The work required to take a unit mass from point $P$ on its axis to infinity is

(A)

\frac{2 GM}{7 R} (42 - 5)

(B)

- \frac{2 GM}{7 R} (42 - 5)

(C)

\frac{GM}{4 R}

(D)

\frac{2 GM}{5 R} (2 - 1)

JEE Advanced 2017

LEVELJEE Advanced

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 $3 \times 1 0^{5}$ times heavier than the Earth and is at a distance $2.5 \times 1 0^{4}$ times larger than the radius of Earth. The escape velocity from Earth's gravitational field is $v_{e} = 11.2 km s^{- 1}$ . The minimum initial velocity ( $v_{s}$ ) 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 other planet)

(A)

v_{s} = 72 km s^{- 1}

(B)

v_{s} = 22 km s^{- 1}

(C)

v_{s} = 42 km s^{- 1}

(D)

v_{s} = 62 km s^{- 1}

JEE Advanced 2003

LEVELJEE Advanced

There is a crater of depth $\frac{R}{100}$ on the surface of the moon (radius $R$ ). A projectile is fired vertically upward from the crater with velocity, which is equal to the escape velocity $v$ from the surface of the moon. Find the maximum height attained by the projectile.

JEE Advanced 1988

LEVELJEE Main

The masses and radii of the Earth and the Moon are $M_{1}, R_{1}$ and $M_{2}, R_{2}$ respectively. Their centres are a distance $d$ apart. The minimum speed with which a particle of mass $m$ should be projected from a point midway between the two centres so as to escape to infinity is _________.

JEE Advanced 1993

LEVELJEE Advanced

A solid sphere of uniform density and radius $4$ units is located with its centre at the origin $O$ of coordinates. Two spheres of equal radii $1$ unit, with their centres at $A (- 2, 0, 0)$ and $B (2, 0, 0)$ respectively, are taken out of the solid leaving behind spherical cavities as shown in figure.

* Multiple Correct Options

(A)

the gravitational field due to this object at the origin is zero

(B)

the gravitational field at the point

B (2, 0, 0)

is zero

(C)

the gravitational potential is the same at all points of circle

y^{2} + z^{2} = 36

(D)

the gravitational potential is the same at all points on the circle

y^{2} + z^{2} = 4

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } 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

The escape velocity of a body depends upon mass as

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Energy required to move a body of mass m from an orbit of radius 2R to 3R is

Inside a uniform spherical shell I. the gravitational field is zero. II. the gravitational potential is zero. III. the gravitational field is same everywhere. IV. the gravitation potential is same everywhere. V. All of the above Choose the most appropriate answer from the options given below .

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } 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

Two bodies of masses $m$ and $4 m$ are placed at a distance $r$ . The gravitational potential at a point on the line joining them, where the gravitational field is zero, is

Energy required to move a body of mass $m$ from an orbit of radius $2 R$ to $3 R$ is

A thin uniform annular disc (see figure) of mass $M$ has outer radius $4 R$ and inner radius $3 R$ . The work required to take a unit mass from point $P$ on its axis to infinity is