Gravitational Force

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Two particles of equal mass $m$ go around a circle of radius $R$ under the action of their mutual gravitational attraction. The speed of each particle with respect to their centre of mass is

\frac{G m}{R}

(B)

\frac{G m}{4 R}

(C)

\frac{G m}{3 R}

(D)

\frac{G m}{2 R}

JEE Main 2019, 12 Jan Shift-I

LEVELJEE Advanced

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 $A + B x^{2}$ , is given by

(A)

G m [A (\frac{1}{a + L} - \frac{1}{a}) - B L]

(B)

G m [A (\frac{1}{a + L} - \frac{1}{a}) + B L]

(C)

G m [A (\frac{1}{a} - \frac{1}{a + L}) + B L]

(D)

G m [A (\frac{1}{a} - \frac{1}{a + L}) - B L]

JEE Main 2021, 1 Sep Shift-I

LEVELJEE Advanced

Four particles each of mass $M$ , move along a circle of radius $R$ under the action of their mutual gravitational attraction as shown in figure. The speed of each particle is

(A)

\frac{1}{2} \frac{GM}{R ( 2 2 + 1 )}

(B)

\frac{1}{2} \frac{GM}{R} (22 + 1)

(C)

\frac{1}{2} \frac{GM}{R} (22 - 1)

(D)

\frac{GM}{R}

JEE Main 2021, 22 July Shift-II

LEVELJEE Main

Two identical particles of mass 1 kg each go round a circle of radius $R$ , under the action of their mutual gravitational attraction. The angular speed of each particle is

(A)

\frac{G}{2 R ^{3}}

(B)

\frac{1}{2} \frac{G}{R ^{3}}

(C)

\frac{1}{2 R} \frac{1}{G}

(D)

\frac{2 G}{R ^{3}}

JEE Main 2020, 3 Sep Shift-II

LEVELJEE Advanced

The mass density of a planet of radius $R$ varies with the distance $r$ from its centre as $ρ (r) = ρ_{0} (1 - \frac{r ^{2}}{R ^{2}})$ . Then, the gravitational field is maximum at

(A)

r = \frac{3}{4} R

(B)

r = R

(C)

r = \frac{5}{9} R

(D)

r = \frac{1}{3} R

JEE Main 2020, 8 Jan Shift-I

LEVELJEE Main

Consider two solid spheres of radii $R_{1} = 1 m$ , $R_{2} = 2 m$ and masses $M_{1}$ and $M_{2}$ , respectively. The gravitational field due to sphere 1 and 2 are shown. The value of $\frac{M _{1}}{M _{2}}$ is

(A)

\frac{1}{3}

(B)

\frac{1}{2}

(C)

\frac{1}{6}

(D)

\frac{2}{3}

JEE Main 2021, 26 Feb Shift-I

LEVELJEE Main

Find the gravitational force of attraction between the ring and sphere as shown in the figure, where the plane of the ring is perpendicular to the line joining the centres. If $8 R$ is the distance between the centres of a ring (of mass $m$ ) and a sphere (of mass $M$ ), where both have equal radius $R$ .

(A)

\frac{8}{9} \cdot \frac{G m M}{R ^{2}}

(B)

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

(C)

\frac{1}{3 8} \cdot \frac{GM m}{R ^{2}}

(D)

\frac{8}{27} \cdot \frac{G m M}{R ^{2}}

JEE Main 2014

LEVELJEE Advanced

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

(A)

\frac{GM}{R}

(B)

22 \frac{GM}{R}

(C)

\frac{GM}{R} (1 + 22)

(D)

\frac{1}{2} \frac{GM}{R} (1 + 22)

LEVELJEE Main

Statement I : For a mass $M$ kept at the centre of a cube of side $a$ , the flux of gravitational field passing through its sides is $4 π GM$ . Statement II : If the direction of a field due to a point source is radial and its dependence on the distance $r$ from the source is given as $1/ r^{2}$ , its flux through a closed surface depends only on the strength of the source enclosed by the surface and not on the size or shape of the surface.

(A)

Statement I is true, Statement II is true; Statement II is the correct explanation of Statement I

(B)

Statement I is true, Statement II is true; Statement II is not the correct explanation of Statement I

(C)

Statement I is true, Statement II is false

(D)

Statement I is false, Statement II is true

JEE Main 2019, 8 April Shift-I

LEVELJEE Advanced

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 other's gravitational field in a circular orbit circumscribing the square ?

(A)

1.35 \frac{GM}{a}

(B)

1.16 \frac{GM}{a}

(C)

1.21 \frac{GM}{a}

(D)

1.41 \frac{GM}{a}

JEE Main 2021, 25 Feb Shift-I

LEVELJEE Advanced

A solid sphere of radius $R$ gravitationally attracts a particle placed at $3 R$ from its centre with a force $F_{1}$ . Now, a spherical cavity of radius $(R /2)$ is made in the sphere (as shown in figure) and the force becomes $F_{2}$ . The value of $F_{1} : F_{2}$ is

(A)

25 : 36

(B)

36 : 25

(C)

50 : 41

(D)

41 : 50

JEE Main 2021, 24 Feb Shift-I

LEVELJEE Advanced

Two stars of masses $m$ and $2 m$ at a distance $d$ rotate about their common centre of mass in free space. The period of revolution is

(A)

\frac{1}{2 π} \frac{3 G m}{d ^{3}}

(B)

2 π \frac{d ^{3}}{3 G m}

(C)

2 π \frac{3 G m}{d ^{3}}

(D)

\frac{1}{2 π} \frac{d ^{3}}{3 G m}

JEE Advanced 1994

LEVELJEE Main

The magnitudes of the gravitational field at distance $r_{1}$ and $r_{2}$ from the centre of a uniform sphere of radius $R$ and mass $M$ are $F_{1}$ and $F_{2}$ , respectively. Then

* Multiple Correct Options

(A)

\frac{F _{1}}{F _{2}} = \frac{r _{1}}{r _{2}}

r_{1} < R

and

r_{2} < R

(B)

\frac{F _{1}}{F _{2}} = \frac{r _{2}^{2}}{r _{1}^{2}}

r_{1} > R

and

r_{2} > R

(C)

\frac{F _{1}}{F _{2}} = \frac{r _{1}^{3}}{r _{2}^{3}}

r_{1} < R

and

r_{2} < R

(D)

\frac{F _{1}}{F _{2}} = \frac{r _{1}^{2}}{r _{2}^{2}}

r_{1} < R

and

r_{2} < R

JEE Advanced 1988

LEVELJEE Advanced

Three particles, each of mass $m$ , are situated at the vertices of an equilateral triangle of side length $a$ . The only forces acting on the particles are their mutual gravitational forces. It is desired that each particle moves in a circle while maintaining the original mutual separation $a$ . Find the initial velocity that should be given to each particle and also the time period of the circular motion.

JEE Advanced 2015

LEVELJEE Advanced

A large spherical mass $M$ is fixed at one position and two identical masses $m$ are kept on a line passing through the centre of $M$ (see figure). The point masses are connected by a rigid massless rod of length $l$ and this assembly is free to move along the line connecting them. All three masses interact only through their mutual gravitational interaction. When the point mass nearer to $M$ is at a distance $r = 3 l$ from $M$ the tension in the rod is zero for $m = k (\frac{M}{288})$ . The value of $k$ is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } 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+Bx2, is given by

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.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Two identical particles of mass 1 kg each go round a circle of radius R, under the action of their mutual gravitational attraction. The angular speed of each particle is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The mass density of a planet of radius R varies with the distance r from its centre as ρ(r)=ρ0​(1−R2r2​). Then, the gravitational field is maximum at

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Two stars of masses m and 2m at a distance d rotate about their common centre of mass in free space. The period of revolution is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The magnitudes of the gravitational field at distance r1​ and r2​ from the centre of a uniform sphere of radius R and mass M are F1​ and F2​, respectively. Then

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 $A + B x^{2}$ , is given by

Four particles each of mass $M$ , move along a circle of radius $R$ under the action of their mutual gravitational attraction as shown in figure. The speed of each particle is

Two identical particles of mass 1 kg each go round a circle of radius $R$ , under the action of their mutual gravitational attraction. The angular speed of each particle is

The mass density of a planet of radius $R$ varies with the distance $r$ from its centre as $ρ (r) = ρ_{0} (1 - \frac{r ^{2}}{R ^{2}})$ . Then, the gravitational field is maximum at

Two stars of masses $m$ and $2 m$ at a distance $d$ rotate about their common centre of mass in free space. The period of revolution is

The magnitudes of the gravitational field at distance $r_{1}$ and $r_{2}$ from the centre of a uniform sphere of radius $R$ and mass $M$ are $F_{1}$ and $F_{2}$ , respectively. Then