Gravitational Field Due to a Ring

Q. A uniform solid sphere of mass $m$ and radius $r$ is surrounded symmetrically by a uniform thin spherical shell of radius $2r$ and mass $m$. Choose the correct statement

1. The gravitational field at a distance of $1.5r$ from the centre is $\frac{2}{9}\frac{Gm}{r^2}$
2. The gravitational field at a distance of $2.5r$ from the centre is $\frac{8}{25}\frac{Gm}{r^2}$
3. The gravitational field at a distance of $1.5r$ from the centre is zero.
4. The gravitational field between the sphere and spherical shell is uniform.

Q. Find the gravitational field strength at the centre of arc of linear mass density $\lambda$ subtending an angle $2\alpha$ at the centre

1. $E=\frac{G\lambda }{R}\sin \alpha$
   
2. $E=\frac{2G\lambda }{R}\sin \alpha$
   
3. $E=\frac{2G\lambda }{R}\cos \alpha$
   
4. $E=\frac{2G\lambda }{R}\sin 2\alpha$

Q. The height above the surface of Earth at which gravitational field intensity is reduced to $1\%$ of its value on the surface of Earth is ($R_e$ is radius of Earth)

1. $100R_e$
2. $10R_e$
3. $99R_e$
4. $9R_e$

Q. Two concentric spherical shells have masses $M_1$ and $M_2$ and radii $R_1$ and $R_2$ $(R_1<R_2)$. What is the force exerted by this system on a particle of mass $m$ if it is at a distance $\frac{(R_1+R_2)}{2}$ from the centre?

1. $\frac{4G(M_1+M_2)m}{(R_1+R_2{)}^2}$
2. $\frac{4G{M}_{2}m}{(R_1+R_2{)}^2}$
3. $\frac{4G{M}_{1}m}{(R_1+R_2{)}^2}$
4. $\frac{4G(M_1-M_2)m}{(R_1+R_2{)}^2}$

Q. Inside a uniform sphere of density $\rho$ there is a spherical cavity whose centre is at a position vector $\overrightarrow{l}$ from the centre of the sphere. Find the strength of gravitational field inside the cavity.

1. $-\frac{4}{3}\pi G\rho \overrightarrow{l}$
2. $-\frac{2}{3}\pi G\rho \overrightarrow{l}$
3. $-\frac{8}{3}\pi G\rho \overrightarrow{l}$
4. $\frac{2}{3}\pi G\rho \overrightarrow{l}$

Q. 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 of two pieces, a spherical shell of negligible thickness of mass $2M/3$ and a point mass $M/3$ at the centre. Change in the force of gravity experienced by the man is

1. 
   $\frac{2}{3}\frac{GMm}{R^2}$
2. $0$
3. 
   $\frac{1}{3}\frac{GMm}{R^2}$
4. 
   $\frac{4}{3}\frac{GMm}{R^2}$

Q. A solid sphere of mass $m$ radius $r$ is placed inside a hollow thin spherical shell of mass $M$ and radius $R$ as shown in figure. A particle of mass $m^{\prime }$ is placed on the line joining the two centres at a distance $x$ from the point of contact of the sphere and the shell. Find the magnitude of the resultant gravitational force on this particle due to the sphere and the shell if $r<x<2r$.

1. $\frac{Gm{m}^{\prime }(2r-x)}{2r^3}$
2. $\frac{{Gmm}^{\prime }(x-r)}{r^3}$
3. $\frac{Gm{m}^{\prime }}{(r-x{)}^2}$
4. $\frac{{Gmm}^{\prime }x}{2r^3}$

Q. A solid sphere of mass $m$ radius $r$ is placed inside a hollow thin spherical shell of mass $M$ and radius $R$ as shown in figure. A particle of mass $m^{\prime }$ is placed on the line joining the two centres at a distance $x$ from the point of contact of the sphere and the shell. Find the magnitude of the resultant gravitational force on this particle due to the sphere and the shell if $2r<x<2R$.

1. $\frac{Gm{m}^{\prime }(2r-x)}{2r^3}$
2. $\frac{{Gmm}^{\prime }x}{2r^3}$
3. $\frac{Gm{m}^{\prime }}{(x-r{)}^2}$
4. $\frac{{Gmm}^{\prime }x}{2r^3}$

Q. A solid sphere of mass $m$ and radius $r$ is placed inside a hollow thin spherical shell of mass $M$ and radius $R$ as shown in figure. A particle of mass $m^{\prime }$ is placed on the line joining the two centres at a distance $x$ from the point of contact of the sphere and the shell. Find the magnitude of the resultant gravitational force on this particle due to the sphere and the shell if $x>2R$.

1. $\frac{2Gm{m}^{\prime }}{(x-r{)}^2}+\frac{GM{m}^{\prime }}{(x-R{)}^2}$
2. $\frac{GM{m}^{\prime }}{(x-R{)}^2}+\frac{2Gm{m}^{\prime }}{(x-r{)}^2}$
3. $\frac{GM{m}^{\prime }}{(x-R{)}^2}+\frac{GM{m}^{\prime }}{(x-r{)}^2}$
4. $\frac{GM{m}^{\prime }}{(x-R{)}^2}+\frac{Gm{m}^{\prime }}{(x-r{)}^2}$

Q. 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 :

1. $\frac{F_1}{F_2}=\frac{r_1}{r_2}$ if $r_1<R$ and $r_2<R$
2. $\frac{F_1}{F_2}=\frac{r_2^2}{r_1^2}$ if $r_1>R$ and $r_2>R$
3. $\frac{F_1}{F_2}=\frac{r_1}{r_2}$ if $r_1>R$ and $r_2>R$
4. $\frac{F_1}{F_2}=\frac{r_1^2}{r_2^2}$ if $r_1<R$ and $r_2<R$

Q. A uniform solid sphere of mass $M$ and radius $R$ is surrounded symmetrically by a uniform thin spherical shell of mass $\frac{M}{2}$ and radius $2R$. Then gravitational field at a distance of $\frac{5}{2}R$ from the centre is

1. $\frac{2}{5}\frac{GM}{R^2}$ towards center
   
2. $\frac{2}{5}\frac{GM}{R^2}$ away from the centre
   
3. $\frac{6}{25}\frac{GM}{R^2}$ away from centre
   
4. $\frac{6}{25}\frac{GM}{R^2}$ towards centre
   

Q. A uniform ring of mass $m$ and radius $a$ is placed directly above a uniform sphere of mass $M$ and of equal radius. The centre of the ring is directly above the centre of the sphere at a distance $a\sqrt{3}$ as shows in the figure. The gravitational force exerted by the sphere on the ring will be

1. $\frac{\sqrt{3}GMm}{4a^2}$
2. $\frac{\sqrt{3}GMm}{8a^2}$
3. $\frac{\sqrt{5}GMm}{4a^2}$
4. $\frac{2GMm}{\sqrt{3}{a}^2}$

Q.

Two spheres each of mass $70kg$ are $0.14m$ apart. Find the force of attraction between them. Given $G=6.67x{10}^{-11}N{m}^{2}k{g}^{-2}.$

Q. A mass $m$ is placed at point $\text{P}$ at a distance $h$ along the normal through the centre $\text{O}$ of a thin circular ring of mass $M$ and radius $r$ as shown in figure. If the mass is moved further away such that, $\text{OP}$ becomes $2h$, by what factor, the force of gravitation will decrease, if $h=r$

1. $\frac{3\sqrt{2}}{4\sqrt{3}}$
2. $\frac{5\sqrt{2}}{\sqrt{3}}$
3. $\frac{4\sqrt{2}}{5\sqrt{5}}$
4. $\frac{4\sqrt{3}}{5}$

Q.

The centers of two similar spheres are 1m apart. If the gravitational force between the two spheres is 1 newton, what is the mass of each sphere?

Q.
The net gravitational field at point $P$ is

1. $\frac{\sqrt{65}}{36}\frac{GM}{R^2}$
   
2. $\frac{\sqrt{65}}{72}\frac{GM}{R^2}$
3. $\frac{1}{9}\frac{GM}{R^2}$
4. $\frac{\sqrt{3}}{2}\frac{GM}{R^2}$

Q. A uniform solid sphere of mass $M$ and radius $R$ is surrounded symmetrically by a uniform thin spherical shell of mass $\frac{M}{2}$ and radius $2R$. Then gravitational field at a distance $\frac{3R}{2}$ from the centre will be

1. $\frac{3}{2}\frac{GM}{R^2}$, towards centre
2. $\frac{4}{9}\frac{GM}{R^2}$, away from centre
3. $\frac{1}{4}\frac{GM}{R^2}$, towards centre
4. $\frac{4}{9}\frac{GM}{R^2}$, towards centre

Q. A uniform ring of mass $m$ is lying at a distance $a$ from the centre of a sphere of mass $M$ just over the sphere (where $a$ is the radius of the ring as well as that of the sphere). Then magnitude of gravitational force between them is
6

1. $\frac{GMm}{2\sqrt{2}{a}^2}$
2. $\sqrt{3}\frac{GMm}{8a^2}$
3. $\frac{GMm}{8a^2}$
4. $\sqrt{2}\frac{GMm}{a^2}$

Q. Assuming the earth to be a sphere of uniform density the gravitational field

1. at a point outside the earth is inversely proportional to the square of its distance from the centre.
2. at a point outside the earth is inversely proportional to its distance from the centre.
3. at a point inside is zero.
4. at a point inside is proportional to square of its distance from the centre.

Q. Mass $M$ is distributed uniformly along a line of length $2L$. A particle of mass $m$ is at a point ($\text{P}$) at distance $a$ above the centre of the line on its perpendicular bisector as shown in figure. The gravitational force that the mass distribution along the line exerts on the particle is

1. $\frac{2GMm}{a\sqrt{L^2+a^2}}$
2. $\frac{GMm}{a\sqrt{L^2+a^2}}$
3. $\frac{GMm}{\sqrt{L^2+a^2}}$
4. $\frac{3GMm}{a\sqrt{L^2+a^2}}$

Q. A solid sphere of mass $m$ radius $r$ is placed inside a hollow thin spherical shell of mass $M$ and radius $R$ as shown in figure. A particle of mass $m^{\prime }$ is placed on the line joining the two centres at a distance $x$ from the point of contact of the sphere and the shell. Find the magnitude of the resultant gravitational force on this particle due to the sphere and the shell if $r<x<2r$.

1. $\frac{Gm{m}^{\prime }(2r-x)}{2r^3}$
2. $\frac{{Gmm}^{\prime }(x-r)}{r^3}$
3. $\frac{Gm{m}^{\prime }}{(r-x{)}^2}$
4. $\frac{{Gmm}^{\prime }x}{2r^3}$

Q.

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

Q. Two concentric spherical shells have masses $M_1$ and $M_2$ and radii $R_1$ and $R_2$ $(R_1<R_2)$. What is the force exerted by this system on a particle of mass $m$ if it is at a distance $\frac{(R_1+R_2)}{2}$ from the centre?

1. $\frac{4G(M_1+M_2)m}{(R_1+R_2{)}^2}$
2. $\frac{4G{M}_{2}m}{(R_1+R_2{)}^2}$
3. $\frac{4G{M}_{1}m}{(R_1+R_2{)}^2}$
4. $\frac{4G(M_1-M_2)m}{(R_1+R_2{)}^2}$

Q. What is the radius of the circular orbit of a stationary satellite which remains motionless with respect to earth's surface ?

1. ${\left(\frac{g{R}^2{T}^2}{4{\pi }^2}\right)}^{\frac{1}{3}}$
2. ${\left(\frac{gRT}{2\pi }\right)}^{\frac{1}{3}}$
3. ${\left(\frac{gR}{4{\pi }^2{T}^2}\right)}^{\frac{1}{2}}$
4. ${\left(\frac{gR}{2\pi T}\right)}^{\frac{1}{2}}$

Q. A uniform ring and uniform sphere are placed as shown in the figure. The gravitational attraction between the sphere and ring ($d=\sqrt{3}R$) is.

1. $\frac{8G{M}^2}{R^2}$
2. $\frac{2G{M}^2}{R^2}$
3. $\frac{3G{M}^2}{2R^2}$
4. $\frac{\sqrt{3}G{M}^2}{R^2}$

Q. Assuming the earth to be a sphere of uniform density the gravitational field

1. at a point outside the earth is inversely proportional to the square of its distance from the centre.
2. at a point outside the earth is inversely proportional to its distance from the centre.
3. at a point inside is zero.
4. at a point inside is proportional to square of its distance from the centre.

Q. Assume a planet is a uniform sphere of radius R that (somehow) has a narrow radial tunnel through its center . Also assume we can position an apple anywhere along the tunnel of outside the sphere. Let $F_R$ be the magnitude of the gravitational force on the apple when it is located at the planet's surface. How far from the surface is there a point where the magnitude is $\frac{1}{2}{F}_R$ if we move the apple (a) away from the planet and (b) into the tunnel?

Q. A uniform ring and uniform sphere are placed as shown in the figure. The gravitational attraction between the sphere and ring ($d=\sqrt{3}R$) is.

1. $\frac{8G{M}^2}{R^2}$
2. $\frac{2G{M}^2}{R^2}$
3. $\frac{3G{M}^2}{2R^2}$
4. $\frac{\sqrt{3}G{M}^2}{R^2}$