Gravitational Field Due to a Solid Sphere

Q. The kinetic energies of a planet in an elliptical orbit about the Sun, at positions $\text{A, B}$ and $\text{C}$ are $K_A$, $K_B$ and $K_C$, respectively. $\text{AC}$ is the major axis and $\text{SB}$ is perpendicular to $\text{AC}$ at the position of the Sun $S$ as shown in the figure. Then

1. $K_A<K_B<K_C$
2. $K_A>K_B>K_C$
3. $K_B>K_A>K_C$
4. $K_B<K_A<K_C$

Q. A planet has mass $\frac{1}{10}$ of that of earth, while radius is $\frac{1}{3}$ that of earth. If a person can throw a stone on earth surface to a height of $90\text{m}$, then he will be able to throw the stone on that planet to a height (in m)Assume that velocity of projection remains same on earth and planet.

Q. A uniform thin rod of mass $m$ and length $R$ is placed normally on surface of earth as shown. The mass of earth is $M$ and its radius is $R$. Then the magnitude of gravitational force exerted by earth on the rod is :

1. $\frac{GMm}{2R^2}$
2. $\frac{4GMm}{9R^2}$
3. $\frac{GMm}{8R^2}$
4. $\frac{GMm}{4R^2}$

Q. The acceleration due to gravity near the earth's surface is $9.8{\text{m/s}}^2$, and the earth's radius is $6400\text{km}$. From this data, calculate the mass of the earth.
$[G=6.67\times {10}^{-11}{\text{Nm}}^2{\text{/kg}}^2]$

1. $6\times {10}^{27}\text{kg}$
2. $3\times {10}^{24}\text{kg}$
3. $3\times {10}^{27}\text{kg}$
4. $6\times {10}^{24}\text{kg}$

Q. A spherically symmetric gravitational system of particles has a mass density $\rho =\{\begin{array}{c}{\rho }_0\text{for}r\le R\\ 0\text{for}r>R\end{array}$
where ${\rho }_0$ is a constant. A test mass can undergo circular motion under the influence of the gravitational field of particles. Its speed $V$ as a function of distance $r(0<r<\infty )$ from the centre of the system is represented by

1. 
2. 
3. 
4. 

Q. The magnitudes of the gravitational force at distances $r_1$ 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. $\frac{F_1}{F_2}=\frac{r_1^2}{r_2^2}$ if $r_1<R$ and $r_2<R$
3. $\frac{F_1}{F_2}=\frac{r_2^2}{r_1^2}$ if $r_1<R$ and $r_2<R$
4. $\frac{F_1}{F_2}=\frac{r_1}{r_2}$ if $r_1>R$ and $r_2>R$

Q. The gravitational field intensity $\overrightarrow{E}$ of earth at any point is defined as the gravitational force per unit mass at that point. It varies from place to place. The variation is shown in column II with position $\overrightarrow{r}$ vs the $\overrightarrow{E}$ in the form of graphs. The variation of $\overrightarrow{r}$ is given in column I. Choose the correct form of graphs for the corresponding variations of $\overrightarrow{r}.$

1. $A\to 1;B\to 2;C\to 3;D\to 4$
2. $A\to 4;B\to 3;C\to 2;D\to 1;$
3. $A\to 4;B\to 1;C\to 3;D\to 2$
4. $A\to 3;B\to 2;C\to 1;D\to 4$

Q. 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 $\frac{R}{2}$ is now made in the sphere as shown in the figure. The sphere with cavity now applies a gravitational force $F_2$ on the same particle. The ratio $\frac{F_2}{F_1}$ is:

1. $\frac{9}{50}$
2. $\frac{41}{50}$
3. $\frac{3}{25}$
4. $\frac{22}{25}$

Q. The figure represents an elliptical orbit of a planet around sun. The planet takes time $T_1$ to travel from $\text{A}$ to $\text{B}$ and it takes time $T_2$ to travel from $\text{C}$ to $\text{D}$ . If the area $\text{CSD}$ is double that of area $\text{ASB}$, then

1. data insufficient
2. $T_1=0.5T_2$
3. $T_1=T_2$
4. $T_1=2T_2$

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 $\sqrt{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 $R$.

1. $\frac{\sqrt{8}}{9}\frac{GmM}{R}$
2. $\frac{\sqrt{8}}{27}\frac{GmM}{R^2}$
3. $\sqrt{2}$ $\frac{GmM}{R^2}$
4. $\frac{1}{3\sqrt{8}}\frac{GmM}{R^2}$

Q. A large spherical mass $M$ is fixed at one position and two identical point masses $m$ are kept on a line passing through the centre of $M$ as shown in the 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 distance $r=3l$ from $M$, the tension in the rod is zero for $m=k\left(\frac{M}{288}\right)$. The value of $k$ is

Q. A ball of mass $m$ is dropped from a height $h$ equal to the radius of the earth above the tunnel dug through the earth as shown in the figure. Choose the correct options. (Mass of earth $=M$)

1. Particle will execute simple harmonic motion.
2. Particle passes the centre of earth with a speed of $\sqrt{\frac{2GM}{R}}$
3. Motion of the particle is periodic
4. Particle will oscillate through the earth to a height $h$ on both sides

Q. Two blocks of masses \(3m\) and \(2m\) are in contact on a smooth table. A force \(P\) is first applied horizontally on block of mass \(3m\) and then on mass \(2m\). The contact forces between the two blocks in the two cases are in the ratio



3m _ 

Q. If the mass of the moon is $\frac{M}{81}$, where $M$ is the mass of the earth, find the distance of the point where the gravitational field due to earth and moon cancel each other, from the centre of the moon. Given that the distance between the centres of the earth and moon is $60R$ where $R$ is the radius of earth.

1. $4R$
2. $8R$
3. $12R$
4. $6R$

Q. Mass density and radius of a solid sphere are ρ, R respectively. Gravitational field at an internal point which is at a distance r from the centre is.

1. $\frac{4\mathrm{\pi \rho Gr}}{3}$
2. $\frac{4{\mathrm{\pi \rho Gr}}^2}{3}$
3. $\frac{4\mathrm{\pi \rho G}{R}^3}{3r^2}$
4. $\frac{4\mathrm{\pi \rho G}{R}^3}{3r}$

Q. A body weights $3.5$ kg wt. on the surface of the earth. What will be its weight on the surface of a planet whose mass is $\frac{1}{7}$th of the mass of the earth and radius half of that of the earth?

Q. Gravitational acceleration on the surface of a planet is $\frac{\sqrt{6}}{11}g,$ 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 $11{\text{kms}}^{-1}$, the escape speed on the surface of the planet in ${\text{kms}}^{-1}$ will be

1. $2{\text{kms}}^{-1}$
2. $3{\text{kms}}^{-1}$
3. $4{\text{kms}}^{-1}$
4. $5{\text{kms}}^{-1}$

Q. A metallic sphere weighing 2.1 kg in the air is held by a string so as to be completely immersed in a liquid of relative density 0.8. The relative density of metallic is 10.5. The tension in the string in Kg-wt is

1. $1.6$
2. $1.94$
3. $3.1$
4. $5.25$

Q. A circular disc of mass $2kg$ and radius $10cm$ rolls without slipping with a speed $2m/s$. The total kinetic energy of disc is

1. $10J$
2. $6J$
3. $2J$
4. $4J$

Q. A spherically symmetric gravitational system of particles has a mass density $\rho =\{\begin{array}{c}{\rho }_0\text{for}r\le R\\ 0\text{for}r>R\end{array}$
where ${\rho }_0$ is a constant. A test mass can undergo circular motion under the influence of the gravitational field of particles. Its speed $V$ as a function of distance $r(0<r<\infty )$ from the centre of the system is represented by

1. 
2. 
3. 
4. 

Q. 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 $\frac{R}{2}$ is now made in the sphere as shown in the figure. The sphere with cavity now applies a gravitational force $F_2$ on the same particle. The ratio $\frac{F_2}{F_1}$ is:

1. $\frac{9}{50}$
2. $\frac{41}{50}$
3. $\frac{3}{25}$
4. $\frac{22}{25}$

Q. A uniform solid sphere of mass $M$ and radius a is surrounded summetrically by a uniform thin spherical shell of equal mass and radius $2\text{a}$. What will be the gravitational field at a distance $\frac{5}{2}$ a from the centre?

1. $\frac{4}{9}\frac{GM}{a^2}$
2. $\frac{4}{16}\frac{GM}{a^2}$
3. $\frac{8}{25}\frac{GM}{a^2}$
4. Zero

Q. A particle of mass m is located inside a spherical shell of mass M and radius R. The gravitational force of attraction between them is

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

Q. A planet has mass $\frac{1}{10}$ of that of earth, while the radius is $\frac{1}{3}$ that of earth. If a person can throw a stone on earth surface to a height of $90\text{m}$, then the will be able to throw the stone on that planet to a height (in metre)

Q. A solid spherical planet of mass $3m$ and radius 'R' has a very small tunnel along its diameter. A small cosmic particle of mass $2m$ is at a distance $2R$ from the center of the planet as shown. Both are initially at rest, and due to gravitational attraction, both start moving towards each other. After sometime, the cosmic particle passes through the centre of the planet. (Assume the planet and the cosmic particle are isolated from other planets)

1. Displacement of the cosmic particle till that instant is $1.2R$.
2. Acceleration of the cosmic particle at that instant is zero.
3. Velocity of the cosmic particle at that instant is $\sqrt{\frac{8Gm}{3R}}$.
4. Total work done by the gravitational force on both the particle is $+\frac{6G{m}^2}{R}$.

Q. A tunnel is dug in the earth across one of its diameters. Two masses $m$ and $2m$ are dropped from the ends of the tunnel. If the masses collide and stick to each other and perform $\text{S.H.M.}$ then amplitude of $\text{S.H.M.}$ will be: $(R=$ radius of the earth$)$

1. $R$
2. $\frac{R}{2}$
3. $\frac{R}{3}$
4. $\frac{2R}{3}$

Q. A ball falling in a lake to a depth $200m$ shows a decrease of $0.1$% in its volume at the bottom. the bulk modulus of the ball is

1. 19.6 x 10${}^8$ N/m${}^2$
2. 19.6 x 10${}^{-10}$ N/m${}^2$
3. 19.6 x 10${}^{10}$ N/m${}^2$
4. 19.6 x 10${}^{-8}$ N/m${}^2$

Q. If Gravitational field due to uniform thin hemispherical shell at point $P$ is $I$, then the magnitude of gravitational field at $Q$ is (Mass of hemisphere is $M$, radius $R$) -

1. $\frac{GM}{4R^2}+I$
2. $\frac{GM}{4R}-I$
3. $2I-\frac{GM}{2R^2}$
4. $\frac{GM}{4R^2}-I$

Q. A planet has a mass of $8.4\times {10}^{24}kg$, which is about wight times the mass of its moon. If the distance between the planet and the moon is about $4.2\times {10}^{5}km$, what is the gravitational pull of the planet on the moon?

1. $2.1\times {10}^{29}N$
2. $3.3\times {10}^{21}N$
3. $2.1\times {10}^{23}N$
4. $3.3\times {10}^{27}N$

Q. Mass density and radius of a solid sphere are ρ, R respectively. Gravitational field at an internal point which is at a distance r from the centre is.

1. $\frac{4\mathrm{\pi \rho Gr}}{3}$
2. $\frac{4{\mathrm{\pi \rho Gr}}^2}{3}$
3. $\frac{4\mathrm{\pi \rho G}{R}^3}{3r^2}$
4. $\frac{4\mathrm{\pi \rho G}{R}^3}{3r}$