Gravitational Field Due to a Disc

Q. A system consists of a disc and square with equal surface mass density as shown in figure. The centre of mass is at the point of contact.The relation between $L$ and $R$ is $L=(2\pi {)}^{n}R$.Then, find $n$.
(Answer up to two decimal places)

Q. A sphere of mass $M$ and radius $R_2$ has a concentric cavity of radius $R_1$ as shown in figure. The force $F$ exerted by the sphere on a particle of mass $m$ located at a distance $r$ from the centre of sphere varies as $(0\le r\le \infty )$

1. 
2. 
3. 
4. 

Q. Consider two solid uniform spherical objects of the same density $\rho .$ One has radius $R$ and the other has radius $2R.$ They are in outer space where the gravitational fields from other objects are negligible. If they are arranged with their surface touching, what is the contact force between the objects due to their traditional attraction ?

1. 
   $G{\pi }^2{R}^4$
2. 
   $\frac{128}{81}G{\pi }^2{R}^4{\rho }^2$
3. 
   $\frac{128}{81}G{\pi }^2$
4. 
   $\frac{128}{87}{\pi }^2{R}^{2}G$

Q. A horizontal disc rotates with a constant angular velocity $\omega$ about a vertical axis passing through its centre. A small body $m$ moves along a diameter with a velocity $v$. Find the force the disc exerts on the body when it is at a distance $r$ located from the rotation axis.

1. $mr{\omega }^2+2mv\omega$
2. $mg+\sqrt{m^2{r}^2{\omega }^4+{\left(2mv\omega \right)}^2}$
3. $\sqrt{m^2{g}^2+{\left(2mv\omega \right)}^2}+mr{\omega }^2$
4. $m\sqrt{g^2+r^2{\omega }^4+{\left(2v\omega \right)}^2}$

Q. A small sphere of radius $R$, is held against the inner surface of a smooth spherical shell of radius $6R$ as shown in the figure. The masses of the shell and small sphere are $4M$ and $M$ respectively. This arrangement is placed on a smooth horizontal table. The small sphere is now released. The x-coordinate of the centre of the shell when the smaller sphere reach the other extreme position is,

1. $R$
2. $2R$
3. $3R$
4. $4R$

Q. A very long (length $L$) cylindrical galaxy is made of uniformly distributed mass and has radius $R(R<<L)$. A star outside the galaxy is orbiting the galaxy in a plane perpendicular to the galaxy and passing through its centre. If the time period of star is $t$ and its distance from the galaxy's axis is $r$, then

1. $T\propto r^2$
2. $T\propto r$
3. $T\propto \sqrt{r}$
4. $T^2\propto r^3$

Q. A planet of radius R is revolving a round sun in an orbit of radius r. If the temperature of sun is Ts and its radius is Rs. the equilibrium temp of the plant assuming both sun and planet as black bodies is $T=Ts\sqrt{\frac{Rs}{nr}}$. find n.

Q. The magnitudes of the gravitational field at distances $r_1$ and $r_2$ from the centre of a uniform sphere of radius $R$ and mass $M$ are $E_1$ and $E_2$ respectively. Then:

1. $\frac{E_1}{E_2}=\frac{r_1}{r_2}$, if $r_1<R$ and $r_2<R$
2. $\frac{E_1}{E_2}=\frac{r_2^2}{r_1^2}$, if $r_1>R$ and $r_2>R$
3. $\frac{E_1}{E_2}=\frac{r_1^3}{r_2^3}$, if $r_1<R$ and $r_2<R$
4. $\frac{E_1}{E_2}=\frac{r_1^2}{r_2^2}$, if $r_1<R$ and $r_2<R$

Q. A solid sphere of mass $M$ and radius $a$ is surrounded by a uniform concentric spherical shell of thickness $2a$ and mass $2M$. The gravitational field at distance $3a$ from the centre will be:

1. $\frac{2GM}{9a^2}$
2. $\frac{GM}{3a^2}$
3. $\frac{GM}{9a^2}$
4. $\frac{2GM}{3a^2}$

Q. A uniform sphere has a mass $M$ and radius $R$. Find the gravitational pressure $P$ inside the sphere, as a function of the distance from its centre.

Q. Consider a spherical gaseous cloud of mass density $\rho \left(r\right)$ in a free space where r is the radial distance from its centre. The gaseous cloud is made of particles of equal mass m moving in circular orbits about their common centre with the same kinetic energy K. The force acting on the particles is their mutual gravitational force. If $\rho \left(r\right)$ is constant in time. The particle number density $n\left(r\right)=\rho \left(r\right)/m$ is? (G$=$ universal gravitational constant)

1. $\frac{K}{6\pi r^2{m}^{2}G}$
2. $\frac{K}{\pi r^2{m}^{2}G}$
3. $\frac{3K}{\pi r^2{m}^{2}G}$
4. $\frac{K}{2\pi r^2{m}^{2}G}$

Q. The path of the cathode rays in an magnetic field can be approximated to a circle. In order to double the radius of the circular path keeping velocity constant :

1. double the magnetic field
2. halve the magnetic field
3. increase the magnetic field to four times
4. triple the magnetic field

Q. Find the magnitude of the gravitational field at distances $r_1$ and $r_2$ from the center of a uniform sphere of radius R and mass M, respectively.

Q. A small photocell is placed at a distance of a 4 m from photosensitive surface. When light falls on the surface the current is 5 mA. If the distance of cell is decreased to 1 m , the current will become:

1. $\left(\frac{5}{16}\right)mA$
2. $1.25mA$
3. $80mA$
4. $20mA$