Gravitational Potential and Field

In older times, people used to think that the Earth was flat. Imagine that the Earth is indeed not a sphere of radius $R$, but an infinite plate of thickness $H$. What value of $H$ is needed to allow the same gravitational acceleration to be experienced as on the surface of the actual Earth? (Assume that the Earth's density is uniform and equal in the two models).

The escape velocity for a planet is $V_e$. A tunnel is dug along a diameter of the planet and a small body is dropped into it at the surface. When the body reaches the centre of the planet, its speed will be

A smooth tunnel is dug along a chord of the earth at a perpendicular distance $\frac{R}{2}$ from the centre of the earth where $R=$ radius of the earth. A particle of mass $m$ is released from one end of the tunnel. During the motion of the particle in the tunnel, [$g=$ acceleration due to gravity at the surface of earth]

There is a concentric hole of radius $R$ in a solid sphere of radius $2\mathrm{R}$. Mass of the remaining portion is $M$. The gravitational potential at centre is $-\frac{xGM}{R},\mathrm{x}=$

A body of mass $m$ is placed on the earth's surface. It is taken from the earth's surface to a height $h=3R$ when $R$ is the radius of the earth. The change in gravitational potential energy of the body is

A small bead can slide without friction on a wooden rod of length $\ell =10.0 \mathrm{m}.$ Initially the rod and the bead both are held motionless with the rod aligned radially with the earth. The left end of the rod is at a distance $r_0=4\times {10}^8 \mathrm{m}$ from the earth centre and the bead is at a distance ${\mathrm{x}}_0=2.0 \mathrm{cm}$ away from the left end. Both the bodies are released simultaneously. Considering gravitational interaction only with the earth, if time after the release, the bead will separate from the rod is $\mathrm{P}\times {10}^4 \sec .$ Find $\mathrm{P}.$ Radius of the earth is $\mathrm{R}=6400 \mathrm{km}$ and acceleration due to gravity on the earth is$\mathrm{g}=10 \mathrm{m}/{\mathrm{s}}^2.$ (Approximate the velue of $P$ to the nearest integer.)

A cavity is made in a fixed sphere of radius$R$ & mass $m$ such that the centre of the cavity is at a distance $\frac{R}{2}$from the centre of the sphere. A small particle of mass $m_1$is released at point $A$. Time taken by particle to reach point $B$ such that $AB$ is the diameter is $\sqrt{\frac{a{R}^3}{Gm}}$. Find a.

In the reported figure of earth, the value of acceleration due to gravity is same at point $A$ and $C$ but it is smaller than that of its value at point $B$ (surface of the earth). The value of OA : AB will be $x : y.$ The value of $x$ is ___________ .

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,

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

The gravitational field due to a mass distribution is, $E=\frac{K}{x^3}$ in the $x$-direction ($K$ is a constant). Taking the gravitational potential to be zero at infinity, its value at a distance $x$ is,

A spiral galaxy can be approximated as an infinitesimally thin disc of a uniform surface mass density (mass per unit area) located at $z=0,$ Two stars $A$ and $B$ start from rest from heights $2z_0$ and $z_0$ ($z_0<<$ radial extent of the disc), respectively and fall towards the disc, cross over to the other side and execute periodic oscillations. The ratio of time periods of $A$ and $B$ is

At some point, the gravitational potential and also the gravitational field due to earth is zero. The point is

Two uniform solid spheres of equal radii $R\text{,}$ but mass $M$ and $4M$ have a centre to centre separation $6R\text{,}$ as shown in the figure. A projectile of mass $m$ is projected from the surface of the sphere of mass $M$ directly towards the centre of the second sphere. The minimum speed of the projectile so that it reaches the surface of the second sphere is

A uniform thin shell of mass $\mathrm{M}$ and radius $\mathrm{R}=1 \mathrm{m}$ has centre at origin and another
uniform solid sphere of mass $4\mathrm{M}$ and radius $\mathrm{R}$ is kept with its centre at $\mathrm{x}=6 \mathrm{m}$ as shown.
Neglect external fields.

At which of the following point is the net gravitational field zero ?

Over a certain region of space, the gravitational potential is given by, $V=3x^{2}y-2yz+x^{3}z$. The magnitude of gravitational field at $\left(0\text{,} 1\text{,} -1\right)$ is,

Suppose a frictionless tunnel is made inside the earth along a diameter. A string of mass $\mathrm{m}$ and length $R/2$ is suspended in the tunnel with one end of the string at the surface of the earth. A force $F$ is applied to pull the string slowly to the surface to the earth. Find the work done by the force $F$ to pull the string completely on the surface of the earth. (Given acceleration due to gravity on the surface of the earth $=g$, radius of the earth $=R)$

At a point above the surface of the earth, the gravitational potential is $-5.12\times {10}^7 \mathrm{J} {\mathrm{kg}}^{-1}$ and the acceleration due to gravity is $6.4 {\mathrm{ms}}^{-2}$.  Assuming the mean radius of the earth to be $6400 \mathrm{km}$, the height of this point above the earth's surface is.

If mass density of earth varies with distance $r$ from centre of earth as $\mathrm{\rho }=kr$ and $\text{'}R\text{'}$ is radius of earth, then find the orbital velocity of an object revolving around earth at a distance $\text{'}R\text{'}$ from its centre.

A thin wire of mass $m$ and length $l$  is bent to form semicircle. The gravitational potential at the centre of semicircle is :