Gravitational Potential and Potential Energy

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 uniform solid sphere of radius $R$ produces a gravitational acceleration of $a_o$ on its surface. The distance of the point from the centre of the sphere where the gravitational acceleration becomes $\frac{a_o}{4}$ is,

The gravitational field in a region is given by $\overrightarrow{l}=5\hat{\mathrm{i}}+12\hat{\mathrm{j}} \mathrm{N}·\mathrm{k}{g}^{-1}.$ The change in the gravitational potential energy of an object of mass $3 \mathrm{kg}$ when it is taken from the origin to a point $(8\mathrm{m}, -2\mathrm{m})$ is $\_\_\_$

An artificial satellite moving in a circular orbit at a distance $h$ from the centre of the earth has a total energy $E_0$. Then its Potential energy is

If $\text{'}R\text{'}$ is the radius of orbit of a satellite, then the kinetic energy of the satellite is________.

Work done in raising a body of mass $m$ from the surface of the earth to a height $\frac{R}{3}$ is ______. (Where $R$ is radius of earth, $M$ is mass of the earth and $G$ is gravitational constant)

A spherical shell is cut into two pieces along a chord as shown in the figure. $P$ is a point on the plane of the chord. The gravitational field at $P$ due to the upper part is $I_1$ and that due to the lower part is $I_2.$ What is the relation between them?

The gravitational field in a region is given by $\overrightarrow{g}=\left(5\hat{\mathrm{i}}+12\hat{\mathrm{j}}\right) \mathrm{N} {\mathrm{kg}}^{-1}$. The change in the gravitational potential energy of a particle of mass $2 \mathrm{kg}$ when it is taken from the origin to a point $\left(7 \mathrm{m},-3 \mathrm{m}\right)$ is

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

Three identical stars, each of mass $M,$ form an equilateral triangle (stars are positioned at the corners) that rotates around the centre of the triangle. The system is isolated and the edge length of the triangle is$L.$The amount of work done, that is required to dismantle the system is

The mass density of a spherical body is given by $\rho \left(r\right)=\frac{k}{r}$ for $r\le R$ and $\rho \left(r\right)=0$ for $r>R,$ where $r$ is the distance from the centre. The correct graph that describes qualitatively the acceleration, $a$ of a test particle as a function of $r$ is :

The mass density of a spherical body is given by $\rho \left(r\right)=\frac{k}{r}$ for $r\le R$ and $\rho \left(r\right)=0$ for $r>R,$ where $r$ is the distance from the centre. The correct graph that describes qualitatively the acceleration, $a$ of a test particle as a function of $r$ is :

Which of the following most closely depicts the correct variation of the gravitation potential, $V\left(r\right)$ with distance $r$ due to a large planet of radius $R$ and uniform mass density? (figures are not drawn to scale)

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 ___________ .

The initial velocity $v_i$ required to project a body vertically upward from the surface of the earth to reach a height of $10 \mathrm{R},$ where $R$ is the radius of the earth, may be described in terms of escape velocity $v_e$ such that $v_i=\sqrt{\frac{x}{y}}\times v_e.$ The value of $x$ will be

An asteroid is moving directly towards the centre of the earth. When at a distance of $10 \mathrm{R}$ ($\mathrm{R}$ is the radius of the earth) from the earths centre, it has a speed of $12 \mathrm{km}/\mathrm{s}.$ Neglecting the effect of earths atmosphere, what will be the speed of the asteroid when it hits the surface of the earth (escape velocity from the earth is $11.2 \mathrm{km}/\mathrm{s}$)? Give your answer to the nearest integer in kilometer/s ....

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