Gravitational Potential and Potential Energy

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

The graph shows the variation with distance $r$ of the gravitational potential $V_g$ (in tera joule per kilogram) due to a plane of radius $2.0 \times {10}^5 \mathrm{m}$.

The graph shows the variation with distance $r$ of the gravitational potential $V_g$ (in tera joule per kilogram) due to a plane of radius $2.0 \times {10}^5 \mathrm{m}$.

Calculate the mass of the planet.

A particle of mass $m$ is rotating in a circular orbit of radius $r$ under the action of gravity in the presence of another stationary particle of very large mass $M\left(M\gg m\right)$. Consider that the gravitational potential energy is zero at infinite separation. If the total energy of the rotating particle is $E$ then, which the following expression correctly represent the angular momentum of the particle?

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

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)

What is intensity of gravitational field at the centre of a spherical shell -

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.

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 ....

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

Three masses $m, 2 \mathrm{m}$ and $3 \mathrm{m}$ are arranged in two triangular configurations as shown in figure $1$ and figure $2$ . Work done by an external agent in changing the configuration from figure $1$ to figure $2$ is

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

Escape velocity from the surface of a planet is $v_e.$ A tunnel is dug across the diameter of the planet and a ball is dropped into it. When the body reaches the centre of the planet, its speed is

Work done by the gravitational force to bring mass $m$ from $nR$ height from earth's surface to earth's surface slowly is $(R$ is radius of earth$)$

A body of mass $\mathrm{m}$ is lifted up from the surface of earth to a height three times the radius of the earth. The change in potential energy of the body is