Gravitational Field and Potential

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

If a missile launched with a velocity less than escape velocity, the sum of its $K.E$. and $P.E$. is always

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

Which of the following is true about the gravitational potential $V$ due to a point mass at a distance $r$ from the point mass?

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

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

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

A spherical portion of radius $\frac{R}{2}$ is removed from a solid sphere of mass $M$ and radius $R$ as shown in the figure. Taking gravitational potential $V=0$ at $r=\infty$, the potential at the centre of the cavity thus formed is $(G=\text{gravitational constant})$,

Two spheres each of mass $\mathrm{M}$ and radius $\mathrm{R}$ are separated by a distance of $\mathrm{r}$ .If a line is joined in between the centres of the spheres, The gravitational potential at the midpoint of that line is

Two bodies of masses ${\mathrm{m}}_1$ and ${\mathrm{m}}_2$ start moving towards each other from rest under mutual gravitational attraction from infinity. Their relative velocity when they are r distance apart is

What is the minimum energy required to launch a satellite of mass $\mathrm{m}$ from the surface of a planet of mass $\mathrm{M}$ and radius $\mathrm{R}$ in a circular orbit at an altitude of $2\mathrm{R}?$

A particle of mass $50 \mathrm{g}$ is kept on the surface of a uniform sphere of mass $120 \mathrm{k}\mathrm{g}$ and radius $50 \mathrm{c}\mathrm{m}$. Find the work to be done against the gravitational force between them (in $\mathrm{n}\mathrm{J}$) to take the particle far away from the sphere. (Take $G=6.67\times {10}^{-11} \mathrm{N} {\mathrm{m}}^2 \mathrm{k}{\mathrm{g}}^{-2}$).

A small $2 \mathrm{k}\mathrm{g}$ mass moved slowly from the surface of earth to a height of $6.4\times {10}^6 \mathrm{m}$ above the earth. Find the work done [in mega joule].(Radius of earth is $6.4\times {10}^6$ $\mathrm{m}$)

Consider a planet moving in an elliptical orbit around the Sun. The work done on the planet by the gravitational force of the Sun:

Which of the following are correct?

The correct graph showing the variation of magnitude of acceleration of object with its distance from centre of earth is,

Gravitational potential $V$ versus distance $r$ graph is represented in figure. The magnitude of gravitational field intensity is equal to:

A person brings a mass of $1 \mathrm{k}\mathrm{g}$ from infinity to a point $A$. Initially the mass was at rest but it moves with a speed of $2 \mathrm{m}/\mathrm{s}$ as it reaches $A$. The work done by the person on the mass is $-3 \mathrm{J}$. The potential at $A$ is:

A uniform metal sphere of radius $a$ and mass $M$ is surrounded by a thin uniform spherical shell of equal mass and radius $4a$. The centre of the shell is on the surface of the inner sphere.

The gravitational field intensity at a point $P_2$ is

A uniform metal sphere of radius $a$ and mass $M$ is surrounded by a thin uniform spherical shell of equal mass and radius $4a$. The centre of the shell falls on the surface of the inner sphere.

The gravitational field intensity at point $P_1$ is