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

Three particles, two with masses $m$ and one with mass$M$, might be arranged in any of the four configurations shown below. Rank the configurations according to the magnitude of the gravitational force on $M$ least to greatest

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?

The gravitational potential difference between a point on the surface of the planet and another point $10 \mathrm{m}$ above is $4 \mathrm{J} {\mathrm{kg}}^{-1}$. Considering gravitational field to be uniform, how much work is done in moving a mass of $2 \mathrm{kg}$ from the surface to a point $5 \mathrm{m}$ above the surface?

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

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?

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:

If both the objects have the same $PE$ curve as shown in the figure, then:

As Pluto moves from the perihelion to the aphelion, the work done by gravitational pull of Sun on Pluto is:

At perihelion, the gravitational potential energy of Pluto in its orbit has:

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

The gravitational field intensity at a point $\mathrm{10,000} \mathrm{km}$ from the centre of the earth is $4.8$ $\mathrm{N}\mathrm{ }\mathrm{k}{\mathrm{g}}^{-1}$. The gravitational potential at that point is,

A non - homogeneous sphere of radius $R$ has the following density variation:
$\rho =\left\{\begin{array}{c}{\rho }_{\mathit{0}} ; 0<r\le R/3\\ {\rho }_{\mathit{0}}/2 ; (R/3)<r\le (\mathit{3}R/4)\\ {\rho }_{\mathit{0}}/8 ; (\mathit{3}R/4)<r\le R \end{array}\right.$
Where ${\rho }_0$ is constant. The gravitational field at a distance $2R$ from from the centre of the sphere is:

An asteroid of mass $m$ is approaching earth, initially at a distance $10R_{\mathrm{E}}$ with speed $v_{\mathrm{i}}$. It hits the earth with a speed $v_{\mathrm{f}}$ ($R_{\mathrm{E}}$ and $M_{\mathrm{E}}$ are radius and mass of earth), then

Six point masses each of mass $\mathrm{m}$ are placed at the vertices of a regular hexagon of side $\mathrm{l}$ .The force acting on any of the masses is

If $g$ is acceleration due to gravity on the earth's surface, then the gain in the potential energy of an object of mass $\mathrm{m}$ raised from the surface of the earth to height equal to the radius $R$ of the earth is

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