Embibe Experts Solutions for Chapter: Gravitation, Exercise 3: Level 3

Mass $m$ is divided into two parts $x \mathrm{m}$ and  $(1-x) \text{m}$. For a given separation, the value of $x$ for which the gravitational attraction between the two pieces becomes maximum is
 

An object is propelled vertically to a maximum height of $4R$ from the surface of a planet of radius $R$ and mass $M.$ The speed of the object when it returns to the surface of the planet is

A planet of mass '$m$' moves in an elliptical orbit around an unknown star of mass '$M$' such that its maximum and minimum distances from the star are equal to $r_1$ and $r_2$ respectively. The angular momentum of the planet relative to the centre of the star is

The mass density of a spherical galaxy varies as $\frac{\mathit{K}}{\mathit{r}}$ over a large distance $r$ from its center. In that region, a small star is in a circular orbit of radius $R$. Then the period of revolution,$T$ depends on $R$ as:

The mass of a spaceship is $1000 \mathrm{kg}$. It is to be launched from the earth's centre out into free space. The value of $g$ and $R$ (radius of earth) are $10 \mathrm{m} {\mathrm{s}}^{-2}$ and $6400 \mathrm{km}$, respectively. The required energy for this work will be,

A planet orbits in an elliptical path of eccentricity $e$ around a massive star considered fixed at one of the foci. The point in space where it is closest to the star is denoted by $P$ and the point where it is farthest is denoted by $A$. Let $v_{\mathrm{P}}$ and $v_{\mathrm{A}}$ be the respective speeds at $P$ and $A$. Then,

Two satellites $S_1$ and $S_2$ are revolving around a planet in the opposite sense in coplanar circular concentric orbits. At time $t=0,$ the satellites are farthest apart. The periods of revolution of $S_1$ and $S_2$ are $3 \mathrm{h}$ and $24 \mathrm{h}$ respectively. The radius of the orbit of $S_1$ is $3\times {10}^4 \mathrm{km}.$ Then the orbital speed of $S_2$ as observed from

A planet revolves about the sun in elliptical orbit. The areal velocity $\left(\frac{\mathrm{d}A}{\mathrm{d}t}\right)$ of the planet is $4.0\times {10}^{16} {\mathrm{m}}^2 {\mathrm{s}}^{-1}$. The least distance between planet and the sun is $2\times 10^{12} \mathrm{m}$. Then the maximum speed of the planet in $\mathrm{km} {\mathrm{s}}^{-1}$ is-