Embibe Experts Solutions for Chapter: Gravitation, Exercise 1: JEE Advanced Paper 2 - 2014

The distance between two stars of masses $3{\mathrm{M}}_S$ and $6{\mathrm{M}}_S$ is $9\mathrm{R}$. Here $R$ is the mean distance between the centres of the Earth and the Sun, and $M_S$ is the mass of the Sun. Two stars orbit around their common centre of mass in circular orbits with period $nT$, where$T$ is the period of Earth's revolution around the Sun.

The value of $n$ is _____.

A rocket is launched normal to the surface of the Earth, away from the Sun, along the line joining the Sun and the Earth. The Sun is $3\times {10}^5$ times heavier than the Earth and is at a distance $2.5\times {10}^4$ times larger than the radius of the Earth. The escape velocity from Earth's gravitational field is $v_e=11.2\mathrm{ }\mathrm{km}\mathrm{ }{\mathrm{s}}^{-1}$ . The minimum initial velocity $\left(v_s\right)$ required for the rocket to be able to leave the Sun-Earth system is closest to

(Ignore the rotation and revolution of the Earth and the presence of any other planet)

A spherical body of radius $\mathrm{R}$ consists of a fluid of constant density and is in equilibrium under its own gravity. If $\mathrm{P}\left(\mathrm{r}\right)$ is the pressure at $\mathrm{r}(\mathrm{r}<\mathrm{R})$ , then the correct option(s) is (are)

A planet of radius $\mathrm{R}=\frac{1}{10}\times$ (radius of Earth) has the same mass density as Earth. Scientists dig a well of depth $\frac{\mathrm{R}}{5}$ on it and lower a wire of the same length and of linear mass density ${10}^{-3}\mathrm{ }\mathrm{k}\mathrm{g} {\mathrm{m}}^{-1}$ into it. If the wire is not touching anywhere, the force applied at the top of the wire by a person holding it in place is (take the radius of Earth $=6\times {10}^6\mathrm{ }\mathrm{m}$ and the acceleration due to gravity of Earth is $10\mathrm{ }\mathrm{m} {\mathrm{s}}^{-2}$)

Two bodies, each of mass $M$, are kept fixed with a separation of $2L$. A particle of mass $m$ is projected from the midpoint of the line joining their centers, perpendicular to the line. The gravitational constant is $G$. The correct statement (s) is (are) :

Consider a spherical gaseous cloud of mass density $\mathrm{\rho }\left(\mathrm{r}\right)$ in free space where $\mathrm{r}$ is the radial distance from its center. The gaseous cloud is made of particles of equal mass m moving in circular orbits about the common center with the same kinetic energy $\mathrm{K}$ . The force acting on the particles is their mutual gravitational force. If $\mathrm{\rho }\left(\mathrm{r}\right)$ is constant in time, the particle number density $\mathrm{n}\left(\mathrm{r}\right)=\mathrm{\rho }\left(\mathrm{r}\right)/\mathrm{m}$ is: [ $\mathrm{G}$ is universal gravitational constant]