Embibe Experts Solutions for Chapter: Gravitation, Exercise 2: Exercise-2

A tunnel is dug along a chord of the Earth at a perpendicular distance $\mathrm{R}/2$ from the Earth's centre. The wall of the tunnel may be assumed to be frictionless. A particle is released from one end of the tunnel. The pressing force by the particle on the wall, and the acceleration of the particle varies with $\mathrm{x}$ (distance of the particle from the centre) according to :

A double star is a system of two stars of masses $\mathrm{m}$ and $2 \mathrm{m}$, rotating about their centre of mass only under their mutual gravitational attraction. If $\mathrm{r}$ is the separation between these two stars then their time period of rotation about their centre of mass will be proportional to :

A solid sphere of uniform density and radius 4 units is located with its centre at the origin $\mathrm{O}$ of coordinates. Two spheres of equal radii 1 unit, with their centres at $\mathrm{A}(-2,0,0)$ and $\mathrm{B}(2,0,0)$ respectively, are taken out of the solid leaving behind spherical cavities as shown in figure. Then :-

The magnitudes of the gravitational field at distance ${\mathrm{r}}_1$ and ${\mathrm{r}}_2$ from the centre of a uniform, sphere of radius $\mathrm{R}$ and mass $\mathrm{M}$ are ${\mathrm{F}}_1$ and ${\mathrm{F}}_2$ respectively. then :-

Mark the correct statement/s :-

Gravitational potential at the centre of curvature of a hemispherical bowl of radius $\mathrm{R}$ and mass $\mathrm{M}$ is $\mathrm{V}$ :-

Suppose a smooth tunnel is dug along a straight line joining two points on the surface of the Earth and a particle is dropped from rest at its one end. Assume that mass of Earth is uniformly distributed over lts volume. Then

A small ball of mass ' $\text{m}$ ' is released at a height ' $\text{R}$ ' above the Earth surface, as shown in the figure. If the maximum depth of the hall to which it goes is $\mathrm{R}/2$ inside the Earth through a narrow grove before coming to rest momentarily. The grove, contain an ideal spring of spring constant $\mathrm{K}$and natural length $\mathrm{R}$, the value of $\text{K}$is $\text{(R}$ is radius of Earth and $\text{M}$ mass of Earth)