Derivation of Kepler's Laws

The sun is losing its mass due to ___________.

As the Sun converts hydrogen to helium through nuclear fusion, it emits energy. This decreases the mass of the Sun and changes the orbits of all the planets.How much lighter will the Sun be when it completes the hydrogen burning phase? By how much will this change the orbit of the Earth? Give your answers in numbers and in per cent..( the sun emits ${10}^3\mathrm{W}/{\mathrm{m}}^2$)
 

If the mass of the earth is $6\times {10}^{24}\mathrm{Kg}$ and the distance between the sun and the earth is $1.5\times {10}^{11}\mathrm{m}$. If the gravitational force between the earth and the sun is $3.5\times {10}^{22}\mathrm{N}$, what is the mass of the sun? ($\left(\mathrm{G}=6.7\times {10}^{-11}{\mathrm{Nm}}^2{\mathrm{kg}}^{-2}\right)$

The time period $\left(\mathrm{T}\right)$of a planet around the sun is related to its distance $\left(\mathrm{r}\right)$ from the sun by the relation,

A Saturn year is $29.5$ times that of the earth. Find the distance of Saturn from the sun if the earth is $1.5\times {10}^8\mathrm{km}$away?

Derive Kepler's second law using gravitation.

What is the mass of the sun?

Determine the mass of the sun applying the laws of gravitation.

Derive Kepler's third law from the law of gravitation

The mass and density of the moon, if acceleration due to gravity on its surface is $1.62 \mathrm{m} {\mathrm{s}}^{-2}$ and its radius is $1.74\times {10}^6 \mathrm{m}$, respectively. $($Take, $G=6.67\times {10}^{-11} \mathrm{N}-{\mathrm{m}}^2 {\mathrm{kg}}^{-2}$)

Two planets have the same average density but their radii are $R_1$ and $R_2.$ If acceleration due to gravity on these planets be $g_1$ and  $g_2$ respectively, then :-

Assuming the earth to be a homogeneous sphere, determine the density of the earth from following data:

$g=9.8 \mathrm{m}/{\mathrm{s}}^2, G= 6.673 \mathrm{x} {10}^{-11} {\mathrm{Nm}}^2/{\mathrm{kg}}^2, R=6400 \mathrm{km}.$

The mass and the diameter of a planet are three times of the respective values for the earth. If the time period of oscillation of a simple pendulum on earth is $2 \mathrm{s}$, then the time period of oscillation of the same pendulum on the surface of the given planet would be

Change in acceleration due to gravity is same upto a height h from each other the earth surface and below depth $\text{x}$ , then relation between x and h is (h and $\text{x}<<{\text{R}}_{\text{e}}$ )

Consider a planet in some solar system which has a mass double the mass of the earth and density equal to the average density of the earth. An object weighing W on the earth will weight

The moon's radius is one fourth that of the earth and its mass is $\frac{1}{80}$ times that of earth, if $g$ represents the acceleration due to gravity on the surface of the earth, then that on the surface of the moon is:

Suppose the earth was covered by an ocean of uniform depth $h$$\left(\mathrm{h}\ll \mathrm{R}\right).\mathrm{ }$ let $\mathrm{\sigma }$ be density of ocean and $\mathrm{\rho }$ be mean density of earth. Let $∆\mathrm{g}$ be the approximate difference of value of net acceleration due to gravity between the bottom of the ocean and top $\Delta g=g_{top}-g_{bottom}$. Choose the correct option.

The acceleration due to gravity '$g$' and mean density of the earth $\rho$ are related by which of the following relations?

(Where, $G$ is the gravitational constant and $R$ is radius of earth.)

The value of $g$ on the surface of earth is $9.8 \mathrm{m} {\mathrm{s}}^{-2}$ & the radius of earth is $6400\mathrm{ }\mathrm{k}\mathrm{m}$. The average density of earth in $\mathrm{k}\mathrm{g} {\mathrm{m}}^{-3}$ will be:

The diameters of two planets are in the ratio of $4:1$ & mean density have ratio $1:2$ , then the ratio of $g$ on the planets will be: