Physics Gravitation

The minus sign in the formula indicates that:

• A gravitational force exists.
• The satellite cannot leave the orbit by itself.
• A minimum threshold of energy will be required to move the satellite from its existing orbit.

According to kepler’s third law of time period –

  ${\left(\frac{T_A}{T_B}\right)}^2={\left(\frac{r_A}{r_B}\right)}^3\Rightarrow \frac{r_A}{r_B}=4^{1/3}$  

$\Rightarrow r_4^3=4r_B^3$  

Kindly go through the solution 

$b=a\sqrt[]{1-e^2}$

$\frac{dA}{dt}=\frac{L}{2m}$

$\frac{A}{T}=\frac{L}{2m}$

$T=\frac{2mA}{L}=\frac{2m\pi ab}{L}=\frac{2m\pi a^2\sqrt[]{1-e^2}}{L}$

$V_{in}=\frac{-GM}{2R}\left[3-{\left(\frac{r}{R}\right)}^2\right],$

$V_{surface}=\frac{-GM}{R},V_{out}=\frac{-GM}{r}$

$T=2\pi \sqrt[]{\frac{r^3}{G{M}_e}}$

$T_B-T_A=\frac{2\pi }{\sqrt[]{G{M}_e}}\left(r_B^{3/2}-r_A^{3/2}\right)=\frac{2\times 3.14}{\sqrt[]{6.67\times 10^{-11}\times 6\times 10^{24}}}\left[{\left(8\times 10^6\right)}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}-{\left(7\times 10^6\right)}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\right]$

  $\frac{1}{2}m{V}_a^2-\frac{GMm}{2x}$  

           =   $\frac{1}{2}m\frac{V_a^2}{4}-\frac{GMm}{4x},\boxed{V_a=\sqrt[]{\frac{2}{3}\frac{GM}{x}}}$

ROC at point – B

$ROC=\frac{V_b^2}{a\perp }=\frac{\frac{2GM}{3R}\times \frac{1}{4}}{\frac{GM}{{\left(4R\right)}^2}}=\frac{8R}{3}$

v = √ (GM/r) (orbital speed)
⇒ GM = v²r ⇒ (GM/R²) = (v²r/R²) ⇒ g = (v²r/R²)

$\mathrm{}g=\frac{G{M}_{\text{in}}}{r^2}$  , where $M_{\text{in}}={\int }_0^r?\rho 4\pi x^2\cdot dx$

$\begin{array}{rr}& M_{\text{in}}=4\pi {\rho }_0{\int }_0^r??\left(1-\frac{x^3}{R^3}\right)x^{2}dx=4\pi {\rho }_0\left(\frac{r^3}{3}-\frac{r^6}{6R^3}\right)\\ & \therefore \mathrm{}g=\frac{4\pi G{\rho }_0}{6}\left(2r-\frac{r^4}{R^3}\right)\end{array}$ ${}_{}{}_{}^{}{}^{}$

For $\mathrm{g}$ $$ to be maximum or minimum or constant $\frac{dg}{dr}=0\Rightarrow 2-\frac{4r^3}{R^3}=0\Rightarrow r=\frac{R}{2^{1/3}}$ $\frac{}{}\frac{{}^{}}{{}^{}}\frac{}{{}^{}}$

Let the origin be at the CM of the particles, let the initial positions of the particles be x = a/2 and x = -a/2 and let the instantaneous positions of the particles be x = r and x = -r
Let the instantaneous velocity of each particle be v
Let the time after which the distance between the particles has