Gravitation

Chapters included in Gravitation carry Newton's law of gravitation, Projection of satellite, Periodic time, Statement of Kepler's laws of motion, Binding energy and escape velocity of a satellite, Weightlessness condition in orbit, depth, and motion, Communication satellite and its uses.

Chapters included in Gravitation carry Newton's law of gravitation, Projection of satellite, Periodic time, Statement of Kepler's laws of motion, Binding energy and escape velocity of a satellite, Weightlessness condition in orbit, depth, and motion, Communication satellite and its uses.

  $\frac{GMm}{{\left(R+x\right)}^2}=\frac{GMm\left(R-x\right)}{R^3}$

->R³ = (R + x)² (R – x)

->R³ = (R²– x²) (R + x)

->x² + Rx – R² = 0

$x=\frac{-R\pm \sqrt[]{R+4R^2}}{2}$

$x=\frac{\left(\sqrt[]{5R}-1\right)}{2}R$

Stress $=\frac{\mathrm{F}}{\mathrm{A}}=\frac{\mathrm{T}}{\mathrm{A}}=\frac{\mathrm{W}}{\mathrm{A}}$

$g=\frac{Ax}{{\left(x^2+a^2\right)}^{3/2}}$  $\frac{}{{\left({}^{}{}^{}\right)}^{}}$

$\Rightarrow \mathrm{}{\int }_v^0?dV=-{\int }_x^{\mathrm{\infty }}?gdx$

$\Rightarrow \mathrm{}O-V=-\left[\int \frac{Ax}{{\left(a^2+x^2\right)}^{3/2}}\right]$

Let, $a^2+x^2=t^2$ ${}^{}{}^{}{}^{}$

$\Rightarrow \mathrm{}2xdx=2tdt$

$\Rightarrow \mathrm{}\mathrm{x}\mathrm{d}x=\mathrm{t}\mathrm{d}\mathrm{t}$

$\Rightarrow \mathrm{}V=\int \frac{Atdt}{t^3}\Rightarrow -\frac{A}{t}\Rightarrow -{\left.\frac{A}{\sqrt[]{a^2+x^2}}\right|}_x^{\mathrm{\infty }}$

$\Rightarrow \mathrm{}V=\frac{A}{\sqrt[]{a^2+x^2}}$

${\left(v_e\right)}_A={\left(v_e\right)}_B\Rightarrow \frac{M_1}{R_1}=\frac{M_2}{R_2}$

R is not correct.

$F_1=\frac{GMm}{9R^2}$

$F_2=\frac{GMm}{9R^2}-\frac{G\frac{M}{8}m}{{\left(\frac{5R}{2}\right)}^2}=\frac{GMm}{R^2}\left(\frac{1}{9}-\frac{1}{50}\right)$

$\frac{F_1}{F_2}=\frac{1}{1-\frac{9}{50}}=\frac{50}{41}$

Apply energy conservation,

$U_i+K_i=U_f+K_f$

$\Rightarrow -\frac{GMm}{R}+K_i=-\frac{GMm}{3R}+\frac{1}{2}m{v}^2$

$\Rightarrow -\frac{GMm}{R}+K_i=-\frac{GMm}{3R}+\frac{1}{2}*m*\frac{GM}{3R}$

$\Rightarrow K_i=-\frac{1}{6}\frac{GMm}{R}+\frac{GMm}{R}$

$K_i=\frac{5}{6}\frac{GMm}{R}$