Derivation of $g = \dfrac{GM}{R^2}$
Step 1: सतह पर वस्तु पर बल: $F = \dfrac{GMm}{R^2}$
Step 2: न्यूटन के द्वितीय नियम से: $F = mg$
Step 3: $mg = \dfrac{GMm}{R^2}$
Step 4: $m$ काटने पर
$$g = \frac{GM}{R^2}$$
Variation of $g$ with Height
Step 1: $g_h = \dfrac{GM}{(R+h)^2} = \dfrac{g}{\left(1+\dfrac{h}{R}\right)^2}$
Step 2: $h \ll R$ के लिए $\left(1+\dfrac{h}{R}\right)^{-2} \approx 1 - \dfrac{2h}{R}$
$$g_h = g\left(1 - \frac{2h}{R}\right)$$
Variation of $g$ with Depth
Step 1: गहराई $d$ पर प्रभावी द्रव्यमान: $M' = \dfrac{4}{3}\pi (R-d)^3 \rho$
Step 2: $\dfrac{g_d}{g} = \dfrac{R-d}{R} = 1 - \dfrac{d}{R}$
$$g_d = g\left(1 - \frac{d}{R}\right)$$
Gravitational Potential Energy
Step 1: $W = \displaystyle\int_{\infty}^{r} \dfrac{GMm}{x^2}\,dx$
Step 2: $W = -GMm\left[\dfrac{1}{x}\right]_{\infty}^{r} = -\dfrac{GMm}{r}$
$$U = -\frac{GMm}{r}$$
Escape Velocity
Step 1: ऊर्जा संरक्षण: $\dfrac{1}{2}mv_e^2 + \left(-\dfrac{GMm}{R}\right) = 0 + 0$
Step 2: $\dfrac{1}{2}v_e^2 = \dfrac{GM}{R}$
$$v_e = \sqrt{\frac{2GM}{R}} = \sqrt{2gR}$$
Orbital Velocity
Step 1: $\dfrac{m v_o^2}{r} = \dfrac{GMm}{r^2}$
Step 2: $v_o^2 = \dfrac{GM}{r}$
$$v_o = \sqrt{\frac{GM}{r}}$$
Total Energy of Satellite
Step 1: $K = \dfrac{1}{2}m v_o^2 = \dfrac{GMm}{2r}$
Step 2: $U = -\dfrac{GMm}{r}$
Step 3: $E = K + U$
$$E = -\frac{GMm}{2r}$$
Relation between Escape & Orbital Velocity
Step 1: $\dfrac{v_e}{v_o} = \dfrac{\sqrt{2GM/R}}{\sqrt{GM/R}} = \sqrt{2}$
$$v_e = \sqrt{2}\,v_o$$
