Gravitation

(1) $F = G {m_1 m_2}/{r^2}$ G = 6.673 x $10^{-11}$ $m^3$/$kg s^2$

(2) $g = {GM}/{R^2}$ i.e. $weight = {GMm}/{R^2}$

(3) $v_c = √{{GM}/{r}}$ = $√{{GM}/{R + h}}$ = $√{{gR^2}/{R + h}}$ = $√{g_{h} (R+h)}$

Also, $v_{c} = √{gR}$

(4) $T^2 = {4π^2}/{GM} r^3$ i.e. $T^2 α r^3$

(5) Gravitational potential: $V = – {GM}/{r}$ (r≥R)

For a satellite, $KE = {GM m }/{2r}$ = ${GMm}/{2(R+h)}$ = $BE$
Total $E = – {GMm}/{2r}$ = – ${GMm}/{2(R+h)}$

(6) For body at rest on Earth’s surface, $v_e = √{{2GM}/{R}}$ = $√{2gR}$

i.e. $v_e = √{2} v_c$

(7) $g_h = g ({R}/{R+h})^2$ [For less than 100 km, $g_h = g {(1 – {2h}/{R})}$]

(8) $g’ = g – w^2 R cos^2 ϕ$

(9) $g_d = g {(1 – {d}/{R})}$