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# What is the radius of moon's orbit around earth assuming the orbit to be circular ($g=9.8 m/s^2$; Radius of earth =6400 km; period of rotation of moon around the earth $=27.3\; days$

$(a)\;6400\;km \quad (b)\;3.8 \times 10^8\;m \quad (c)\;1000\;km \quad (d)\;10000\;km$

Let orbital radius of moon be $R_0$ and radius of earth be R. mass of earth M mass of moon m, orbital velocity of moon v and its orbital time period be T
For moon resolving around earth
$\large\frac{GMm}{{R_0}^2}=\frac{mv^2}{R_0}$
Gravitational force =centrepetal force
$GM=gR^2$
$\large\frac{gR^2}{R_0}$$=v^2 =\bigg(\large\frac{2 \pi R_0}{T}\bigg)^2$
$\qquad= \large\frac{4 \pi^2 {R_0}^2}{T^2}$
${R_0}^3=\large\frac{gR^2T^2}{4 \pi ^2}$
$R_0=\bigg[\large\frac{g \pi ^2 T^2}{4 \pi ^2}\bigg]^{\large\frac{1}{3}}$
$R_0=\bigg[\large\frac{9.8 (6.4 \times 10^6)^2 (27.3 \times 86400)^2 }{4(3.14)^2}\bigg]^{\Large\frac{1}{3}}$
$\qquad= 3.83 \times 10^8 \;m$
Hence b is the correct answer.

edited Feb 17, 2014 by meena.p