Question:
The moon takes about $27.3$ days to revolve round the earth in a nearly circular orbit of radius $3.84 \times 105 \mathrm{~km}$. Calculate the mass of the earth from these data.
Solution:
Time period of the moon around the earth is given by,
$T=2 \pi \sqrt{\frac{r^{\mathrm{g}}}{G M}}$
$\mathrm{m}=$ mass of earth
$r=$ Distance between center of the moon and earth
$27.3 \times 86400=2 \times 3.14 \sqrt{\frac{\left(3.84 \times 10^{5} \times 10^{\mathrm{s}}\right)^{\mathrm{s}}}{6.67 \times 10^{-11} \times M}}$
$M=6.02 \times 10^{24} \mathrm{~kg}$
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