Question

The period of moon revolving under the gravitational force of earth is 27.3days.Find the distance of the moon from centre of earth if mass of earth is 5.97×10^24 kg.

Answer 1

Let's moon revolves around the earth is a circular path.

then, centripetal force is balanced by gravitational force between them.

centripetal force = $m\omega^2r$

gravitational force = $\frac{GmM}{r^2}$

where, r is the separation between moon and earth, m is mass of moon and M is mass of the earth .

so, $\cancel{m}\omega r=\frac{G\cancel{m}M}{r^2}$

or, $\omega^2=\frac{GM}{r^3}$

we know, $\omega=\frac{2\pi}{T}$ , where T is time period.

so, $T^2= 4\pi^2\frac{r^3}{GM}$

or, $r^3=\frac{GMT^2}{4\pi^2}$

now putting T = 27.3 days = 27.3 × 24 × 3600 sec

G = 6.67 × 10^-11 Nm²/Kg²

and M = 5.97 × 10²⁴ kg

now, r³ = {6.67 × 10^-11 × 5.97 × 10²⁴ × (27.3 × 24 × 3600)²}/(4 × 3.14²)

= 5.6 × 10^25 m

now, r = 382,586,237 m = 382,586.237 km

hence, distance between earth and moon is 382,586.237 km