Question

Calculate the height of the communication satellite. [Given $G=6.67\times {10}^{-11}N{m}^2/k{g}^2,M=6\times {10}^{24}kg,R=6400km$ (Ans:$35.9\times {10}^{6}m$)

Answer 1

Gravitational force = Centripetal force

$\frac{GMm}{r^2}=\frac{m{V}^2}{r}$

Now V = $\frac{2\pi r}{T}$

$\frac{GMm}{r^2}=m\frac{(\frac{2\pi r}{T}{)}^2}{r}$

$\frac{GM}{r}=(\frac{2\pi r}{T}{)}^2$

$r^3=(R+h{)}^3=\frac{GM{T}^2}{\left(4{\pi }^2\right)}$

For communication satellite T = 24 hrs = $24\times$ 3600 s

$r^3=(R+h{)}^3=6.67\times {10}^{-11}\times 6\times {10}^{24}\frac{(24\times 3600{)}^2}{\left(4{\pi }^2\right)}$

= $7575046127.63\times {10}^{13}{m}^3$

r = $4231.18\times {10}^{4}m$ = 42312 km

r = R + h = 42312 km

h = 42312 - 6400

= 35912 km

= $35.9\times {10}^{6}m$