An earth satellite moves in a circular orbit with a speed of $6.2\;km/s$. Find the time of one revolution and its centripetal acceleration.
Given,
Orbital velocity $v_o = 6.2\;km/s = 6200\;m/s$
Radius of earth $(R) = 6400000\;m$
Time Period $(T) = \; ?$
Centripetal Acceleration $(a) = \;?$
We have,
$ v_{0} = R \sqrt { \frac{g}{R+h}} $
or, $6200 = 6400000\;*\; \sqrt{\frac{10}{6400000 \; + \; h}}$
or, $9.38 \; *\; 10^{-7} = \sqrt{\frac{10}{6400000 + h}}$
or, $h = 10655301 - 6400000$
h = $4255301\;m$
Now,
$T = 2 \pi \; \sqrt{\frac{(R + h)^3}{g * R^2}}$
or, $2\pi \; \sqrt{\frac{(4255301 + 6400000)^3}{10 * 6400000}}$
or, $2 \; \pi \;*\;1718.57 = 10792.65\;Sec = 2.99\;hrs$
∴ Time of revolution = $2.99\;hrs$
Again,
Centripetal Acceleration $a = \frac{v_0^2}{h + R} = \frac{(6200)^2}{4255301 + 6400000} = 3.6\;m/s^2$
∴ Centripetal Acceleration = $3.6\;m/s^2$
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