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I am unable to understand one particular thing about geostationary satellite. Let us assume that I am on earth at a particular place and remain there (moving with the place during rotation) such that the satellite is just above me all the time.

Now my orbital velocity will be

$$ v_o = \sqrt{GM_e/R_e} $$

Talking about the geostationary satellite at a height $h$ we can say that the orbital velocity is

$$ v_{geo} = \sqrt{GM_e/(R_e+h) } $$

This showa that the velocity of the satellite has decreased and simultaneously its radius has increased.

But it must have my angular velocity to have the same time period.

$\omega =v/r$

For the satellite $v$ has decreased and $r $ has increased. How can the angular velocity of satellite be equal to that of mine.

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You wrote $v_o = \sqrt{G M_e / R_e}$ for your orbital velocity. But you're not in orbit! Rather, you're going in a circle with the angular frequency of the Earth, $\Omega \approx \frac{2 \pi}{1\;\text{day}}$. This makes it possible to set your angular frequency equal to that of the satellite.

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