Problem geostationary satellite

[fluidistic]

Homework Statement

At what height (from the ground of the Earth) must be placed a geostationary satellite?

Homework Equations

The Attempt at a Solution

I'm not sure I understand well the question. I guess they are asking for a satellite in such a position that he would not deviate from the point over the ground it is situated. Well, I believe that the height doesn't matter at all and what matters is the velocity you put the satellite in orbit that matters. But the answer of the question is $$3.58 \cdot 10^5 \text{km}$$. I have no idea of how to get the solution. Do you have an idea?

 

[Redbelly98]

They mean that the orbital period is 1 day, so that the satellite is always directly over the same place on Earth.

Also, assume a circular orbit. What equations do you know that apply to:
1. gravity
2. circular motion

 

[fluidistic]

They mean that the orbital period is 1 day, so that the satellite is always directly over the same place on Earth.

Also, assume a circular orbit. What equations do you know that apply to:
1. gravity
2. circular motion

Hmm... I guess they want me to use $$F_c=m_ca_c=m_c\frac{v^2}{r}$$. Also $$F_c=\frac{Gm_cM_E}{r^2}$$ which led me to conclude that $$r=\frac{GM_E}{v^2}$$. But the velocity is unknown so I'm stuck. At last I could calculate it but only in function of the altitude (using the formula $$v=\omega r$$) which is precisely what I'm looking for...

 

[alphysicist]

Hmm... I guess they want me to use $$F_c=m_ca_c=m_c\frac{v^2}{r}$$. Also $$F_c=\frac{Gm_cM_E}{r^2}$$ which led me to conclude that $$r=\frac{GM_E}{v^2}$$. But the velocity is unknown so I'm stuck. At last I could calculate it but only in function of the altitude (using the formula $$v=\omega r$$) which is precisely what I'm looking for...

What is $\omega$ for this satellite?

 

[Redbelly98]

$$r=\frac{GM_E}{v^2}$$. But the velocity is unknown so I'm stuck. At last I could calculate it but only in function
of the altitude (using the formula $v=\omega r$) which is precisely what I'm looking for...

You're on the right track. What happens if you substitute $v=\omega r$ into the previous equation here?

 

[fluidistic]

Oh... Thanks to both! I didn't think about replacing $$v$$ by $$\omega r$$!