What is wrong with this reasoning about angular velocity $\omega$?

Question

Satellite $X$ orbits a planet with orbital radius R. Satellite Y orbits the same planet with orbital radius 2R. Satellites X and Y have the same mass.

What is $\frac{\text{centripetal acceleration of X}}{\text{centripetal acceleration of Y}}?$

My reasoning is that both satellites orbit the same planet, they have the same angular velocity $\omega$.

As such, $a_{c_x} = \frac{v_{x}^{2}}{r}$ where $v = \omega \cdot r$. Thus,

$$a_{c_x} = \frac{\omega^2 \cdot r^2}{r} \implies = \omega^2 \cdot r$$.

Doing the same thing for $a_{c_y} = \omega^2 \cdot 2r$.

Then, $$a_{c_x}/ a_{c_y} = \frac{1}{2}$$

Answer 1

From X and Y orbiting the planet, it does not follow that they have the same angular velocity. The centripetal acceleration is what keeps the satellites orbiting. It is the acceleration, which corresponds to the gravitational force of the planet, which is constant for both satellites, as they are orbiting the same planet. As both satellites have the same mass $m$ we get: $$a_{c_x}=a_{c_y}=\frac{F_g}{m}=g,$$ where $g$ is the acceleration caused by the planets mass. So what we can calculate is the angular momentum that the satellites will have, because of this gravitational acceleration $g$: $$g=a_c=\omega^2r$$ $$\Leftrightarrow \,\,\,\,\,\,\omega_x=\sqrt{\frac{g}{R}} \,\,\,\,\,\,\wedge\,\,\,\,\,\, \omega_y=\sqrt{\frac{g}{2R}},$$ where $R$ is the radius of satellite X. Thus $$\frac{\omega_x}{\omega_y}=\sqrt{2},$$ is something you could calculate here.