Question

What is the orbital radius and speed of a synchronous satellite which orbits the earth once every 24h? Take G = 6:67 x 10À11Nm2=kg2
, Mass of the earth is 5:98 x 1024kg

Answer 1

Answer on Question 51080, Physics, Mechanics | Kinematics | Dynamics

Question:

What is the orbital radius and speed of a synchronous satellite which orbits the Earth once every $24h$? Take $G = 6.67 \cdot 10^{-11} \frac{Nm^2}{kg^2}$, mass of the Earth is $5.98 \cdot 10^{24} kg$.

Solution:

1) When the satellite orbits the Earth, the centripetal force acts on it:

$$F_c = \frac{m_{sat} v^2}{R_{sat}},$$

where, $m_{sat}$ is the mass of the satellite, $v$ is the orbital speed of the satellite and $R_{sat}$ is the orbital radius of the satellite.

From the other hand, the gravitational force attracts the satellite towards the Earth, and we can write:

$$F_{grav} = G \frac{m_{sat} M_E}{R_{sat}^2},$$

where, $G = 6.67 \cdot 10^{-11} \frac{Nm^2}{kg^2}$ is the gravitational constant, $M_E = 5.98 \cdot 10^{24} kg$ is the mass of the Earth.

Since, $F_c = F_{grav}$, we obtain:

$$\frac{v^2}{R_{sat}} = G \frac{M_E}{R_{sat}^2}.$$

Because the satellite travels around the entire circumference of the circle which is $2\pi R_{sat}$ in the period $T$, this means that the orbital speed must be $v = \frac{2\pi R_{sat}}{T}$. Substituting the expression for the orbital speed into the last equation we get:

$$\frac{\left(\frac{2\pi R_{sat}}{T}\right)^2}{R_{sat}} = G \frac{M_E}{R_{sat}^2}.$$

Finally, after simplification we get the formula for the orbital speed of the synchronous satellite:

$$R_{sat} = \sqrt[3]{\frac{GM_E T^2}{4\pi^2}} = \sqrt[3]{\frac{6.67 \cdot 10^{-11} \frac{Nm^2}{kg^2} \cdot 5.98 \cdot 10^{24} kg \cdot (24 \cdot 3600s)^2}{4 \cdot (3.14)^2}} = 4.226476 \cdot 10^7 m.$$

2) In order to find the orbital speed we use the formula $v = \frac{2\pi R_{sat}}{T}$:

$$v = \frac{2\pi R_{sat}}{T} = \frac{2 \cdot 3.14 \cdot 4.226476 \cdot 10^7 m}{24 \cdot 3600s} = 3072 \frac{m}{s}.$$

**Answer:**

1) The orbital radius of the synchronous satellite is $R_{sat} = 4.226476 \cdot 10^7 m$.

2) The orbital speed of the synchronous satellite is $v = 3072 \frac{m}{s}$.

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