Question

Find the satellite's orbital period

Answer 1

Answer on Question#39469 – Physics - Other

Find the satellite's orbital period.

Solution:

Consider a satellite with mass $m$ orbiting a central body with a mass of mass $M$. The central body could be a planet, the sun or some other large mass capable of causing sufficient acceleration on a less massive nearby object. If the satellite moves in circular motion, then the net centripetal force acting upon this orbiting satellite is given by the relationship

$$F_{\text{centr}} = \frac{m v^2}{r}$$

This net centripetal force is the result of the gravitational force that attracts the satellite towards the central body and can be represented as

$$F_{\text{grav}} = G \frac{m \cdot M}{r^2}$$ $$(1) = (2)$$ $$\frac{m v^2}{r} = G \frac{m \cdot M}{r^2}$$ $$m v^2 r = G m M$$ $$v = \sqrt{\frac{G M}{r}}$$

Now we can find the orbital period:

$$T = \frac{2 \pi r}{v} = \frac{2 \pi r}{\sqrt{\frac{G M}{r}}} = 2 \pi \sqrt{\frac{r^3}{G M}}$$

where $r$ is the distance from the center of the Earth (Earth's radius + altitude), $G$ is the gravitational constant, and $M$ is the mass of the Earth.

Answer: orbital period $T = 2\pi \sqrt{\frac{r^3}{GM}}$.