# Ethys, one of Saturn's moons, travels in a circular orbit at a speed of 1.1 10 4 m / s . The...

## Question:

Ethys, one of Saturn's moons, travels in a circular orbit at a speed of {eq}1.1\times10^4 \; m/s {/eq}. The mass of Saturn is {eq}5.67\times10^{26} \;kg {/eq}. Calculate

a) the orbital radius in kilometers.

b) the orbital period in Earth days.

## Orbital Speed

The speed with which a satellite orbit around a planet is called orbital speed. The orbital speed of a satellite depends upon the mass of the planet and radius of the orbit of the satellite. Mathematically

{eq}\begin{align} v_o = \sqrt{\frac{GM}{r}} \end{align} {/eq}

Where M is the mass of the planet and r is the radius of the orbit.

Data Given

• Orbital speed of the Ethys {eq}v_o = 1.1 \times 10^4 \ \rm m/s {/eq}
• The mass of the Saturn {eq}M = 5.67 \times 10^{26} \ \rm kg {/eq}

Part A) We know that orbital speed is given by

{eq}\begin{align} v_o = \sqrt{\frac{GM}{r}} \end{align} {/eq}

{eq}\begin{align} v_o^2 = \frac{GM}{r} \end{align} {/eq}

{eq}\begin{align} r = \frac{GM}{v_o^2} \end{align} {/eq}

{eq}\begin{align} r = \frac{6.67 \times 10^{-11} \ \rm N.m^2kg^{-2} \times 5.67 \times 10^{26} \ \rm kg}{(1.1 \times 10^4 \ \rm m/s)^2} \end{align} {/eq}

{eq}\begin{align} \color{blue}{\boxed{r = 3.13 \times 10^8 \ \rm m = 3.13 \times 10^5 \ \rm km}} \end{align} {/eq}

Part B) Orbital period is given by

{eq}\begin{align} T = \frac{2 \pi r}{v_o} \end{align} {/eq}

{eq}\begin{align} T = \frac{2 \pi \times 3.13 \times 10^8 \ \rm m }{1.1 \times 10^4 \ \rm m/s} \end{align} {/eq}

{eq}\begin{align} \color{blue}{\boxed{T = 178330 \ \rm s = 0.07 \ \rm day}} \end{align} {/eq}