Question

Miranda, one of the moons of the planet Uranus, orbits the planet with an average speed of $6.5\text{km}{\text{s}}^{-1}$, while experiencing a centripetal force of $2\times {10}^{19}\text{N}$. What is the distance (in $\text{km}$) between Uranus and Miranda, if the moon has a mass of $6.4\times {10}^{19}\text{kg}$?

• A. 135200

Answer 1

The correct option is A 135200
Given:
Mass of the moon Miranda, $\text{m}=6.4\times {10}^{19}\text{kg}$
Average orbital speed of Miranda, $\text{v}=6.5\text{km}{\text{s}}^{-1}$
Centripetal force:
${\text{F}}_{\text{c}}=2\times {10}^{19}\text{N}$
${\text{F}}_{\text{c}}=2\times {10}^{19}\text{kg}\text{m}{\text{s}}^{-2}$
${\text{F}}_{\text{c}}=2\times {10}^{16}\text{kg}\text{km}{\text{s}}^{-2}$

Centripetal force acting on Miranda is given by:
${\text{F}}_{\text{c}}=\frac{\text{m}{\text{v}}^2}{\text{r}}$,
where:
$\text{r}$ is the radius of the moon's orbit, i.e. the distance between Miranda and Uranus.
$\text{r}=\frac{\text{m}{\text{v}}^2}{{\text{F}}_{\text{c}}}$
$\text{r}=\frac{6.4\times {10}^{19}\times 6.5\times 6.5}{2\times {10}^{16}}\text{km}$
$\text{r}=1,35,200\text{km}$