Chapter 5: Q22. (page 131)

The mass of the “planet” Pluto was not known until it was discovered to have a moon. Explain how this enabled an estimate of Pluto’s mass.

Expert verified

The mass of planet Pluto is $m' = \frac{4 π^{2} r^{3}}{G T^{2}}$ .

Step by step solution

In this problem, the relation of gravitational force and centripetal force will be used in order to estimate the mass of the plant Pluto.

The relation of the centripetal force is given by,

$F = \frac{m v^{2}}{r}$

Here, m is the mass of the moon, v is the velocity of the moon and r is the distance.

The relation of the gravitational force from Pluto is given by,

$F = \frac{G m m'}{r^{2}}$

Here, $m'$ is the mass of the planet and G is the gravitational constant.

On equating the above two relations.

$\begin{matrix} \frac{m v^{2}}{r} = \frac{G m m'}{r^{2}} \\ v^{2} = \frac{G m'}{r} \\ m' = \frac{v^{2} r}{G} \end{matrix}$

The relation of velocity of the moon is given by,

$v = \frac{2 π r}{T}$

Here, T is the time period.

On plugging the values in the above relation.

$\begin{array}{l} m' = \frac{{(\frac{2 π r}{T})}^{2} r}{G} \\ m' = \frac{4 π^{2} r^{3}}{G T^{2}} \end{array}$

Thus, $m' = \frac{4 π^{2} r^{3}}{G T^{2}}$ is the mass of the Pluto.

Unlock Step-by-Step Solutions & Ace Your Exams!

Over 30 million students worldwide already upgrade their learning with Vaia!

Recommended explanations on Physics Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.