Question

Pluto’s diameter is approximately 2370 km, and the diameter of its satellite Charon is 1250 km. Although the distance varies, they are often about 19,700 km apart, center to center. Assuming that both Pluto and Charon have the same composition and hence the same average density, find the location of the center of mass of this system relative to the center of Pluto.

Answer 1

Answer:

$r_{cm} = 2520.5 km$

Explanation:

As we know that mass is the product of volume and density

so we will have

$M = \rho V$

here we have

$M = \rho(\frac{4}{3}\pi r^3)$

so we will have

$\frac{M_p}{M_c} = (\frac{r_p}{r_c})^3$

so we will have

$\frac{M_p}{M_c} = (\frac{2370}{1250})^3$

$M_p = 6.81 M_c$

now let the position of Pluto is at origin so we have

$r_{cm} = \frac{M_p (0) + M_c(19700)}{M_p + M_c}$

$r_{cm} = \frac{19700}{\frac{M_p}{M_c} + 1}$

$r_{cm} = \frac{19700}{6.81 + 1}$

$r_{cm} = 2520.5 km$

Answer 2

The location of the center of mass of this system is 2,522.41 km.

Given the following data:

Pluto’s diameter = 2370 km.

Satellite's diameter = 1250 km.

Distance = 19,700 km.

How to determine the center of mass of this system.

First of all, we would derive an expression for the ratio of the mass of Pluto to the mass of its satellite Charon:

$\rho = \frac{M}{V} \\\\M=\rho V\\\\M=\rho(\frac{4\pi r^3}{3} )\\\\\frac{M_p}{M_s} =(\frac{d_p}{d_s})^3\\\\\frac{M_p}{M_s} =(\frac{2370}{1250})^3\\\\M_p = 6.81M_s$

From the origin, Pluto's position is given by this expression:

$C=\frac{M_s d + M_p}{M_s +M_p} \\\\C=\frac{M_s (19700) + M_p(0)}{M_s +6.81M_s} \\\\C=\frac{19700M_s}{7.81M_s} \\\\C=\frac{19700}{7.81}$

C = 2,522.41 km.

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