Question

Center of Mass ProblemPluto and Charon. Pluto’s diameter is approximately 2370 km, and the diameter of itssatellite Charon is 1250 km. Although the distance varies, they are often about 19,700 kmapart, center to center. Assuming that both Pluto and Charon have the same compositionand hence the same average density, find the location of the center of mass of this systemrelative to the center of Pluto.

Answer 1

Answer:

2520.54463 km

Explanation:

The  location  of  the  center  of  mass  is  given  by

$r_{cm}=\frac{\sum m_j\vecr}{\sum m_j}$

Using Coordinates  with  the  origin  at  the  center  of  Pluto  and  the  +x direction  toward Charon, so

$x_p$ = 0

$x_c$ = 19700 km

$m=\frac{4}{3}\pi r^3=\frac{1}{6}\pi d^3$ = Mass

$d_c$ = Diameter of Charon

$d_p$ = Diameter of Pluto

$\\\Rightarrow x_{cm}=\frac{m_px_p+m_cx_c}{m_p+m_c}\\\Rightarrow x_{cm}=\frac{m_c}{m_p+m_c}x_c=\frac{\frac{1}{6}\rho \pi d_c^3}{\frac{1}{6}\rho \pi d_p^3+\frac{1}{6}\rho \pi d_c^3}\\\Rightarrow x_{cm}=\frac{d_c^3}{d_p^3+d_c^3}x_c\\\Rightarrow x_{cm}=\frac{1250^3}{2370^3+1250^3}19700\\\Rightarrow x_{cm}=2520.54463\ km$

The location of the center of mass of this systemrelative to the center of Pluto is 2520.54463 km