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

2520.54463 km

Step-by-step explanation:

The location of the center of mass is given by

`r_(cm)=(\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=(4)/(3)\pi r^3=(1)/(6)\pi d^3` = Mass

`d_c` = Diameter of Charon

`d_p` = Diameter of Pluto

`\\\Rightarrow x_(cm)=(m_px_p+m_cx_c)/(m_p+m_c)\\\Rightarrow x_(cm)=(m_c)/(m_p+m_c)x_c=((1)/(6)\rho \pi d_c^3)/((1)/(6)\rho \pi d_p^3+(1)/(6)\rho \pi d_c^3)\\\Rightarrow x_(cm)=(d_c^3)/(d_p^3+d_c^3)x_c\\\Rightarrow x_(cm)=(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