Find Center of Mass of Pluto & Charon System

[elsternj]

Homework Statement

Pluto's diameter is approximately 2370km , and the diameter of its satellite Charon is 1250km . Although the distance varies, they are often about 1.93×10⁴km 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.

Homework Equations

x_cm=m₁x₁+m₂x₂ / m₁ + m₂

The Attempt at a Solution

Well the center of Pluto and its satellite would be half the diameter so it's center of mass, I am assuming, would be

Pluto: 1185km
Charon: 625km

where i am confused is that i don't have the masses or any actual coordinates. I'm assuming that the masses would end up canceling out? But can somebody guide me as to where to go from here? How do I get the coordinates with the distance they are from each other? Is my equation even right?

 

[PhanthomJay]

The equation is correct, where x1, x2, and xcm are the distances from any arbitrarilly chosen reference point. It is simplest to choose that reference point at the center of one of the masses. Note that the mass of a sphere is is a function of its diameter cubed.

 

[elsternj]

So am I to assume that this is one dimensional? ..For I have no idea the angle at which these spheres are from one another. Say I choose Pluto to be the reference point. Make that x position to be 0 and then Charon's position to be the given distance?

 

[PhanthomJay]

So am I to assume that this is one dimensional? ..For I have no idea the angle at which these spheres are from one another. Say I choose Pluto to be the reference point. Make that x position to be 0 and then Charon's position to be the given distance?

Yes and Yes.

 

[rwisz]

To find the center of mass of the Pluto and Charon system, we will need to use the formula you provided, which is correct. However, we will also need to consider the masses of each object in the system. According to NASA, the mass of Pluto is 1.31 x 10^22 kg and the mass of Charon is 1.52 x 10^21 kg.

Using these values and the distance between the two objects, we can calculate the center of mass as follows:

xcm = (m1*x1 + m2*x2) / (m1 + m2)

xcm = (1.31 x 10^22 * 1185km + 1.52 x 10^21 * 625km) / (1.31 x 10^22 + 1.52 x 10^21)

xcm = 1067.9 km from the center of Pluto

This means that the center of mass of the Pluto and Charon system is located 1067.9 km from the center of Pluto and 1302.1 km from the center of Charon.

It is important to note that the center of mass is not necessarily located at the midpoint between the two objects, as it takes into account the masses of each object. In this case, Pluto's higher mass results in the center of mass being closer to its center.

I hope this helps clarify the concept of center of mass and how to calculate it for the Pluto and Charon system.