Question

A common, though incorrect, statement is, "The Moon orbits the Earth." That creates an image of the Moon?s orbit that looks like that shown in the figure. (Figure 1) The Earth's gravity pulls on the Moon, causing it to orbit. However, by Newton?s third law, it is known that the Moon exerts a force back on the Earth. Therefore, the Earth should move in response to the Moon. Thus a more accurate statement is, "The Moon and the Earth both orbit the center of mass of the Earth-Moon system." In this problem, you will calculate the location of the center of mass for the Earth-Moon system, and then you will calculate the center of mass of the Earth-Moon-Sun system. The mass of the Moon is 7.35?1022kg , the mass of the Earth is 6.00?1024kg , and the mass of the sun is 2.00?1030kg . The distance between the Moon and the Earth is 3.80?105km . The distance between the Earth and the Sun is 1.50?108km . Part A Calculate the location x_cm of the center of mass of the Earth-Moon system. Use a coordinate system in which the center of the Earth is at x=0 and the Moon is located in the positive x direction.

Answer 1

Answer:

$x_{cm}$ = 4.6 10⁶ m

Explanation:

The definition of mass center is

    $x_{cm}$ = 1/M ∑ xi mi

Where M is the total mass, mi and xi the mass and position of each body

Let us apply this equation to our case, as they indicate that we take the center of reference to Earth, its distance is zero; let's write the data they give us

Earth

    M = 6.00 10²⁴ kg

    r1 = 0

Moon

    m = 7.35 10²² kg

    r = 3.80 10⁵ km (1000m / 1km) = 3.80 10⁸ m

Let's calculate

        $x_{cm}$ = 1 /(m + M) (0 + m r)

        $x_{cm}$ = m / (m + M) r

        $x_{cm}$ = 7.35 10²² / (7.35 10²² + 600 10²²) 3.80 10⁸

        $x_{cm}$ = 4.6 10⁶ m