Calculate the position of the center of mass of the following pairs of objects. use a coordinate system where 0 is at the center of the more massive object. give your answer not in meters but as a fraction of the radius as requested. you can find the data you need in the inside of the back cover of your textbook. (a) the earth and the moon. give the answer as a fraction of the earth's radius.

the center of mass is  Rcm = 0.725 R Earth

Explanation:

The concept of center of mass is of great importance in all physics, since on this all external forces of the system can be considered applied, in the collision analysis it gives an ideal reference system for resolution

The equation for the center of mass is

        Rcm = ∑ mi ri / Mt

With Rcm and ri being vector quantities, "m" is body mass and MT total mass

If we write this equation for our system we have

       Rcm = (M1 r1 + M2 r2) / (M1 + M2)

M1 and M2 are the mass of the Earth and the Moon respectively, r1 and r2 are the distances measured from our reference system. As they indicate that we place the most massive reference system we place it on planet Earth.  The data taken a textbook.

        M1 = 5.98 10²⁴  Kg

        M2 = 7.36 10²² Kg

       Distance from Earth to the moon on average 3.84 10⁸ m

How the coordinate system is placed on Earth

        r1 = 0

        r2 = 3.84 10 8 m

      Rcm = 0 + 7.36 10²²  3.84 10⁸ / (5.98 10²⁴ + 7.36 10²²)

R cm = 2.826 10³¹ / (598 10²² + 7.36 10²²)

Rcm = 2.826 10³¹ / 611.34 10²²

R cm = 4.62 10⁶  m

we are asked to give this distance as a fraction of the radius of the Earth, we divide the two quantities

      R earth = 6.37 10⁶ m

     Fraction = R cm / R Earth

     Fraction = 4.62 10⁶ / 6.37 10⁶

     Fraction = 0.725

In summary the center of mass is

     Rcm = 0.725 R Earth

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