Calculate the position of the center of mass of the following pairs of objects. Use acoordinate system where the origin is at the center of the more massive object. Give youranswer not in meters but as a fraction of the radius as requested. Get data from insidethe front cover of your text.
Earth Radius = 6.37x10^6 m
Moon Radius = 1.74x10^6 m
Jupiter Radius = 1.90x10^7 m
Earth to moon distance = 3.84x10^8 m
earth to sun 1.496x10^11m
jupiter to sun 7.78x10^11 m
(a) The Earth and the Moon. Give the answer as a fraction of the earth's radius.
(b) The Sun and the Earth. Give the answer as a fraction of the sun's radius.
(c) The Sun and Jupiter. Give the answer as a fraction of the sun's radius.
please show how my answers arent even getting close
To calculate the position of the center of mass for each pair of objects, we can use the formula:
Center of Mass = (m1 * r1 + m2 * r2) / (m1 + m2)
where m1 and m2 are the masses of the two objects, and r1 and r2 are the distances of the objects' centers from the reference point (in this case, the origin).
(a) The Earth and the Moon:
Given that the Earth's radius is 6.37x10^6 m and the Moon's radius is 1.74x10^6 m, we need to find the distance between their centers. The Earth-Moon distance is given as 3.84x10^8 m.
The distance between the centers is the sum of the Earth's radius and the Moon's radius: 6.37x10^6 m + 1.74x10^6 m = 8.11x10^6 m.
Now, let's calculate the center of mass in terms of the Earth's radius:
Center of Mass = (m1 * r1 + m2 * r2) / (m1 + m2)
= (m1 * 0 + m2 * (r1 + r2)) / (m1 + m2)
= (m2 * (r1 + r2)) / (m1 + m2)
= (1 * (1.74x10^6 m + 8.11x10^6 m)) / (1 + 1)
= 9.85x10^6 m / 2
To convert this to a fraction of the Earth's radius, divide the center of mass by the Earth's radius:
Center of Mass in terms of Earth's radius = (9.85x10^6 m / 2) / (6.37x10^6 m)
= 0.772
So, the position of the center of mass of the Earth and the Moon is 0.772 times the Earth's radius.
(b) The Sun and the Earth:
Given that the Earth-Sun distance is 1.496x10^11 m, to calculate the center of mass in terms of the Sun's radius, we need to find the distance between the centers of the Sun and the Earth.
The distance between the centers is the sum of the Earth-Sun distance and the Earth's radius: 1.496x10^11 m + 6.37x10^6 m = 1.4960637x10^11 m.
Now, let's calculate the center of mass in terms of the Sun's radius:
Center of Mass = (m1 * r1 + m2 * r2) / (m1 + m2)
= (m1 * r1 + m2 * (r1 + r2)) / (m1 + m2)
= (m1 * r1 + m2 * r1 + m2 * r2) / (m1 + m2)
= (r1 * (m1 + m2) + m2 * r2) / (m1 + m2)
= r1 + (m2 * r2) / (m1 + m2)
= 1 + (m2 * r2) / (m1 + m2)
Let's plug in the values for the Sun and the Earth:
Center of Mass in terms of Sun's radius = 1 + (m2 * r2) / (m1 + m2)
= 1 + ((5.97x10^24 kg) * (6.37x10^6 m)) / ((1.99x10^30 kg) + (5.97x10^24 kg))
Evaluating the expression gives a position of the center of mass in terms of the Sun's radius.
(c) The Sun and Jupiter:
Given that the Jupiter-Sun distance is 7.78x10^11 m, we can follow the same procedure as above to calculate the center of mass in terms of the Sun's radius.