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 to get it i am really stuck.
Just think of the planets as point masses on a lever. We want the lever to balance, where the fulcrum is the center of mass.
So, m1d1 = m2d2
or, m1/m2 = d2/d1
For earth-moon,
d = 3.84x10^8 = d1+d2
earth mass = 5.9742x10^24
moon mass = 7.3477x10^22
m1/m2 = 81.307
so, 82.307 * d1 = 3.84x10^8
d1 = 4.7228x10^6
So, the center of mass for the earth-moon system is 4.7228x10^6m from the center of the earth.
4.7228x10^6/3.84x10^8 = 0.0123 of earth radius
do similar work for the other systems
Oops. I figured as a fraction of the earth-moon distance, not the earth radius.
4.7228x10^6/6.37x10^6 = 0.74 of earth radius
Oh, don't worry, I'm here to help you out of this gravitational dilemma! Let's calculate the position of the center of mass for each pair of objects:
(a) Earth and Moon:
To find the position of the center of mass, we need to consider the masses of the two objects. Given that the mass of the Earth is much greater than the Moon, we can assume that the center of mass is located very close to the center of the Earth.
Since the origin is the center of the Earth, the distance to the center of the Moon is the Earth-Moon distance, which is 3.84x10^8 m. The radius of the Earth is 6.37x10^6 m.
To find the position of the center of mass as a fraction of the Earth's radius, we divide the Earth-Moon distance by the Earth's radius:
Center of mass position = Earth-Moon distance / Earth's radius
Center of mass position = (3.84x10^8 m) / (6.37x10^6 m)
Center of mass position ≈ 60.28
So, the position of the center of mass for the Earth and Moon pair is approximately 60.28 times the radius of the Earth.
(b) Sun and Earth:
Again, assuming that the Sun is much more massive than the Earth, we can consider the center of mass located very close to the center of the Sun.
Since the origin is the center of the Sun, the distance to the center of the Earth is the Earth-Sun distance, which is 1.496x10^11 m. The radius of the Sun is 1.90x10^7 m.
To find the position of the center of mass as a fraction of the Sun's radius, we divide the Earth-Sun distance by the Sun's radius:
Center of mass position = Earth-Sun distance / Sun's radius
Center of mass position = (1.496x10^11 m) / (1.90x10^7 m)
Center of mass position ≈ 7873.68
So, the position of the center of mass for the Sun and Earth pair is approximately 7873.68 times the radius of the Sun.
(c) Sun and Jupiter:
Similarly, let's assume the Sun is much more massive than Jupiter, meaning the center of mass is located close to the center of the Sun.
Since the origin is the center of the Sun, the distance to the center of Jupiter is the Jupiter-Sun distance, which is 7.78x10^11 m. The radius of the Sun is 1.90x10^7 m.
To find the position of the center of mass as a fraction of the Sun's radius, we divide the Jupiter-Sun distance by the Sun's radius:
Center of mass position = Jupiter-Sun distance / Sun's radius
Center of mass position = (7.78x10^11 m) / (1.90x10^7 m)
Center of mass position ≈ 40,947.37
So, the position of the center of mass for the Sun and Jupiter pair is approximately 40,947.37 times the radius of the Sun.
Hope that clears things up! If gravity worked differently, we might have had a circus in space!
To calculate the position of the center of mass, we use the formula:
Center of mass = (m1 * r1 + m2 * r2) / (m1 + m2)
Where:
- m1 and m2 are the masses of the objects
- r1 and r2 are the distances from the center of mass to each object
Now, let's calculate the position of the center of mass for each pair of objects:
(a) Earth and Moon:
- Mass of Earth = 5.97x10^24 kg
- Mass of Moon = 7.35x10^22 kg
- Distance from Earth to Moon = 3.84x10^8 m
To calculate the position as a fraction of Earth's radius, we divide the distance by the Earth's radius:
Distance from Earth to Moon in terms of fraction of Earth's radius = (3.84x10^8 m) / (6.37x10^6 m) = 60.26
Therefore, the position of the center of mass is approximately 60.26 Earth radii from the center of the Earth towards the Moon.
(b) Sun and Earth:
- Mass of Sun = 1.99x10^30 kg
- Mass of Earth = 5.97x10^24 kg
- Distance from Sun to Earth = 1.496x10^11 m
To calculate the position as a fraction of the Sun's radius, we divide the distance by the Sun's radius:
Distance from Sun to Earth in terms of fraction of Sun's radius = (1.496x10^11 m) / (1.90x10^7 m) = 7880
Therefore, the position of the center of mass is approximately 7880 Sun radii from the center of the Sun towards the Earth.
(c) Sun and Jupiter:
- Mass of Sun = 1.99x10^30 kg
- Mass of Jupiter = 1.90x10^27 kg
- Distance from Sun to Jupiter = 7.78x10^11 m
To calculate the position as a fraction of the Sun's radius, we divide the distance by the Sun's radius:
Distance from Sun to Jupiter in terms of fraction of Sun's radius = (7.78x10^11 m) / (1.90x10^7 m) = 40947
Therefore, the position of the center of mass is approximately 40947 Sun radii from the center of the Sun towards Jupiter.