## Mastering Physics Solutions: The Center of Mass of the Earth-Moon-Sun System

The Center of Mass of the Earth-Moon-Sun System

Part A = 4600 (yes, that’s different than the # of sig figs MP says it wants)
Part B = The center of mass is inside the Earth.
Part C = 456km

Solutions Below:

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 7.35×1022 kg, the mass of the Earth is 6.00×1024 kg, and the mass of the sun is 2.00×1030 kg. The distance between the Moon and the Earth is 3.80×105 km. The distance between the Earth and the Sun is 1.50×108 km.

Part A

Calculate the location xcm 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.

The center mass is given by the formula:

xcm = (x1m1 + x2m2) / (m1 + m2)

Using the Earth as “1″ and the Moon as “2″ (note that the x coordinate for the Earth or whatever our frame of reference is will be zero since there is no distance between the ‘Earth’ and ‘Earth’). For this problem, I’ve converted all distances to meters since I prefer to work in those units, but this isn’t required. The calculator uses km, and you can too, just keep track of whichever units you’re using. To follow my tutorial, below, start by multiplying km by 1000 to get the proper units:

xcm = ((0*6.00×1024) + (3.80×105 * 7.35×1022)) / (6.00×1024 + 7.35×1022)
xcm = (0 + 2.793×1028) / (6.0735×1024)
xcm = (2.793×1031) / (6.0735×1024)
xcm = 4598666m

Now convert back to km since mastering physics is picky:

xcm = 4598.666km

Since MP wants the answer in 3 significant figures, you have to round. You’d normally use 4.60×103 so that you’d have 3 significant figures, but MP’s answer is 4600km- yes, that’s right, MP asks for 3 significant figures and gives an answer with 2. Aren’t you glad you’re reading this now?

xcm = 4600 km

If MP won’t accept that answer, then try it in scientific notation (4.60×103).

Part B

Where is the center of mass of the Earth-Moon system?

The radius of the Earth is 6378 km and the radius of the Moon is 1737 km. Select one of the answers below:

A. The center of mass is exactly in the center between the Earth and the Moon.
B. The center of mass is nearer to the Moon than the Earth, but outside the radius of the Moon.
C. The center of mass is nearer to the Earth than the Moon, but outside the radius of the Earth.
D. The center of mass is inside the Earth.
E. The center of mass is inside the Moon.

We already have the center of mass from Part A, which is 4,600km. Since the radius of the Earth is 6378km,

D. The center of mass is inside the Earth.

Part C

Calculate the location of the center of mass of the Earth-Moon-Sun system during a full Moon. A full Moon occurs when the Earth, Moon, and Sun are lined up as shown in the figure. Use a coordinate system in which the center of the sun is at x=0 and the Earth and Moon both lie along the positive x direction.

For this problem, we use a similar formula as in Part A, but we add in the mass of the Sun and distance from Earth to the Sun:

xcm = (x1m1 + x2m2 + x3m3) / (m1 + m2 + m3)

And the problem tells us to use a coordinate system where the center of the sun is zero and the Earth and Moon are both in the positive direction. So start by getting the distance between the Sun and the Earth, which is given (I’ve converted everything to meters for the tutorial, but you don’t have to and the calculator uses km, so just keep track of whichever units you’re using):

x3 = 1.50×108 km
x3 = 1.50×1011 m

Now get the distance between the Sun and the Moon. The diagram puts Earth in the middle, so just add the distance between the Sun and the Earth to the distance between the Earth and the Moon:

x2 = 1.50×108 km + 3.80×105 km
x2 = 1.5034×108 km
x2 = 1.5034×1011 m

Now solve (“1″ = Sun, “2″ = Sun/Moon, and “3″ = Sun/Earth):

xcm = (x1m1 + x2m2 + x3m3) / (m1 + m2 + m3)
xcm = ((0 * 2.00×1030) + (1.5034×1011 * 7.35×1022) + (1.50×1011 * 6.00×1024)) / (m1 + m2 + 6.00×1024)
xcm = (0 + 1.105×1034 + 9.00×1035) / (2.00×1030 + 7.35×1022 + 6.00×1024)
xcm = (9.1105×1035) / (2.00×1030)
xcm = 455525m

Remember to convert back to km for Mastering Physics:

xcm = 456km

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### 4 Responses to Mastering Physics Solutions: The Center of Mass of the Earth-Moon-Sun System

1. Steve says:

Thank you so much, this helped me understand how to approach the problem, very detailed, Thanks again Mastering Physics Solutions!

• Mastering Physics Solutions says:

Thank you! Glad to hear it!

2. Tim says:

You guys rock! Wish I would have found this weeks ago!

• Mastering Physics Solutions says:

Thanks! We’re glad you finally found us!