# Find the position of the center of mass?

Find the position of the center of mass of the Sun and Jupiter. Does the center of mass lie inside or outside the sun ?

M(sun) = 1.99*10^30

M(Jupiter) = 1.90*10^27

Distance from Sun to Jupter = 7.78 * 10^8 km

r(cm) = (m1r1 + m2r2)/(m1+m2)

Relevance

When I do a center of mass problem, I think about a teeter totter. A heavier person is one end, and a lighter person is on the other end. To balance the teeter totter board, the board must be moved closer to the heavier person.

This is a torque problem. The product of the weight time distance from the center of mass must be the same for both people.

Let x = distance from Sun to center of mass

7.78 * 10^8 – x = distance from Jupiter to center of mass

M sun * x = M Jupiter * (7.78 * 10^8 – x)

1.99 * 10^30 * x = 1.90 * 10^27 * (7.78 * 10^8 – x)

1.99 * 10^30 * x = (1.90 * 10^27 * 7.78 * 10^8) – (1.90 * 10^27 * x)

Add (1.90 * 10^27 * x) to both sides

(1.90 * 10^27 * x) + 1.99 * 10^30 * x = (1.90 * 10^27 * 7.78 * 10^8)

1.9919 * 10^30 = 1.4782 * 10^36

x = 1.4782 * 10^36 ÷ 1.9919 * 10^30 ≈ 7.421 * 10^5

The distance from Sun to center of mass = 7.42 * 10^5 km

The distance from Jupiter to center of mass = 7.78 * 10^8 – 7.421 * 10^5 = 7.77 * 10^8 km

Check:

7.42 * 10^5 * 1.99*10^30 = 1.47658 * 10^36

7.77 * 10^8 * 1.90*10^27 = 1.4763 * 10^36

These 2 numbers are the answers for the clockwise and counter clockwise torque. The torques are almost equal, so the center of mass answers are correct.

I like this method, because I do not have to memorize the center of mass equation, shown below.

r(cm) = (m1r1 + m2r2)/(m1+m2)

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