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Center of mass

The center of mass of a group of points is defined as the weighted mean of the points' positions. The weight applied to each point is the point's mass. The centre of mass of an object is the point through which any plane divides the mass of the object in half. It is also called the center of inertia.

For mass that is distributed according to a continuous, nonnegative density $\rho(\mathbf{x}) \ge 0$ over a body V in space, the center of mass is:

$\bar{\mathbf{x}} = \frac{1}{M} \int_{V} \rho(\mathbf{x})\mathbf{x}\,dV$, where total mass $M = \int_{V} \rho(\mathbf{x})\,dV$.

In $\mathcal{R}^3$, $\bar{\mathbf{x}} = (\bar{x}, \bar{y}, \bar{z})$. Each of the center of mass's components can be computed by

$\bar{x} = \frac{1}{M} \int_{V} x\rho(x, y, z)\,dx dy dz$

$\bar{y} = \frac{1}{M} \int_{V} y\rho(x, y, z)\,dx dy dz$

$\bar{z} = \frac{1}{M} \int_{V} z\rho(x, y, z)\,dx dy dz$.

The origin from which positions are calculated has no effect on the result. As long as the same unit is used for all the points, any length and mass unit can be used.

• Point A: position 2m, mass 1kg. Point B: position 4m, mass 2kg (assume positions are distances along a straight line from some origin). Center of mass:

$\frac{2 \times 1 + 4 \times 2}{1+2} = 3.33\ {\rm m}$

• Solid homogenous sphere (ideally divided in a high number of points of equal mass): each point averages with its opposite. Center of mass is at the center.

• Sphere with spherically symmetric density: center of mass is at the center. This approximately applies to the Earth: the density varies considerably, but it mainly depends on depth and less on the other two coordinates.

• Human being: It varies according to the body's position, but often it's somewhere inside the abdomen

• A sports car: engineers try hard to make the car as light as possible, and then add weight on the bottom. This way, the center of mass is nearer to the street, and the car handles better.

When talking about celestial bodies, the center of mass has a special relevance: when a moon orbits around planet, or a planet orbits around a star, both of them are actually orbiting around their center of mass, called the barycenter. There are some interesting consequences:

• Earth-Moon system: the Moon's mass is 1/81 of Earth. Put Earth in position 0, mass 1 (here we use an arbitrary mass unit. It does not matter, provided that we use the same unit for the Moon). Moon position 400,000km, masss 1/81. Center of mass is at:

$\frac{0 \times 1 + 400,000 \times \frac{1}{81}}{1 + \frac{1}{81}} = 4,877\ {\rm km}$

from the Earth's center. We can see that the Earth is far from standing "still" and the Moon moving: both of them move around a point more than 1,000km below the Earth surface.

• Sun-Earth system: put Sun in position 0, mass=333,000 times the Earth. Earth in position 150,000,000 km, mass=1. Center of mass is 450 km from the Sun center. Here, the large mass difference between the two bodies makes the center of mass lie almost where we were expecting it.

• Sun-Jupiter system: put Sun in position 0, mass = 333,000 Earths. Jupiter in position 778,000,000 km, mass=318 Earths. Center of mass is 742,000 km from the Sun center. It's actually outside its surface! As Jupiter does its 11 year orbit, the Sun majestically does a full 1.5 million km orbit around the center of mass.

• To calculate the actual motion of the Sun, you would need to sum all the influences from all the planets, comets, asteroids, etc. of the solar system. But, only Jupiter manages to pull the center of mass so far, thanks to its large mass. If all the planets would align on the same side of the Sun, the combined center of mass would lie about 500,000 km outside the Sun surface.

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