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Centre 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. It is also called the center of inertia.

For mass that is distributed according to a density ρ(x) over a body V in space, the center of mass is

<math>\bar{\mathbf{x}} = \frac{1}{M} \int_{V} \rho(\mathbf{x})\mathbf{x}\,dV<math>

Here <math>M<math> is the total mass, given by

<math>M = \int_{V} \rho(\mathbf{x})\,dV<math>

In <math>\mathcal{R}^3<math>, the components of the center of mass are given by:

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

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

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

For a system of point masses m₁, m₂, ..., the integrals are replaced by sums:

<math>\bar{\mathbf{x}} = \frac{1}{M} \sum m_i\mathbf{x}_i<math>

Where the total mass M is the sum of the constituent masses.

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

Motion of the Center of Mass

The following equations of motion assume that there is a system of particles governed by internal and external forces. An internal force is a force caused by the interaction of the particles within the system. An external force is a force that originates from outside the system, and acts on one or more particles within the system. The external force need not be due to a uniform field.

For any system with no external forces, the center of mass moves with constant velocity. This applies for all systems with classical internal forces, including magnetic fields, electric fields, chemical reactions, and so on. More formally, this is true for any internal forces that satisfy the weak form of Newton's Third Law.

The total momentum for any system of particles is given by

<math>\mathbf{p}=M\mathbf{v}_\mathrm{cm}<math>

Where M indicates the total mass, and v_cm is the velocity of the center of mass. This velocity can be computed by taking the time derivative of the position of the center of mass.

An analogue to the famous Newton's Second Law is

<math>\mathbf{F} = M\mathbf{a}_\mathrm{cm}<math>

Where F indicates the sum of all external forces on the system, and a_cm indicates the acceleration of the center of mass.

The angular momentum vector for a system is equal to the angular momentum of all the particles around the center of mass, plus the angular momentum of the center of mass, as if it were a single particle of mass <math>M<math>:

<math>\mathbf{L}_\mathrm{sys} = \mathbf{L}_\mathrm{cm} + \mathbf{L}_\mathrm{around\mbox{ }cm}<math>

Examples

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

<math>\frac{2\mbox{ m} \times 1\mbox{ kg} + 4\mbox{ m} \times 2\mbox{ kg}}{1\mbox{ kg}+2\mbox{ kg}} = 3.33\mbox{ m}<math>

• 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 400,000 km, mass 1/81. Center of mass is at:

<math>\frac{0 \times 1 + 400,000\mbox{ km} \times (1/81)}{1 + (1/81)} = 4,877\mbox{ km}<math>

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 1,000 km 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.

See also: Center of gravity, Pappus's centroid theorem