# Center of mass

The center of mass of an object is a point at which the object's mass can be assumed, for many purposes, to be concentrated.

## Example

For example, an object can balance on a point only if its center of mass is directly above the point. Alternatively, if you hang an object from a string, the object's center of mass will be directly below the string.

## Comparison with center of gravity

In physics, the center of gravity (CoG) of an object is the average location of its weight. In a uniform gravitational field it coincides with the object's center of mass, but in a nonuniform gravitational field it can be located elsewhere; see space elevators for one example of a case where the two points can be drastically different. The path of an object in orbit depends only on its center of gravity. Most astronomical objects are radially symmetric, causing both the center of gravity and the center of mass to coincide at the center of the sphere.

## Definition

The center of mass is defined as the weighted average of position, with in the discrete case the masses as weights, and in the continous case the density function as the weight function.

Thus the center of mass of an object is the position vector given by: $\displaystyle \frac{\int x \; dm}{m}$ .

## More formulas

In the discrete case:

$\displaystyle \mathbf{s}_{\hbox{CM}} = {1 \over m_{\hbox{total}}} \sum_{i = 0}^{n} m_i \mathbf{s}_i$

where $\displaystyle n$ is the number of mass particles.

Or in the continuous case:

$\displaystyle \mathbf{s}_{\hbox{CM}} = {1 \over m_{\hbox{total}}} \int \rho(\mathbf{s}) dV$

where ρ(s) is the scalar mass density as a function of the position vector.

If an object has uniform density then the center of mass is the same thing as the centroid.

## Aeronautical significance

The center of mass is an important point on an aircraft, as it defines the amount of mass forward or behind the center of gravity that needs to be moved in order to pitch the plane up or down without applying any external forces.

In conventional designs the center of mass is often located very near the line 1/3rd back from the front of the wing. That is the line where most wings generate their lift, known as the center of pressure, so by balancing the plane at that point, the lift and weight balance out with no net torque. The center of mass is sometimes moved slightly to the rear of this line in order to provide the plane with a natural "nose up" tendency when lift increases (like when applying more power).

If the balance of the plane is moved too far from the center of mass the control surfaces may have trouble controlling the plane. The actual force generated by the surfaces is typically quite small (a few pounds) but due to their location at the end of the tail (typically) they generate considerable torque to pitch the plane. If the center of mass starts to move away from the center of pressure there will be an increasing amount of constant torque they have to counteract, and if it moves too far, it may be more than the controls can counter.

## 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

$\displaystyle \mathbf{p}=M\mathbf{v}_\mathrm{cm}$

Where M indicates the total mass, and vcm 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

$\displaystyle \mathbf{F} = M\mathbf{a}_\mathrm{cm}$

Where F indicates the sum of all external forces on the system, and acm 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 $\displaystyle M$ :

$\displaystyle \mathbf{L}_\mathrm{sys} = \mathbf{L}_\mathrm{cm} + \mathbf{L}_\mathrm{around\,cm}$

## 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:
$\displaystyle \frac{2\mbox{ m} \times 1\mbox{ kg} + 4\mbox{ m} \times 2\mbox{ kg}}{1\mbox{ kg}+2\mbox{ kg}} = 3.33\mbox{ m}$
• 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.
• 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.

## Barycenter

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, see two-body problem.

The barycenter (from the Greek βαρύκεντρον) is the center of mass of two or more bodies which are orbiting each other, and is the point around which both of them orbit. It is an important concept in the fields of astronomy, astrophysics, and the like.

In the case where one of the two objects is much larger and more massive than the other, the barycenter will be located within the larger object. Rather than appearing to orbit it will simply be seen to "wobble" slightly. This is the case for the Moon and Earth, where the barycenter is located on average 4,671 km from Earth's center, well within the planet's radius of 6,378 km. When the two bodies are of similar masses (or at least the mass ratio is less extreme), however, the barycenter will be located outside of either of them and both bodies will follow an orbit around it. This is the case for Pluto and Charon, Jupiter and the Sun, and many binary asteroids and binary stars.

The distance from the center of a body (thought of as a point-mass) to the barycenter in a simple two-body case can be calculated as follows:

$\displaystyle r_1 = r_{\rm tot} {m_2 \over m_1 + m_2}$

where :

r1 is the distance from body 1 to the barycenter
rtot is the distance between the two bodies
m1 and m2 are the masses of the two bodies.

Some examples:

• Earth-Moon system: the Moon's mass is 0.0123 that 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). The Moon is at an average distance of 384400 km from the Earth. Then the center of mass is at:
$\displaystyle \frac{0 \times 1 + 384400\mbox{ km} \times 0.0123}{1 + 0.0123} = 4671\mbox{ km}$
from the Earth's center. Thus, as opposed to the Earth standing "still" and the Moon moving, both of them move around a point about 1700 km below the Earth's 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 at the center of the Sun.
• 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, 96,000 km outside its surface. As Jupiter does its 11 year orbit, the Sun does a 1.5 million km orbit around the center of mass.

Note that the distance from the Sun's center to the center of mass of a two-body system consisting of the Sun and another celestial body, hence the size of the Sun's orbit around this center of mass, is approximately proportional to the product of the mass of that other body, and the distance between the two, even though gravity decreases with distance. That orbit is largest with Jupiter, its large mass more than compensates its smaller distance to the Sun than several other planets. 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.

### Animations

Images are representative, not simulated.

File:Orbit1.gif
Two bodies of similar mass orbiting around a common barycenter.
File:Orbit2.gif
Two bodies with a slight difference in mass orbiting around a common barycenter.
File:Orbit3.gif
Two bodies with a major difference in mass orbiting around a common barycenter.
File:Orbit4.gif
Two bodies with an extreme difference in mass orbiting around a common barycenter.
File:Orbit5.gif
Two bodies with similar mass orbiting around a common barycenter with elliptic orbits.