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Barycenter is sometimes called a centre of gravity.


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Barycenter of a system of particles is a specific point at which, for many purposes, the system's mass behaves as if it were concentrated. The Barycenter is a function only of the positions and masses of the particles that comprise the system. In the case of a rigid body, the position of its Barycenter is fixed in relation to the object (but not necessarily in contact with it). In the case of a loose distribution of masses in free space, such as shot from a shotgun, the position of the Barycenter is a point in space among them that may not correspond to the position of any individual mass. In the context of a uniform gravitational field, the Barycenter is also known as a centre of mass.

Barycenter.
Barycenter: Moment planimeter.

The Barycenter of a body does not always coincide with its intuitive geometric center, and one can exploit this freedom. Engineers try hard to make a sport car as light as possible, and then add weight on the bottom; this way, the Barycenter is nearer to the street, and the car handles better. When high jumpers perform a "Fosbury Flop", they bend their body in such a way that it is possible for the jumper to clear the bar while his or her Barycenter does not.

The Barycenter frame (also called the center of momentum frame) is an inertial frame defined as the frame in which the Barycenter of a system is at rest.

Definition of Barycenter.

The Barycenter of a system of particles is defined as the average of their positions , weighted by their massesmi:

whereM is the total mass of the system, equal to the sum of the particle masses.

For a continuous distribution with mass density , the sum becomes an integral:

If an object has uniform density then its Barycenter is the same as the centroid of its shape.

Examples of a Barycenter.

  • The Barycenter of a two-particle system lies on the line connecting the particles (or, more precisely, their individual centers of mass). The Barycenter is closer to the more massive object; for details, see barycenter below.
  • The Barycenter of a ring is at the center of the ring (in the air).
  • The Barycenter of a solid triangle lies on all three medians and therefore at the centroid, which is also the average of the three vertices.
  • The Barycenter of a rectangle is at the intersection of the two diagonals.
  • In a spherically symmetric body, the Barycenter 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.
  • More generally, for any symmetry of a body, its Barycenter will be a fixed point of that symmetry.

History of Barycenter.

The concept of center of gravity was first introduced by the ancient Greek mathematician, physicist, and engineer Archimedes of Syracuse. Archimedes showed that the Torque exerted on a lever by weights resting at various points along the lever is the same as what it would be if all of the weights were moved to a single point - their center of gravity. In work on floating bodies he demonstrated that the orientation of a floating object is the one that makes its center of gravity as low as possible. He developed mathematical techniques for finding the centers of gravity of objects of uniform density of various well-defined shapes, in particular a triangle, a hemisphere, and a frustum of a circular paraboloid.

Locating the Barycenter of an arbitrary 2D physical shape

This method is useful when you wish to find the center of gravity of a complex planar object with unknown dimensions.

Barycenter.
Barycenter.
Barycenter.
Step 1: An arbitrary 2D shape. Step 2: Suspend the shape from a location near an edge. Drop a plumb line and mark on the object. Step 3: Suspend the shape from another location not too close to the first. Drop a plumb line again and mark. The intersection of the two lines is the center of gravity.

Locating the Barycenter of a composite shape

This method is useful when you wish to find the center of gravity of an object which is easily divided into elementary shapes, whose centers of mass are easy to find (see List of centroids). We will only be finding the Barycenter in the x direction here. The same procedure may be followed to locate the Barycenter in the y direction.

Image:COG_1.png The shape. It is easily divided into a square, triangle, and circle. Note that the circle will have negative area.

Image:COG_2.png From the List of centroids, we note the coordinates of the individual centroids.

Image:COG_3.png From equation 1 above: \frac{-3 \times \pi \times 2.5^2 + 5 \times 10^2 + 13.33 \times \frac{10^2}{2}}{ -\pi \times 2.5^2 + 10^2 + \frac{10^2}{2}} \approx 8.5 units.

The centre of mass of this figure is at a distance of 8.5 units from the left corner of the figure.

Barycenter motion.

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 Barycenter 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

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

An analogue to the famous Newton's second law is

Where F indicates the sum of all external forces on the system, and acm indicates the acceleration of the Barycenter.

Rotation and centers of gravity

fulcrum.
Diagram of an educational toy that balances on a point: the CM (C) settles below its support (P). Any object whose CM is always below the fulcrum will never topple.

The Barycenter is often called the center of gravity because any uniform gravitational field g acts on a system as if the mass M of the system were concentrated at the Barycenter R. This is seen in at least two ways:

  • The gravitational potential energy of a system is equal to the potential energy of a point particle having the same mass M located at R.
  • The gravitational Torque on a system equals the torque of a force Mg acting at R:
    .

If the gravitational field acting on a body is not uniform, then the Barycenter does not necessarily exhibit these convenient properties concerning gravity. As the situation is put in Feynman's influential textbook The Feynman Lectures on Physics:

"The Barycenter is sometimes called the center of gravity, for the reason that, in many cases, gravity may be considered uniform. ...In case the object is so large that the nonparallelism of the gravitational forces is significant, then the center where one must apply the balancing force is not simple to describe, and it departs slightly from the Barycenter. That is why one must distinguish between the Barycenter and the center of gravity."

Later authors are often less careful, stating that when gravity is not uniform, "the center of gravity" departs from the CM. This usage seems to imply a well-defined "center of gravity" concept for non-uniform fields, but there is no such thing. Even when considering tidal forces on planets, it is sufficient to use centers of mass to find the overall motion. In practice, for non-uniform fields, one simply does not speak of a "center of gravity".

Barycenter CM frame.

The angular momentum vector for a system is equal to the angular momentum of all the particles around the Barycenter, plus the angular momentum of the Barycenter, as if it were a single particle of massM:

This is a corollary of the Parallel Axis Theorem.

Barycenter in engineering: Aeronautical significance.

The Barycenter is an important point on an aircraft, which significantly affects the stability of the aircraft. To ensure the aircraft is safe to fly, it is critical that the center of gravity fall within specified limits. This range varies by aircraft, but as a rule of thumb it is centered about a point one quarter of the way from the wing leading edge to the wing trailing edge (the quarter chord point). If the Barycenter is ahead of the forward limit, the aircraft will be less maneuverable, possibly to the point of being unable to rotate for takeoff or flare for landing. If the Barycenter is behind the aft limit, the moment arm of the elevator is reduced, which makes it more difficult to recover from a stalled condition. The aircraft will be more maneuverable, but also less stable, and possibly so unstable that it is impossible to fly.

Barycenter.

The barycenter (or barycentre; from the Greek) is the point between two objects where they balance each other. In other words, the center of gravity where two or more celestial bodies orbit each other. When a moon orbits a planet, or a planet orbits a star, both bodies are actually orbiting around a point which lies outside the center of the greater body. For example, it's not the case that the moon orbits the exact center of the earth, but--like how the center of a see-saw would have to be moved closer to the larger of an adult or a child playing on it in order for them to balance each other--the moon orbits a point outside the earth's center where their respective masses balance each other. The barycenter is one of the foci of the elliptical orbit of each body. This is an important concept in the fields of astronomy, Astrophysics, and the like (see two-body problem).

In a simple two-body case, r1, the distance from the center of the first body to the barycenter is given by:

where:

a is the shortest distance between the two bodies;
m1 and m2 are the masses of the two bodies.

r1 is essentially the Semi-major axis of the first body's orbit around the barycenter - and r2 = a - r1 the semi-major axis of the second body's orbit. Where the barycenter is located within the more massive body, that body will appear to "wobble" rather than following a discernable orbit.

The following table sets out some examples from our Solar System. Figures are given rounded to three significant figures. The last two columns show R1, the radius of the first (more massive) body, and r1/R1, the ratio of the distance to the barycenter and that radius: a value less than one shows that the barycenter lies inside the first body.

Examples of Barycenter.
Larger
body
m1
(mE=1)
Smaller
body
m2
(mE=1)
a
(km)
r1
(km)
R1
(km)
r1/R1Remarks
Earth 1 Moon 0.0123 384,000 4,670 6,380 0.732 The Earth has a perceptible "wobble"
Pluto 0.0021 Charon 0.000,254
(0.121 mPluto)
19,600 2,110 1,150 1.83 Both bodies have distinct orbits around the barycenter, and as such Pluto and Charon were considered as a double planet by many before the redefinition of planet in August 2006.
Sun 333,000 Earth 1 150,000,000
(1 AU)
449 696,000 0.000,646 The Sun's wobble is barely perceptible
Sun 333,000 Jupiter 318 778,000,000
(5.20 AU)
742,000 696,000 1.07 The Sun orbits a barycentre just above its surface

If m1 >> m2 - which is true for the Sun and any planet - then the ratio r1/R1 approximates to:

Hence, the barycenter of the Sun-planet system will lie outside the Sun only if:

That is, where the planet is heavy and far from the Sun.

If Jupiter had Mercury's orbit (57,900,000 km, 0.387 AU), the Sun-Jupiter barycenter would be only 5,500 km from the center of the Sun (r1/R1 ~ 0.08). But even if the Earth had Eris' orbit (68 AU), the Sun-Earth barycenter would still be within the Sun (just over 30,000 km from the center).

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 (see many-body problem). If all the planets were aligned on the same side of the Sun, the combined Barycenter would lie about 500,000 km above the Sun's surface.

The calculations above are based on the mean distance between the bodies and yield the mean value r1. But all celestial orbits are eliptical, and the distance between the bodies varies between the apses, depending on the eccentricity, e. Hence, the position of the barycenter varies too, and it is possible in some systems for the barycenter to be sometimes inside and sometimes outside the more massive body. This occurs where:

{1 \over {1-e}} > {r_1 \over R_1} > {1 \over {1+e}}

Note that the Sun-Jupiter system, with eJupiter = 0.0484, just fails to qualify: 1.05 ? 1.07 > 0.954.

Barycenter animations.

Images are representative, not simulated.

common barycenter.
Two bodies of similar mass orbiting around a common barycenter.
Two bodies orbiting.
Two bodies with a difference in mass orbiting around a common barycenter, as in the Pluto-Charon system.
orbit common barycenter.
Two bodies with a major difference in mass orbiting around a common barycenter (similar to the Earth-Moon system).
mass orbiting around a common barycenter.
Two bodies with an extreme difference in mass orbiting around a common barycenter (similar to the Sun-Earth system).
elliptic orbits.
Two bodies with similar mass orbiting around a common barycenter with elliptic orbits (a common situation for Binary stars).

References to Barycenter.

  • Feynman, Richard, Robert Leighton, Matthew Sands (1963). The Feynman Lectures on Physics. Addison Wesley. ISBN 0-201-02116-1.
  • Goldstein, Herbert, Charles Poole, John Safko (2002). Classical Mechanics, 3e, Addison Wesley. ISBN 0-201-65702-3.
  • Kleppner, Daniel, Robert Kolenkow (1973). An Introduction to Mechanics, 2e, McGraw-Hill. ISBN 0-07-035048-5.
  • Marion, Jerry, Stephen Thornton (1995). Classical Dynamics of Particles and Systems, 4e, Harcourt. ISBN 0-03-097302-3.
  • Murray, Carl, Stanley Dermott (1999). Solar System Dynamics. Cambridge UP. ISBN 0-521-57295-9.
  • Serway, Raymond A.; Jewett, John W. (2004). Physics for Scientists and Engineers (6th ed.). Brooks/Cole. ISBN 0-534-40842-7.
  • Tipler, Paul (2004). Physics for Scientists and Engineers: Mechanics, Oscillations and Waves, Thermodynamics (5th ed.). W. H. Freeman. ISBN 0-7167-0809-4.



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