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Encyclopedia > Center of mass

In physics, the center of mass 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 center of mass 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 center of mass 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 center of mass is a point in space among them that may not correspond to the position of any individual mass. In the context of an entirely uniform gravitational field, the center of mass is often called the center of gravity- the point where gravity can be said to act. Physics (Greek: (phÃºsis), nature and (phusikÃ©), knowledge of nature) is the science concerned with the fundamental laws of the universe and their precise formulation in a mathematical framework. ... Unsolved problems in physics: What causes anything to have mass? The U.S. National Prototype Kilogram, which currently serves as the primary standard for measuring mass in the U.S. Mass is the property of a physical object that quantifies the amount of matter and energy it is equivalent to. ... In physics, a rigid body is an idealization of a solid body of finite size in which deformation is neglected. ... In physics, free space is a concept of electromagnetic theory, corresponding roughly to the vacuum, the baseline state of the electromagnetic field, or the replacement for the electromagnetic aether. ... Lead shot is small balls of lead, traditional made using a shot tower. ... A pump-action and two semi-automatic action Remington 1100 shotguns, 20 boxes of shotgun shells, a clay trap, and three boxes of clay pigeons. ... Space has been an interest for philosophers and scientists for much of human history. ...

The center of mass 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 center of mass 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 center of mass does not. A Honda NSX sports car A TVR Tuscan sports car A sports car is a car designed for sporting performance above utility. ... Car handling and vehicle handling is a description of the way wheeled vehicles perform transverse to their direction of motion, particularly during cornering and swerving. ... Gold medal winner Ethel Catherwood of Canada scissors over the bar at the 1928 Summer Olympics. ... Richard Douglas Dick Fosbury (born March 6, 1947) is an American athlete who revolutionised the high jump using a back-first technique, now known as the Fosbury flop. ...

The center of mass frame (also called the center of momentum frame) is an inertial frame defined as the frame in which the center of mass of a system is at rest. The center of mass frame (also called the center of momentum frame, CM frame, or COM frame) is defined as being the particular inertial frame in which the center of mass of a system of interest, is at rest (has zero velocity). ... The center of mass frame (also called the center of momentum frame, CM frame, or COM frame) is defined as being the particular inertial frame in which the center of mass of a system of interest, is at rest (has zero velocity). ... In physics, an inertial frame of reference, or inertial frame for short (also descibed as absolute frame of reference), is a frame of reference in which the observers move without the influence of any accelerating or decelerating force. ...

The center of mass $mathbf{R}$ of a system of particles is defined as the average of their positions $mathbf{r}_i$, weighted by their masses mi: In mathematics, an average or central tendency of a set (list) of data refers to a measure of the middle of the data set. ... A weight function is a mathematical device used when performing a sum, integral, or average in order to give some elements more of a weight than others. ... Unsolved problems in physics: What causes anything to have mass? The U.S. National Prototype Kilogram, which currently serves as the primary standard for measuring mass in the U.S. Mass is the property of a physical object that quantifies the amount of matter and energy it is equivalent to. ...

$mathbf{R} = frac 1M sum m_i mathbf{r}_i$

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

For a continuous distribution with mass density $rho(mathbf{r})$, the sum becomes an integral: In mathematics, a continuous function is one in which arbitrarily small changes in the input produce arbitrarily small changes in the output. ...

If an object has uniform density then its center of mass is the same as the centroid of its shape. In physics, density is defined as mass m per unit volume V. For the common case of a homogeneous substance, it is expressed as: where, in SI units: Ï (rho) is the density of the substance, measured in kgÂ·m-3 m is the mass of the substance, measured in kg... Centroid of a triangle In geometry, the centroid or barycenter of an object in -dimensional space is the intersection of all hyperplanes that divide into two parts of equal moment about the hyperplane. ...

## Examples

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

The triangle medians and the centroid. ... Adjectives: Terrestrial, Terran, Telluric, Tellurian, Earthly Atmosphere Surface pressure: 101. ...

## History

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. Archimedes (Greek: c. ... It has been suggested that this article or section be merged with Moment (physics). ...

## Locating the center of mass of an arbitrary 2D physical shape

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

 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.

Image File history File links Center_gravity_0. ... Image File history File links Center_gravity_1. ... Image File history File links Center_gravity_2. ... A plumb line is a reference line guided by a string or cord weighted at the end with a large weight known as a plumb bob. ...

## Locating the center of mass 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 center of mass in the x direction here. The same procedure may be followed to locate the center of mass in the y direction. External links http://www. ...

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

From the List of centroids, we note the coordinates of the individual centroids. Image File history File links COG_2. ... External links http://www. ...

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. Image File history File links COG_3. ...

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

## 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 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. Newtons First and Second laws, in Latin, from the original 1687 edition of the Principia Mathematica. ... Newtons laws of motion are the three scientific laws which Isaac Newton discovered concerning the behaviour of moving bodies. ...

The total momentum for any system of particles is given by

$mathbf{p}=Mmathbf{v}_mathrm{cm}$

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 Newtons First and Second laws, in Latin, from the original 1687 edition of the Principia Mathematica. ...

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

## Rotation and centers of gravity

Diagram of an educational toy that balances on a point: the CM (C) settles below its support (P). Any object whose CM is below the fulcrum will not topple.

The center of mass 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 center of mass R. This is seen in at least two ways: Image File history File links CoG_stable. ... Image File history File links CoG_stable. ... Look up Fulcrum in Wiktionary, the free dictionary Fulcrum may refer to one of the following. ... Isaac Newtons theory of universal gravitation states the following: Every single point mass attracts every other point mass by a force heading along the line combining the two. ...

• 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:
$Mmathbf{g}times mathbf{R}=sum_im_imathbf{g}times mathbf{r}_i.$

If the gravitational field acting on a body is not uniform, then the center of mass does not necessarily exhibit these convenient properties concerning gravity. As the situation is put in Feynman's influential textbook The Feynman Lectures on Physics: Potential energy is the energy that is by virtue of the relative positions (configurations) of the objects within a physical system. ... It has been suggested that this article or section be merged with Moment (physics). ... Richard Phillips Feynman (May 11, 1918 â€“ February 15, 1988; surname pronounced ) was an American physicist known for expanding the theory of quantum electrodynamics, the physics of the superfluidity of supercooled liquid helium, and particle theory. ... The Feynman Lectures on Physics, by Richard Feynman, is perhaps his most accessible technical work for anyone with an interest in physics and today is considered to be the classic introduction to modern physics, including lectures on mathematics, electromagnetism, Newtonian physics, quantum physics, and even the relation of physics to...

"The center of mass 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 center of mass. That is why one must distinguish between the center of mass 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". Comet Shoemaker-Levy 9 after breaking up under the influence of Jupiters tidal forces. ... The eight planets and three dwarf planets of the Solar System. ...

## CM frame

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 M: This gyroscope remains upright while spinning due to its angular momentum. ...

$mathbf{L}_mathrm{sys} = mathbf{L}_mathrm{cm} + mathbf{L}_mathrm{around,cm}$

This is a corollary of the Parallel Axis Theorem. The parallel axes rule can be used to determine the moment of inertia of a rigid object about any axis, given the moment of inertia of the object about the parallel axis through the objects center of mass and the perpendicular distance between the axes. ...

## Engineering

### Aeronautical significance

The center of mass 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 center of mass 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 center of mass 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. The center-of-gravity (CG) is the point at which an aircraft would balance if it were possible to suspend it at that point. ... An Airbus A380, currently the worlds largest passenger airliner An aircraft is any vehicle or craft capable of atmospheric flight. ... For other meanings of elevator see Elevator (disambiguation). ... In aerodynamics, a stall is a condition in which an excessive angle of attack causes loss of lift due to disruption of airflow. ...

## Barycenter

The barycenter (or barycentre; from the Greek βαρύκεντρον) 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, the moon does not orbit the exact center of the earth, instead orbiting a point outside the earth's center (but well below the surface of the Earth) 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). Centroid of a triangle In geometry, the centroid or barycenter of an object in -dimensional space is the intersection of all hyperplanes that divide into two parts of equal moment about the hyperplane. ... In physics, an orbit is the path that an object makes, around another object, whilst under the influence of a source of centripetal force, such as gravity. ... Moons of the Solar System scaled to Earths Moon A natural satellite is an object that orbits a planet or other body larger than itself and which is not man-made. ... The eight planets and three dwarf planets of the Solar System. ... This article is about the astronomical object. ... In geometry, the focus (pl. ... Two bodies with similar mass orbiting around a common barycenter with elliptic orbits. ... A giant Hubble mosaic of the Crab Nebula, a supernova remnant Astronomy is the science of celestial objects (such as stars, planets, comets, and galaxies) and phenomena that originate outside the Earths atmosphere (such as auroras and cosmic background radiation). ... Spiral Galaxy ESO 269-57 Astrophysics is the branch of astronomy that deals with the physics of the universe, including the physical properties (luminosity, density, temperature, and chemical composition) of celestial objects such as stars, galaxies, and the interstellar medium, as well as their interactions. ... In classical mechanics, the two-body problem is to determine the motion of two point particles that interact only with each other. ...

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

$r_1 = a cdot {m_2 over m_1 + m_2} = {a over 1 + m_1/m_2}$

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 discernible orbit. Unsolved problems in physics: What causes anything to have mass? The U.S. National Prototype Kilogram, which currently serves as the primary standard for measuring mass in the U.S. Mass is the property of a physical object that quantifies the amount of matter and energy it is equivalent to. ... The semi-major axis of an ellipse In geometry, the term semi-major axis (also semimajor axis) is used to describe the dimensions of ellipses and hyperbolae. ...

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. Major features of the Solar System (not to scale, from left to right): Pluto, Neptune, Uranus, Saturn, Jupiter, the asteroid belt, the Sun, Mercury, Venus, Earth & Moon, and Mars. ... This article or section is not written in the formal tone expected of an encyclopedia article. ...

Examples
Larger
body
m1
(mE=1)
Smaller
body
m2
(mE=1)
a
(km)
r1
(km)
R1
(km)
r1/R1 Remarks
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 barycenter just above its surface

If m1 >> m2 - which is true for the Sun and any planet - then the ratio r1/R1 approximates to: km redirects here. ... Adjectives: Terrestrial, Terran, Telluric, Tellurian, Earthly Atmosphere Surface pressure: 101. ... Apparent magnitude: up to -12. ... Atmospheric characteristics Atmospheric pressure 0. ... Charon (shair-É™n or kair-É™n (key), IPA , Greek Î§Î¬ÏÏ‰Î½) is the largest moon of Pluto, discovered in 1978. ... Pluto and Charon are sometimes informally considered to be a double (dwarf) planet. ... The eight planets and three dwarf planets of the Solar System. ... The Sun is the star at the center of the Solar System. ... The astronomical unit (AU or au or a. ... Adjectives: Jovian Atmosphere Surface pressure: 20â€“200 kPa[4] (cloud layer) Composition: ~86% H2 ~13% Helium 0. ...

${a over R_1} cdot {m_2 over m_1}$

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). Note: This article contains special characters. ... Eris (IPA pronunciation ) or ), also designated (136199) Eris or 136199 Eris (See Minor planet names), is the largest known dwarf planet in the solar system. ...

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 center of mass would lie about 500,000 km above the Sun's surface. The eight planets and three dwarf planets of the Solar System. ... Comet Hale-Bopp Comet McNaught as seen from Swifts Creek, Victoria, Australia on 23 January 2007 A comet is a small body in the solar system that orbits the Sun and (at least occasionally) exhibits a coma (or atmosphere) and/or a tail â€” both primarily from the effects of... 253 Mathilde, a C-type asteroid. ... Major features of the Solar System (not to scale, from left to right): Pluto, Neptune, Uranus, Saturn, Jupiter, the asteroid belt, the Sun, Mercury, Venus, Earth & Moon, and Mars. ... This article is about the many-body problem in quantum mechanics. ...

The calculations above are based on the mean distance between the bodies and yield the mean value r1. But all celestial orbits are elliptical, 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: A diagram of Keplerian orbital elements. ... In astrodynamics, under standard assumptions any orbit must be of conic section shape. ...

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

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

### Animations

Images are representative, not simulated.

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

Image File history File links Two bodies of similar mass orbiting around a common barycenter (red cross) with circular orbits. ... Image File history File links Two bodies of similar mass orbiting around a common barycenter (red cross) with circular orbits. ... Image File history File links Two bodies with a slight difference in mass orbiting around a common barycenter (red cross) with circular orbits. ... Image File history File links Two bodies with a slight difference in mass orbiting around a common barycenter (red cross) with circular orbits. ... Adjectives: Plutonian Atmosphere Surface pressure: 0. ... Charon (shair-É™n or kair-É™n (key), IPA , Greek Î§Î¬ÏÏ‰Î½), discovered in 1978, is, depending on the definition employed, either the largest moon of Pluto or one member of a double dwarf planet with Pluto being the other member. ... Image File history File links Two bodies with a major difference in mass orbiting around a common barycenter (red cross) with circular orbits. ... Image File history File links Two bodies with a major difference in mass orbiting around a common barycenter (red cross) with circular orbits. ... Adjectives: Terrestrial, Terran, Telluric, Tellurian, Earthly Atmosphere Surface pressure: 101. ... Apparent magnitude: up to -12. ... Image File history File links Two bodies with an extreme difference in mass orbiting around a common barycenter (red cross) with circular orbits. ... Image File history File links Two bodies with an extreme difference in mass orbiting around a common barycenter (red cross) with circular orbits. ... The Sun is the star at the center of the Solar System. ... Adjectives: Terrestrial, Terran, Telluric, Tellurian, Earthly Atmosphere Surface pressure: 101. ... Image File history File links Two bodies with similar mass orbiting around a common barycenter (red cross) with elliptic orbits. ... Image File history File links Two bodies with similar mass orbiting around a common barycenter (red cross) with elliptic orbits. ... Two bodies with similar mass orbiting around a common barycenter with elliptic orbits. ... This article is about the astronomical phenomenon. ...

Weight distribution refers to the apportioning of weight within a vehicle, but is used most often to refer to cars, airplanes, and watercraft. ... The center of percussion is the point on a bat, racquet, sword or other long thin object where a perpendicular impact will produce translational and rotational forces which perfectly cancel each other out at some given pivot point. ... The Center of Pressure (or CoP) is the point on a body where the sum of the total pressure acts. ... Ship Stability diagram, showing Center of Gravity (G), Center of Buoyancy (B), and Metacenter (M) with ship upright and heeled over to one side. ... The roll center of a vehicle is the notional point at which the cornering forces in the suspension are reacted to the vehicle body. ... Centroid of a triangle In geometry, the centroid or barycenter of an object in -dimensional space is the intersection of all hyperplanes that divide into two parts of equal moment about the hyperplane. ...

## References

• 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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