Center Of Mass

Information about Center Of Mass

This article is about Center of gravity. For the military concept, see Center of gravity (military). For the aviation concept, see Center of gravity (aircraft).

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, say, 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.

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 design a sport car center of gravity as low as possible to make the car handle 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.^[1]

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.

Definition

The center of mass $\text{[math]}$ of a system of particles is defined as the average of their positions $\text{[math]}$ , weighted by their masses $\text{[math]}$ :

$\text{[math]}$

where $\text{[math]}$ is the total mass of the system, equal to the sum of the particle masses.

For a continuous distribution with mass density $\text{[math]}$ , the sum becomes an integral:

$\text{[math]}$

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

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.

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.

In the Middle Ages, theories on the center of gravity were further developed by Abū Rayhān al-Bīrūnī, al-Razi (Latinized as Rhazes), Omar Khayyám, and al-Khazini.^[2]

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.

The total momentum for any system of particles is given by

$\text{[math]}$

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

$\text{[math]}$

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

Rotation and centers of gravity

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:

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:
: $\text{[math]}$

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:

"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".

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 $\text{[math]}$ :

$\text{[math]}$

This is a corollary of the Parallel Axis Theorem.

Engineering

Aeronautical significance

Main article: Center of gravity (aircraft)

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.

Barycenter in astronomy

For barycenters in geometry, see .

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 that 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).

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

$\text{[math]}$

where:

a is the distance between the two bodies' centers;

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.

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**
Larger body	m₁ (m_E=1)	Smaller body	m₂ (m_E=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 m_Pluto)	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:

$\text{[math]}$

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

$\text{[math]}$

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 n-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 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:

$\text{[math]}$

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

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.

Locating center of mass

This is a method of determining the center of mass of an L-shaped object.

Divide the shape into two rectangles, as shown in fig 2. Find the center of masses of these two rectangles by drawing the diagonals. Draw a line joining the center of masses. The center of mass of the shape must lie on this line AB.
Divide the shape into two other rectangles, as shown in fig 3. Find the center of masses of these two rectangles by drawing the diagonals. Draw a line joining the center of masses. The center of mass of the L-shape must lie on this line CD.
As the center of mass of the shape must lie along AB and also along CD, it is obvious that it is at the intersection of these two lines, at O. The point O might not lie inside the L-shaped object.

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 that 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.

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

From the List of centroids, we note the coordinates of the individual centroids.

From equation 1 above: $\text{[math]}$ units.

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

Locating the center of mass by tracing around the perimeter of the shape

A direct development of the Planimeter known as an integraph, or integerometer, can be used to establish the position of the center of mass of an irregular shape. A better term is probably moment planimeter. This method can be applied to a shape with an irregular, smooth or complex boundary where other methods are too difficult. It was regularly used by ship builders to ensure the ship would not capsize. See Locating the center of mass by mechanical means.

Notes

1. ^ Van Pelt, Michael (2005). Space Tourism: Adventures in Earth Orbit and Beyond. Springer, 185. ISBN 0387402136.
2. ^ Salah Zaimeche PhD (2005). Merv, Foundation for Science Technology and Civilization.

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.

External links

Motion of the Center of Mass shows that the motion of the center of mass of an object in free fall is the same as the motion of a point object.
Center of Gravity Encyclopaedia Britannica
barycenter fold by Paul Niquette
Measuring Center of Gravity Space Electronics, manufacturer of center of gravity measurement instruments
http://web.mat.bham.ac.uk/C.J.Sangwin/Publications/integrometer.pdf Locating the center of mass by mechanical means

The center of gravity (CoG) is a concept developed by Carl von Clausewitz, a Prussian military theorist, in his work On War.

The definition of CoG according to the United States Department of Defense (DoD): "those characteristics, capabilities, or locations from
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The center-of-gravity (CG) is the point at which an aircraft would balance if it were possible to suspend it at that point. It is the mass center of the aircraft, or the theoretical point at which the entire weight of the aircraft is assumed to be concentrated.
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Physics is the science of matter^[1] and its motion^[2]^[3], as well as space and time^[4]^[5] —the science that deals with concepts such as force, energy, mass, and charge.
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Mass is a fundamental concept in physics, roughly corresponding to the intuitive idea of "how much matter there is in an object". Mass is a central concept of classical mechanics and related subjects, and there are several definitions of mass within the framework of relativistic
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rigid body is an idealization of a solid body of finite size in which deformation is neglected. In other words, the distance between any two given points of a rigid body remains constant in time regardless of external forces exerted on it.
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In physics, free space is a concept of electromagnetic theory, corresponding to a theoretical "perfect vacuum".

Definition

Free space simply means that there is no material or other physical phenomenon
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Lead shot is a collective term for small balls of lead. It is used primarily as projectiles in shotguns, but is also used for a variety of other purposes. It was traditionally made using a shot tower.
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shotgun (also known as a fowling piece^[1] or scattergun^[2]) is a firearm typically used to fire a number of small spherical pellets called shot.

Characteristics

Shotguns come in a wide variety of forms, from rimfire models with .
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The term SPACE (capitalized) can refer to:

, a Canadian science-fiction channel
The Society for Promotion of Alternative Computing and Employment
DSPACE, a term in computational complexity theory

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sports car is an automobile designed for performance driving. Most sports cars are rear-wheel drive, have two seats, two doors, and are designed for precise handling, acceleration, and aesthetics.
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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. It also includes their stability when moving in a straight line.
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high jump is an athletics (track and field) event in which competitors must jump over a horizontal bar placed at measured heights without aid of any devices. It has been contested since the Olympic Games of ancient Greece.
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Fosbury Flop is a technique in the low high jump that contrasts with the Eastern Roll and was first used by Dick Fosbury, whose gold medal in the 1968 Summer Olympics made it the dominant technique of the event as it remains today.
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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).
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An inertial frame of reference, or inertial reference frame, is one in which Newton's first and second laws of motion are valid. Newton's laws are valid in any reference frame that is neither rotating nor accelerating relative to the sun and other stars.
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In mathematics, an average, or central tendency of a data set refers to a measure of the "middle" or "expected" value of the data set. There are many different descriptive statistics that can be chosen as a measurement of the central tendency of the data items.
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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. They occur frequently in statistics and analysis, and are closely related to the concept of a measure.
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Continuity may mean:

In mathematics:

Parametric continuity
Geometric continuity
Continuous function with real or complex values
Continuous probability distribution or random variable in probability and statistics
Continuity theorem

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In physics, density is mass m per unit volume V—how heavy something is compared to its size. A small, heavy object, such as a rock or a lump of lead, is denser than a lighter object of the same size or a larger object of the same weight, such as pieces of
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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. Informally, it is the "average" of all points of .
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median of a triangle is a line joining a vertex to the midpoint of the opposite side. It divides the triangle into two parts of equal area. The three medians intersect in the triangle's centroid or center of mass, and two-thirds of the length of each median is between the vertex
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EARTH was a short-lived Japanese vocal trio which released 6 singles and 1 album between 2000 and 2001. Their greatest hit, their debut single "time after time", peaked at #13 in the Oricon singles chart.
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Archimedes of Syracuse (Greek: Άρχιμήδης c. 287 BC – c. 212 BC) was an ancient Greek mathematician, physicist and engineer.
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torque (or often called a moment) can informally be thought of as "rotational force" or "angular force" which causes a change in rotational motion. This force is defined by linear force multiplied by a radius.

The SI unit for torque is the newton meter (N m). In U.S.
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lever (from French lever, "to raise", c.f. a levant) is a rigid object that is used with an appropriate fulcrum or pivot point to multiply the mechanical force that can be applied to another object.
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Middle Ages form the middle period in a traditional schematic division of European history into three "ages": the classical civilization of Antiquity, the Middle Ages and Modern Times.
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Persian scholar
Medieval era

Name: Al-Razi
Birth: 865
Death: 925
School/tradition:
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Latin}}}
Official status
Official language of: Vatican City
Used for official purposes, but not spoken in everyday speech
Regulated by: Opus Fundatum Latinitas
Roman Catholic Church
Language codes
ISO 639-1: la
ISO 639-2: lat
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Center Of Mass

Information about Center Of Mass

Definition

Examples

History

Motion

Rotation and centers of gravity

CM frame

Engineering

Aeronautical significance

Barycenter in astronomy

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

Locating center of mass

Locating the center of mass of a composite shape

Locating the center of mass by tracing around the perimeter of the shape

See also

Notes

References

External links

Definition

Characteristics