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. Physics (Greek Physis - φύσις in everyday terms is the Science of Matter and its motion. Mass is a fundamental concept in Physics, roughly corresponding to the Intuitive idea of how much Matter there is in an object 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 Physics, a rigid body is an idealization of a solid body of finite size in which Deformation is neglected 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 Classical physics, free space is a concept of Electromagnetic theory, corresponding to a theoretically "perfect" Vacuum, and sometimes 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 A shotgun (also known as a scattergun) is a Firearm that is usually designed to be fired from the shoulder which uses the energy of a fixed shell to fire a number Space is the extent within which Matter is physically extended and objects and Events have positions relative to one another 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 to design a sport car center of gravity as low as possible to make the car handle better. A sports car is a term used to describe a class of Automobile. 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 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. The high jump is an Athletics (track and field event in which competitors must jump over a horizontal bar placed at measured heights without the aid of any devices The Fosbury Flop is a style used in the athletics event of High jump. ^[1]

The so-called center of gravity frame (a less-preferred term for the center of momentum frame) is an inertial frame defined as the inertial frame in which the center of mass of a system is at rest. A center of momentum frame (or zero-momentum frame or COM frame of a system is any Inertial frame in which the Center of mass is at rest (has zero velocity A center of momentum frame (or zero-momentum frame or COM frame of a system is any Inertial frame in which the Center of mass is at rest (has zero velocity In Physics, an inertial frame of reference is a Frame of reference which belongs to a set of frames in which Physical laws hold in the same and simplest

Definition

The center of mass of a system of particles is defined as the average of their positions , weighted by their masses m_i:

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

If an object has uniform density then its center of mass is the same as the centroid of its shape. In Mathematics, an average, or central tendency of a Data set refers to a measure of the "middle" or " expected " value of 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 Mass is a fundamental concept in Physics, roughly corresponding to the Intuitive idea of how much Matter there is in an object Continuity may refer to In mathematics: Continuous probability distribution or random variable in probability and statistics For The density of a material is defined as its Mass per unit Volume: \rho = \frac{m}{V} Different materials usually have different In Geometry, the centroid or barycenter of an object X in n- Dimensional space is the intersection of all Hyperplanes

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. In Geometry, a median of a Triangle is a Line segment joining a vertex to the Midpoint of the opposing side
• 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. EARTH was a short-lived Japanese vocal trio which released 6 singles and 1 album between 2000 and 2001
• 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 of Syracuse ( Greek:) ( c. 287 BC – c 212 BC was a Greek mathematician, Physicist, Engineer 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. A torque (τ in Physics, also called a moment (of force is a pseudo- vector that measures the tendency of a force to rotate an object about 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. Latin ( lingua Latīna, laˈtiːna is an Italic language, historically spoken in Latium and Ancient Rome. For the Thoroughbred racehorse see Omar Khayyam (horse Ghiyās od-Dīn Abol-Fath Omār ibn Ebrāhīm Khayyām Neyshābūri (غیاث الدین Abd al-Rahman al-Khazini ( عبدالرحمن الخزيني) (flourished 1115–1130 was a Muslim scientist, physicist, astronomer, biologist ^[2]

Derivation of 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. Newton's laws of motion are three Physical laws which provide relationships between the Forces acting on a body and the motion of the Newton's laws of motion are three Physical laws which provide relationships between the Forces acting on a body and the motion of the

The total momentum for any system of particles is given by

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 Newton's Second Law is

Where F indicates the sum of all external forces on the system, and a_cm indicates the acceleration of the center of mass. Newton's laws of motion are three Physical laws which provide relationships between the Forces acting on a body and the motion of the

Letting the total internal force of the system.

where is the total mass of the system and is a vector yet to be defined, since:

and

then

We therefore have a vectorial definition for center of mass in terms of the total forces in the system. This is particularly useful for two-body systems.

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. Newton 's law of universal Gravitation is a physical law describing the gravitational attraction between bodies with mass 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. Potential energy can be thought of as Energy stored within a physical system
• 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 center of mass does not necessarily exhibit these convenient properties concerning gravity. A torque (τ in Physics, also called a moment (of force is a pseudo- vector that measures the tendency of a force to rotate an object about 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. Richard Phillips Feynman (ˈfaɪnmən May 11 1918 – February 15 1988 was an American Physicist known for the Path integral formulation of quantum The Feynman Lectures on Physics by Richard Feynman, Robert Leighton, and Matthew Sands is perhaps Feynman's most accessible technical work . . . 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. The tidal force is a secondary effect of the Force of Gravity and is responsible for the Tides It arises because the gravitational acceleration experienced A planet, as defined by the International Astronomical Union (IAU is a celestial body Orbiting a Star or stellar remnant that is In practice, for non-uniform fields, one simply does not speak of a "center of gravity".

CM frame

Main article: Center of mass 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 is a corollary of the Parallel Axis Theorem. A center of momentum frame (or zero-momentum frame or COM frame of a system is any Inertial frame in which the Center of mass is at rest (has zero velocity In Physics, the angular momentum of a particle about an origin is a vector quantity equal to the mass of the particle multiplied by the Cross product of the position In Physics, the parallel axis theorem can be used to determine the Moment of inertia of a Rigid body about any axis given the moment of inertia of the

Engineering

Aeronautical significance

Main article: Center of gravity of an aircraft

The center of mass is an important point on an aircraft, which significantly affects the stability of the aircraft. The center-of-gravity (CG is the point at which an aircraft would balance if it were possible to suspend it at that point 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. Elevators are control surfaces usually at the rear of an Aircraft, which control the aircraft's orientation by changing the pitch of the aircraft and so also For other uses see Stall. In Aerodynamics, a stall is a sudden reduction in the lift forces generated by an Airfoil The aircraft will be more maneuverable, but also less stable, and possibly so unstable that it is impossible to fly.

Barycenter in astronomy

The barycenter (or barycentre; from the Greek βαρύκεντρον) is the point between two objects where they balance each other. Greek (el ελληνική γλώσσα or simply el ελληνικά — "Hellenic" is an Indo-European language, spoken today by 15-22 million people mainly In other words, the center of gravity where two or more celestial bodies orbit each other. In Physics, an orbit is the gravitationally curved path of one object around a point or another body for example the gravitational orbit of a planet around a star 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. A natural satellite or moon is a Celestial body that Orbits a Planet or smaller body which is called the primary. A planet, as defined by the International Astronomical Union (IAU is a celestial body Orbiting a Star or stellar remnant that is A star is a massive luminous ball of plasma. The nearest star to Earth is the Sun, which is the source of most of the Energy on Earth 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. In Geometry, the foci (singular focus) are a pair of special points used in describing Conic sections The four types of conic sections are the Circle In Astrodynamics or Celestial mechanics an elliptic orbit is a Kepler orbit with the eccentricity greater than 0 and less than 1 This is an important concept in the fields of astronomy, astrophysics, and the like (see two-body problem). Astronomy (from the Greek words astron (ἄστρον "star" and nomos (νόμος "law" is the scientific study Astrophysics is the branch of Astronomy that deals with the Physics of the Universe, including the physical properties ( Luminosity, 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:

where:

a is the distance between the two bodies' centers;

m₁ and m₂ are the masses of the two bodies. Mass is a fundamental concept in Physics, roughly corresponding to the Intuitive idea of how much Matter there is in an object

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. In Geometry, the semi-major axis (also semimajor axis) is used to describe the dimensions of ellipses and hyperbolae 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. The Solar System consists of the Sun and those celestial objects bound to it by Gravity. Figures are given rounded to three significant figures. The significant figures (also called significant digits and abbreviated sig figs) of a number are those digits that carry meaning contributing to its accuracy 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.

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. A planet, as defined by the International Astronomical Union (IAU is a celestial body Orbiting a Star or stellar remnant that is A comet is a small Solar System body that orbits the Sun and when close enough to the Sun exhibits a visible coma (atmosphere or a tail — Asteroids, sometimes called Minor planets or planetoids', are bodies—primarily of the inner Solar System —that are smaller than planets but of the solar system (see n-body problem). The Solar System consists of the Sun and those celestial objects bound to it by Gravity. The n -body problem is the problem of finding given the initial positions masses and velocities of n bodies their subsequent motions as determined by 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. In Celestial mechanics, an apsis, plural apsides (ˈæpsɨdɪːz is the point of greatest or least distance of the Elliptical orbit of an object from In Astrodynamics, under standard assumptions, any Orbit must be of Conic section shape 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:

Note that the Sun-Jupiter system, with e_Jupiter = 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. (similar to the 90 Antiope system)	Two bodies with a difference in mass orbiting around a common barycenter, as in the Pluto-Charon system. 90 Antiope (ænˈtaɪəpi an-tye'-ə-pee) is an Asteroid discovered on October 1, 1866 by Robert Luther. Charon (ˈʃærən; also, as in Χάρων) discovered in 1978 is either the largest Moon of Pluto or the smaller member of a double	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)

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. EARTH was a short-lived Japanese vocal trio which released 6 singles and 1 album between 2000 and 2001 The Sun (Sol is the Star at the center of the Solar System. EARTH was a short-lived Japanese vocal trio which released 6 singles and 1 album between 2000 and 2001 In Astrodynamics or Celestial mechanics an elliptic orbit is a Kepler orbit with the eccentricity greater than 0 and less than 1 A binary star is a Star system consisting of two Stars orbiting around their Center of mass.

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. A plumb-bob or a plummet is a weight with a pointed tip on the bottom that is suspended from a string and used as a vertical reference line	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.

1. 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.
2. 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.
3. 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). The following diagrams depict a list of Centroids. A centroid of an object X in n- Dimensional space is the intersection of 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. The following diagrams depict a list of Centroids. A centroid of an object X in n- Dimensional space is the intersection of

From equation 1 above: 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 planimeter is a Measuring instrument used to measure the area of an arbitrary two-dimensional 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.

See also

• Weight distribution
• Center of percussion
• Center of pressure
• Metacentric height
• Roll center

Notes

References

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
• [1] Locating the center of mass by mechanical means