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, 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 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. [cite book |title=Space Tourism: Adventures in Earth Orbit and Beyond |first=Michael |last=Van Pelt |publisher=Springer |year=2005 |isbn=0387402136 |pages=185]

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.

Definition

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, $m\_i$::$mathbf\{R\}\; =\; \{\; sum\; m\_i\; mathbf\{r\}\_i\; over\; sum\; m\_i\; \}$

For a continuous distribution with mass density $ho(mathbf\{r\})$ and total mass $M$, the sum becomes an integral:: $mathbf\; R\; =frac\; 1M\; int\; mathbf\{r\}\; ;\; dm\; =\; frac\; 1M\; int\; ho(mathbf\{r\}),\; mathbf\{r\}\; dV\; =frac\{int\; ho(mathbf\{r\}),\; mathbf\{r\}\; dV\}\{int\; ho(mathbf\{r\})\; dV\}$

If an object has uniform density then its "not the 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. [Salah Zaimeche PhD (2005). [http://www.muslimheritage.com/uploads/Merv.pdf Merv] , Foundation for Science Technology and Civilization.]

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.

The total momentum for any system of particles is given by

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

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

: $mathbf\{F\}\; =\; Mmathbf\{a\}\_mathrm\{cm\}$

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

Letting the total internal force of the system.

$mathbf\{F\}=mathbf\{M\}ddotmathbf\{R\}$

where $mathbf\{M\}$ is the total mass of the system and $mathbf\{R\}$ is a vector yet to be defined, since:

$mathbf\{p\}\; =\; sum\{m\_jdot\{r\}\_j\}$

and

$mathbf\{F\}=dotmathbf\{p\}$

then

$mathbf\{R\}=frac\{1\}\{mathbf\{Msum\{m\_jr\_j\}$

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

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 "M"g acting at R:
*:$mathbf\{R\}\; imes\; Mmathbf\{g\}=sum\_im\_i\; mathbf\{r\}\_i\; imes\; mathbf\{g\}.$

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":fact
date=October 2008:"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."

Many authors have been 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. Symon, in his textbook "Mechanics," shows that the center of gravity of an extended body must always be defined relative to an external point, at which location resides a point mass that is exerting a gravitational force on the object in question. In fact, as Symon says::"For two extended bodies, no unique centers of gravity can in general be defined, even relative to each other, except in special cases, as when the bodies are far apart, or when one of them is a sphere....The general problem of determining the gravitational forces between bodies is usually best treated by means of the concepts of the field theory of gravitation..."

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". [Symon, K. R. (1971). "Mechanics", 3rd ed., Reading: Addison-Wesley.fact
date=October 2008]

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 $M$:

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

This is a corollary of the Parallel Axis Theorem.

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.

Barycenter in astronomy

The barycenter (or barycentre; from the Greek "βαρύκεντρον") is the point between two objects where they balance each other. In other words, it is 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, "r"₁, 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 distance between the two bodies' centers;:"m"₁ and "m"₂ are the masses of the two bodies.

"r"₁ is essentially the semi-major axis of the first body's orbit around the barycenter — and "r"₂ = "a" - "r"₁ 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 "R"₁, the radius of the first (more massive) body, and "r"₁/"R"₁, 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.

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.

, we note the coordinates of the individual centroids.

From equation 1 above:$frac\{3\; imes\; -pi2.5^2\; +\; 5\; imes\; 10^2\; +\; 13.33\; imes\; frac\{10^2\}\{2\{\; -pi2.5^2\; +\; 10^2\; +\; frac\{10^2\}\{2\; approx\; 8.5$ 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 (though a better term is probably moment planimeter), can be used to establish the position of the center of mass of an irregular shape. 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 [http://web.mat.bham.ac.uk/C.J.Sangwin/Publications/integrometer.pdf Locating the center of mass by mechanical means] .

ee also

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

Notes

References

*cite book | last = Feynman | first = Richard | coauthors = Robert Leighton, Matthew Sands | year = 1963 | title = The Feynman Lectures on Physics | publisher = Addison Wesley | id = ISBN 0-201-02116-1
*cite book | last = Goldstein | first = Herbert | coauthors = Charles Poole, John Safko | year = 2002 | title = Classical Mechanics | edition = 3e | publisher = Addison Wesley | id = ISBN 0-201-65702-3
*cite book | last = Kleppner | first = Daniel | coauthors = Robert Kolenkow | year = 1973 | title = An Introduction to Mechanics | edition = 2e | publisher = McGraw-Hill | id = ISBN 0-07-035048-5
*cite book | last = Marion | first = Jerry | coauthors = Stephen Thornton | year = 1995 | title = Classical Dynamics of Particles and Systems | edition = 4e | publisher = Harcourt | id = ISBN 0-03-097302-3
*cite book | last = Murray | first = Carl | coauthors = Stanley Dermott | year = 1999 | title = Solar System Dynamics | publisher = Cambridge UP | id = ISBN 0-521-57295-9
*cite book | author=Serway, Raymond A.; Jewett, John W. | title=Physics for Scientists and Engineers (6th ed.) | publisher=Brooks/Cole | year=2004 | id=ISBN 0-534-40842-7
*cite book |last= Symon |first= Keithe R. |title= Mechanics | edition = 3rd edition |publisher= Addison-Wesley |year= 1971 |isbn= 0-201-07392-7
*cite book | author=Tipler, Paul | title=Physics for Scientists and Engineers: Mechanics, Oscillations and Waves, Thermodynamics (5th ed.) | publisher=W. H. Freeman | year=2004 | id=ISBN 0-7167-0809-4

External links

* [http://www.kettering.edu/~drussell/Demos/COM/com-a.html 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.
* [http://www.britannica.com/eb/article-9037797/centre-of-gravity Center of Gravity] Encyclopaedia Britannica
* [http://niquette.com/puzzles/barycntp.htm barycenter fold] by Paul Niquette
* [http://www.space-electronics.com 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
* [http://orbitsimulator.com/gravity/articles/ssbarycenter.html The solar system's barycenter] Simulations showing the effect each planet contributes to the solar system's barycenter