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In classical physics, the center of gravity of an extended body of total mass M is that point in space which, to an observer at P, is the apparent source of gravitational attraction equivalent to a point particle of mass M ^[1]^[2] .

In general the center of gravity of an extended body will depend on the reference location P and for this reason it is more precise to state the position as the center of gravity of the body relative to the point P. In contrast, the center of mass of an extended body is independent of the reference location from which it is determined.

Computation

Consider an extended body of total mass M comprised of a system of particles of masses m_i at vector points r_i. The net gravitational force on a reference particle of mass m at a point r due to the extended body is obtained from Newton's law of gravitation,

\mathbf {F(r)} =Gm\displaystyle \sum {\frac {m_{i}\mathbf {(r_{i}-r)} }{\mathbf {|r_{i}-r|^{3}} }}\qquad ;(continuous,\quad \mathbf {F(r)} =Gm\displaystyle \iiint {\frac {\mathbf {(r'-r)} \rho (r')dV'}{\mathbf {|r'-r|^{3}} }})

(1)

By above description the center of gravity of the extended body with respect to the reference body at r is that point r_cg for which,

{GmM{\frac {\mathbf {(r_{cg}-r)} }{\mathbf {|r_{cg}-r|^{3}} }}}=\mathbf {F(r)}

(2)

Let $\mathbf {|F|}$ be the scalar magnitude of the vector $\mathbf {F}$ , with $\mathbf {{\hat {F}}=F/|F|}$ a unit vector in the direction of $\mathbf {F}$ , then

\mathbf {r_{cg}-r} ={\sqrt {\frac {GmM}{|F|}}}\mathbf {\hat {F}}

(3)

Note that $\mathbf {|F|}$ is proportional to the quantity Gm so that, contrary to superficial appearance, the result is independent of either the reference mass or the gravitational constant.

References

^ Symon, Keith R. Mechanics, Addison-Wesley (1964) p. 258 (ISBN 60-5164).
^ “Centre of Gravity”Encyclopædia Britannica.

[1] Symon, Keith R. Mechanics, Addison-Wesley (1964) p. 258 (ISBN 60-5164).

[2] “Centre of Gravity”Encyclopædia Britannica.

[1]

[2]

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 {{about|the classical physics definition|the military term|Center of gravity (military)|aircraft|Center of gravity of an aircraft|ships|Ship's center of gravity}}
-{{Disputed|date=May 2011}}
-In [[physics]], many textbooks define a '''center of gravity''' to be a generalization on the [[center of mass]] of a body. In sufficiently symmetric configurations of bodies, gravitational effects such as [[potential energy]], [[force]], or [[torque]] can be calculated by applying [[Newton's law of universal gravitation]], and pretending that the mass of each gravitating body is concentrated at its center of mass. The center of gravity seeks to generalize such calculations to less symmetric configurations. Different authors generalize the concept in different ways.
+In classical physics, the '''center of gravity''' of an extended body of total mass M is that point in space which, to an observer at P,  is the apparent source of gravitational attraction equivalent to a point particle of mass M <ref> Symon, Keith R. ''Mechanics'', Addison-Wesley (1964) p. 258  ('''ISBN 60-5164''').</ref><ref>[http://www.britannica.com/EBchecked/topic/242556/centre-of-gravity  “Centre of Gravity”]Encyclopædia Britannica.</ref> .
-==Body in an external field==
-===Torque===
-''[[The Feynman Lectures on Physics]]'' {{Harv|Feynman|Leighton|Sands|1963}} characterizes the center of gravity of a body in a non-uniform field as the point at which one would need to balance it against the force of gravity so that there is no net torque on the body.<ref>{{Harvnb|Feynman|Leighton|Sands|1963|p=19-3}}: "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."</ref> This approach is also taken by   {{Harvtxt|Tipler|Mosca|2004}},<ref>{{Harvnb|Tipler|Mosca|2004|pp=371–372}}: "Conveniently, the net torque due to gravity about any point can be calculated as if the entire weight <math>\vec W</math> were applied at a single point, the '''center of gravity'''..."</ref> {{Harvtxt|Pollard|Fletcher|2005}},<ref>{{Harvnb|Pollard|Fletcher|2005}}: "It is convenient from a computational point of view to replace the action of all the body forces, {{math|'''w'''<sub>''i''</sub>}}, with the resultant force acting at a ''center of gravity''. The center of gravity and the center of mass are coincident if the gravity field over the body is uniform in magnitude."</ref> {{Harvtxt|Rosen|Gothard|2009}},<ref>{{Harvnb|Rosen|Gothard|2009|pp=75–76}}: "The center of gravity of a body is the point at which the total weight of the body, the sum of the forces of gravity on all the components of the body, can be thought to be acting. Thus, since the line of action of a body's weight force passes through the center of gravity, gravity exerts no net torque on a body with respect to the body's center of gravity."</ref> and {{Harvtxt|Pytel|Kiusalaas|2010}}.<ref>{{Harvnb|Pytel|Kiusalaas|2010|pp=442–443}}: "The resultant of the gravity forces acting on a body, which we know as the ''weight'' of the body, acts through a point called the ''center of gravity'' of that body."</ref>
+In general the center of gravity of an extended body will depend on the reference location P and for this reason it is more precise to state the position as the ''center of gravity of the body relative to the point P''.  In contrast, the [[Center of mass|center of mass]] of an extended body is independent of the reference location from which it is determined.
-{{Harvtxt|Millikan|1902}} defines the center of gravity of a body to be the unique point in the body, if it exists, that satisfies the following stronger requirement: There is no torque about the point for any positioning of the body in the field of force in which it is placed. Millikan observes that this center of gravity exists only when the force is uniform, in which case it coincides with the center of mass.<ref>{{Harvnb|Millikan|1902|pp=34–35}}: "Now the ''center of gravity'' of a body is defined as that point which is so situated that, whatever the position of the body, a plane which passes through this point and is parallel to the direction of the force always satisfies the condition {{math|Σ''fr'' = 0.}} It is evident that if a body has any center of gravity at all this point must always coincide with its center of mass; and yet a body possesses a center of gravity only when it lies in a ''uniform'' field of force."</ref> This approach dates back to [[Archimedes]].{{sfn|Shirley|Fairbridge|1997|p=92}}
+== Computation ==
-===Potential energy===
-{{Harvtxt|Goldstein|Poole|Safko|2002}} describe the center of gravity as a point where the potential energy of the body may be concentrated.<ref>{{Harvnb|Goldstein|Poole|Safko|2002|p=185}}: "...the potential energy in a uniform gravitational field will depend only upon the Cartesian vertical coordinate of the center of gravity.* *The center of gravity of course coincides with the center of mass in a uniform gravitational field."</ref>
+Consider an extended body of total mass ''M'' comprised of a system of particles of masses ''m''<sub>i</sub> at vector points '''r'''<sub>i</sub>. The net gravitational force on a reference particle of mass m at a point '''r''' due to the extended body is obtained from [[Newton's law of universal gravitation|Newton's law of gravitation]],
-===Weighted average===
-{{Harvtxt|Asimov|1988}} writes that a body in the earth's gravitational field has a center of gravity that is lower than its center of mass, because its lower portion is more strongly influenced by the earth's gravity.<ref>{{Harvnb|Asimov|1988|p=77}}: "Suppose next that a body is falling toward the earth. Every particle of the body is being pulled by the force of gravity, but the body behaves as if all that force were concentraed at one point within the body; that point is the ''center of gravity''. If the body were in a uniform gravitational field, the center of gravity would be identical with the center of mass. However, the lower portion of a body is somewhat closer to the center of the earth than is the upper, and the lower portion is therefore more strongly under gravitational influence. The center of gravity is consequently very slightly below the center of mass..."</ref> {{Harvtxt|Frautschi|Olenick|Apostol|Goodstein|1986}} gives a similar treatment, for which the center of gravity of the moon is closer to the earth than its center of mass.<ref>{{Harvnb|Frautschi|Olenick|Apostol|Goodstein|1986|p=269}}: "It is important to realize that center of mass is a different concept from center of gravity. The center of gravity of an extended body is the point at which the resultant of gravitational forces on the body acts. ...for a body sufficiently large that the nonuniformity of the gravitational acceleration must be taken into account, {{math|'''r'''<sub>CG</sub>}} is shifted from <math>\bar \mathbf r</math> toward the side of the body where gravity is strongest. For example, the moon's center of gravity lies somewhat closer to the earth than the moon's center of mass, because the gravitational force {{math|''GM''<sub>E</sub>''m''<sub>''i''</sub>/''r''<sup>2</sup>}} on a particle of mass {{math|''m''<sub>''i''</sub>}} is stronger nearer the earth."</ref>
-{{Harvtxt|Beatty|2006|p=48}} defines this notion more precisely: the center of gravity of the body is a certain weighted average of the locations of its particles. Instead of integrating over the mass density, which defines the center of mass, he integrates over the weight density, in which the mass of each particle is multiplied by the (scalar) gravitational field strength. The same definition is found in {{Harvtxt|Jong|Rogers|1995}}.{{sfn|Jong|Rogers|1995|pp=212–213}} Similarly, {{Harvtxt|Serway|Jewett|2006}} states that "The center of gravity is the average position of the gravitational forces on all parts of the object"{{sfn|Serway|Jewett|2006|p=244}} and {{Harvtxt|De Silva|2005}} contains the definition "Center of gravity: The point representing the average position of weight in a body."{{sfn|De Silva|2005|p=G-9}}
+{{NumBlk|::|<math>\mathbf{F(r)} =  G m \displaystyle \sum { \frac {m_i \mathbf{(r_i - r)}} {\mathbf{|r_i - r|^3}} } \qquad ; (continuous , \quad \mathbf{F(r)} =  G m  \displaystyle  \iiint { \frac { \mathbf{(r' - r)} \rho (r') dV'} {\mathbf{|r' - r|^3}} }) </math>|{{EquationRef|1}}}}
-==Field generated by a body==
-===Force===
-{{Harvtxt|Symon|1964}} defines the center of gravity of an extended body of total mass {{mvar|M}} as that point in space which, to an observer at a point {{mvar|P}}, is the apparent source of gravitational attraction equivalent to a point particle of mass {{mvar|M}}.<ref>{{Harvnb|Symon|1964|p=258}}: "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..."</ref> Consider an extended body of total mass {{mvar|M}} comprised of a system of particles of masses {{math|''m''<sub>''i''</sub>}} at vector points {{math|'''r'''<sub>''i''</sub>}}. The center of gravity of the body relative to the point {{math|'''r'''}} is a point that satisfies the equation:
-:<math>\frac {M \mathbf{(r_{cg} - r)}} {\mathbf{|r_{cg} - r|^3}} = \sum_i \frac {m_i \mathbf{(r_i - r)}} {\mathbf{|r_i - r|^3}}.</math>
+By above description  the center of gravity of the extended body with respect to the reference body at '''r''' is that point '''r'''<sub>cg</sub> for which,
-If the right-hand side of the equation is nonzero, then {{math|'''r'''}} exists and is uniquely determined. The same definition is given by {{Harvtxt|Hamill|2009}}.<ref>{{Harvnb|Hamill|2009|pp=494–496}}: "The center of gravity of an extended body of mass {{mvar|M}} is defined in terms of its interaction with a particle (say {{mvar|m}}). ...Suppose the extended body shrinks to a point. Where should this point mass be located to produce the same force (and torque) as the extended body? The location of this hypothetical point mass is called the center of gravity of {{mvar|M}} relative to {{mvar|m}}."</ref>
-==Notes==
-{{Reflist}}
+{{NumBlk|::|<math> { G m M \frac { \mathbf{(r_{cg} - r)}} {\mathbf{|r_{cg} - r|^3}} } = \mathbf{F(r)}  </math>|{{EquationRef|2}}}}
-==References==
-*{{Citation |last=Asimov |first=Isaac |authorlink=Isaac Asimov |year=1988 |origyear=1966 |title=[[Understanding Physics]] |publisher=Barnes & Noble Books |isbn=0-88029-251-2}}
-*{{Citation |last=Beatty |first=Millard F. |year=2006 |title=Principles of Engineering Mechanics, Volume 2: Dynamics—The Analysis of Motion |publisher=Springer |series=Mathematical Concepts and Methods in Science and Engineering |volume=33 |isbn=0-387-23704-6}}
+Let  <math>\mathbf{|F|}</math>  be the scalar magnitude of the vector <math>\mathbf{F}</math>, with  <math>\mathbf{\hat F = F / |F|}</math>  a unit vector in the direction of <math>\mathbf{F}</math>, then
-*{{Citation |last=Feynman |first=Richard |authorlink=Richard Feynman |last2=Leighton |first2=Robert B. |author2-link=Robert B. Leighton (physicist) |last3=Sands |first3=Matthew |author3-link=Matthew Sands |year=1963 |title=[[The Feynman Lectures on Physics]] |volume=1 |edition=Sixth printing, February 1977 |publisher=Addison-Wesley |isbn=0-201-02010-6-H}}
-*{{Citation |last=Frautschi |first=Steven C. |authorlink=Steven Frautschi |last2=Olenick |first2=Richard P. |last3=Apostol |first3=Tom M. |author3-link=Tom M. Apostol |last4=Goodstein |first4=David L. |author4-link=David Goodstein |year=1986 |title=The Mechanical Universe: Mechanics and heat, advanced edition |publisher=Cambridge University Press |isbn=0-521-30432-6}}
-*{{Citation |last=Goldstein |first=Herbert |authorlink=Herbert Goldstein |last2=Poole |first2=Charles |last3=Safko |first3=John |year=2002 |title=[[Classical Mechanics (book)|Classical Mechanics]] |edition=3rd |publisher=Addison-Wesley |isbn=0-201-65702-3}}
+{{NumBlk|::|<math> \mathbf{r_{cg} - r} = \sqrt {\frac {GmM} {|F|}} \mathbf{\hat {F}}</math>|{{EquationRef|3}}}}
-*{{Citation |last=Hamill |first=Patrick |year=2009 |title=Intermediate Dynamics |publisher=Jones & Bartlett Learning |isbn=978-0-7637-5728-1}}
-*{{Citation |last=Jong |first=I. G. |last2=Rogers |first2=B. G. |year=1995 |title=Engineering Mechanics: Statics |publisher=Saunders College Publishing |isbn=0-03-026309-3}}
+Note that <math>\mathbf{|F|}</math> is proportional to the quantity ''Gm'' so that, contrary to superficial appearance, the result is independent of either the reference mass or the gravitational constant.
-*{{Citation |last=Millikan |first=Robert Andrews |authorlink=Robert Andrews Millikan |year=1902 |title=Mechanics, molecular physics and heat: a twelve weeks' college course |publisher=Scott, Foresman and Company |location=Chicago |url=http://books.google.com/books?id=X0tBAAAAYAAJ |accessdate=25 May 2011}}
-*{{Citation |last=Pollard |first=David D. |last2=Fletcher |first2=Raymond C. |year=2005 |title=Fundamentals of structural geology |publisher=Cambridge University Press |isbn=0-521-83927-3}}
+== See also ==
-*{{Citation |last=Pytel |first=Andrew |last2=Kiusalaas |first2=Jaan |year=2010 |title=Engineering Mechanics: Statics |volume=1 |edition=3rd |publisher=Cengage Learning |isbn=978-0-495-29559-4}}
+* [[Center of Mass]]
-*{{Citation |last=Rosen |first=Joe |last2=Gothard |first2=Lisa Quinn |year=2009 |title=Encyclopedia of Physical Science |publisher=Infobase Publishing |isbn=978-0-8160-7011-4}}
+* [[Newton's law of universal gravitation]]
-*{{Citation |last=Serway |first=Raymond A. |last2=Jewett |first2=John W. |year=2006 |title=Principles of physics: a calculus-based text |volume=1 |edition=4th |publisher=Thomson Learning |isbn=0-534-49143-X}}
-*{{Citation |last=Shirley |first=James H. |last2=Fairbridge |first2=Rhodes Whitmore |year=1997 |title=Encyclopedia of planetary sciences |publisher=Springer |isbn=0-412-06951-2}}
+== References ==
-*{{Citation |last=De Silva |first=Clarence W. |year=2002 |title=Vibration and shock handbook |publisher=CRC Press |isbn=978-0-8493-1580-0}}
+{{reflist}}
-*{{Citation |last=Symon |first=Keith R. |year=1964 |title=Mechanics |publisher=Addison-Wesley |isbn=60-5164}}
-*{{Citation |last=Tipler |first=Paul A. |last2=Mosca |first2=Gene |year=2004 |title=Physics for Scientists and Engineers |volume=1A |edition=5th |publisher=W. H. Freeman and Company |isbn=0-7167-0900-7}}
 [[Category:Classical mechanics]]

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