Center of mass

This toy uses the principles of center of mass to keep balance when sitting on a finger.

Inphysics,thecenter of massof a distribution ofmassinspace(sometimes referred to as thebarycenterorbalance point) is the unique point at any given time where theweightedrelativepositionof the distributed mass sums to zero. This is the point to which a force may be applied to cause alinear accelerationwithout anangular acceleration.Calculationsinmechanicsare often simplified when formulated with respect to the center of mass. It is a hypothetical point where the entire mass of an object may be assumed to be concentrated to visualise its motion. In other words, the center of mass is the particle equivalent of a given object for application ofNewton's laws of motion.

In the case of a singlerigid body,the center of mass is fixed in relation to the body, and if the body has uniformdensity,it will be located at thecentroid.The center of mass may be located outside thephysical body,as is sometimes the case forhollowor open-shaped objects, such as ahorseshoe.In the case of a distribution of separate bodies, such as theplanetsof theSolar System,the center of mass may not correspond to the position of any individual member of the system.

The center of mass is a useful reference point for calculations inmechanicsthat involve masses distributed in space, such as thelinearandangular momentumof planetary bodies andrigid body dynamics.Inorbital mechanics,the equations of motion of planets are formulated aspoint masseslocated at the centers of mass (seeBarycenter (astronomy)for details). Thecenter of mass frameis aninertial framein which the center of mass of a system is at rest with respect to the origin of thecoordinate system.

History

The concept of center of gravity orweightwas studied extensively by the ancient Greekmathematician,physicist,andengineerArchimedes of Syracuse.He worked with simplified assumptions about gravity that amount to a uniform field, thus arriving at the mathematical properties of what we now call the center of mass. Archimedes showed that thetorqueexerted on aleverby 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 mass. In his workOn Floating Bodies,Archimedes demonstrated that the orientation of a floating object is the one that makes its center of mass as low as possible. He developed mathematical techniques for finding the centers of mass of objects of uniform density of various well-defined shapes.

Other ancient mathematicians who contributed to the theory of the center of mass includeHero of AlexandriaandPappus of Alexandria.In theRenaissanceandEarly Modernperiods, work byGuido Ubaldi,Francesco Maurolico,Federico Commandino,Evangelista Torricelli,Simon Stevin,Luca Valerio,Jean-Charles de la Faille,Paul Guldin,John Wallis,Christiaan Huygens,Louis Carré,Pierre Varignon,andAlexis Clairautexpanded the concept further.

Newton's second lawis reformulated with respect to the center of mass inEuler's first law.

Definition

The center of mass is the unique point at the center of a distribution of mass in space that has the property that the weighted positionvectorsrelative to this point sum to zero. In analogy to statistics, the center of mass is the mean location of a distribution of mass in space.

A system of particles

In the case of a system of particlesP_i,i= 1, ..., n ,each with massm_ithat are located in space with coordinatesr_i,i= 1, ..., n ,the coordinatesRof the center of mass satisfy the condition $$\sum _{i=1}^{n}m_{i}(\mathbf {r} _{i}-\mathbf {R} )=\mathbf {0}.$$

Solving this equation forRyields the formula $$\mathbf {R} ={\sum _{i=1}^{n}m_{i}\mathbf {r} _{i} \over \sum _{i=1}^{n}m_{i}}.$$

A continuous volume

If the mass distribution is continuous with the density ρ(r) within a solidQ,then the integral of the weighted positioncoordinatesof the points in thisvolumerelative to the center of massRover the volumeVis zero, that is $$\iiint _{Q}\rho (\mathbf {r} )\left(\mathbf {r} -\mathbf {R} \right)dV=0.$$

Solve this equation for the coordinatesRto obtain $$\mathbf {R} ={\frac {1}{M}}\iiint _{Q}\rho (\mathbf {r} )\mathbf {r} \,dV,$$ WhereMis the total mass in the volume.

If a continuous mass distribution has uniformdensity,which means thatρis constant, then the center of mass is the same as thecentroidof the volume.

Barycentric coordinates

Further information:Barycentric coordinate system

The coordinatesRof the center of mass of a two-particle system,P₁andP₂,with massesm₁andm₂is given by $$\mathbf {R} ={{m_{1}\mathbf {r} _{1}+m_{2}\mathbf {r} _{2}} \over m_{1}+m_{2}}.$$

Let thepercentageof the total mass divided between these twoparticlesvary from 100%P₁and 0%P₂through 50%P₁and 50%P₂to 0%P₁and 100%P₂,then the center of massRmoves along the line fromP₁toP₂.The percentages of mass at each point can be viewed as projective coordinates of the pointRon this line, and are termedbarycentric coordinates.Another way of interpreting the process here is the mechanical balancing of moments about an arbitrary point. The numerator gives the total moment that is then balanced by an equivalent total force at the center of mass. This can be generalized to three points and four points to define projective coordinates in the plane, and in space, respectively.

Systems with periodic boundary conditions

For particles in a system withperiodic boundary conditionstwo particles can be neighbours even though they are on opposite sides of the system. This occurs often inmolecular dynamicssimulations, for example, in which clusters form at random locations and sometimes neighbouring atoms cross the periodic boundary. When a cluster straddles the periodic boundary, a naive calculation of the center of mass will be incorrect. A generalized method for calculating the center of mass for periodic systems is to treat each coordinate,xandyand/orz,as if it were on a circle instead of a line.The calculation takes every particle'sxcoordinate and maps it to an angle, $$\theta _{i}={\frac {x_{i}}{x_{\max }}}2\pi$$ wherex_maxis the system size in thexdirection and$x_{i}\in [0,x_{\max })$.From this angle, two new points$(\xi _{i},\zeta _{i})$can be generated, which can be weighted by the mass of the particle$x_{i}$for the center of mass or given a value of 1 for the geometric center: $${\begin{aligned}\xi _{i}&=\cos(\theta _{i})\\\zeta _{i}&=\sin(\theta _{i})\end{aligned}}$$

In the$(\xi,\zeta )$plane, these coordinates lie on a circle of radius 1. From the collection of$\xi _{i}$and$\zeta _{i}$values from all the particles, the averages${\overline {\xi }}$and${\overline {\zeta }}$are calculated.

$${\begin{aligned}{\overline {\xi }}&={\frac {1}{M}}\sum _{i=1}^{n}m_{i}\xi _{i},\\{\overline {\zeta }}&={\frac {1}{M}}\sum _{i=1}^{n}m_{i}\zeta _{i},\end{aligned}}$$ whereMis the sum of the masses of all of the particles.

These values are mapped back into a new angle,${\overline {\theta }}$,from which thexcoordinate of the center of mass can be obtained:

$${\begin{aligned}{\overline {\theta }}&=\operatorname {atan2} \left(-{\overline {\zeta }},-{\overline {\xi }}\right)+\pi \\x_{\text{com}}&=x_{\max }{\frac {\overline {\theta }}{2\pi }}\end{aligned}}$$

The process can be repeated for all dimensions of the system to determine the complete center of mass. The utility of the algorithm is that it allows the mathematics to determine where the "best" center of mass is, instead of guessing or usingcluster analysisto "unfold" a cluster straddling the periodic boundaries. If both average values are zero,$\left({\overline {\xi }},{\overline {\zeta }}\right)=(0,0)$,then${\overline {\theta }}$is undefined. This is a correct result, because it only occurs when all particles are exactly evenly spaced. In that condition, theirxcoordinates are mathematically identical in aperiodic system.

Center of gravity

"Center of gravity" redirects here. For other uses, seeCenter of gravity (disambiguation).

Main article:Centers of gravity in non-uniform fields

Diagram of an educational toy that balances on a point: the center of mass (C) settles below its support (P)

A body's center of gravity is the point around which theresultant torquedue to gravity forces vanishes. Where a gravity field can be considered to be uniform, the mass-center and the center-of-gravity will be the same. However, for satellites in orbit around a planet, in the absence of other torques being applied to a satellite, the slight variation (gradient) in gravitational field between closer-to and further-from the planet (stronger and weaker gravity respectively) can lead to a torque that will tend to align the satellite such that its long axis is vertical. In such a case, it is important to make the distinction between the center-of-gravity and the mass-center. Any horizontal offset between the two will result in an applied torque.

The mass-center is a fixed property for a given rigid body (e.g. with no slosh or articulation), whereas the center-of-gravity may, in addition, depend upon its orientation in a non-uniform gravitational field. In the latter case, the center-of-gravity will always be located somewhat closer to the main attractive body as compared to the mass-center, and thus will change its position in the body of interest as its orientation is changed.

In the study of the dynamics of aircraft, vehicles and vessels, forces and moments need to be resolved relative to the mass center. That is true independent of whether gravity itself is a consideration. Referring to the mass-center as the center-of-gravity is something of a colloquialism, but it is in common usage and when gravity gradient effects are negligible, center-of-gravity and mass-center are the same and are used interchangeably.

In physics the benefits of using the center of mass to model a mass distribution can be seen by considering theresultantof the gravity forces on a continuous body. Consider a bodyQof volumeVwith densityρ(r) at each pointrin the volume. In a parallel gravity field the forcefat each pointris given by, $$\mathbf {f} (\mathbf {r} )=-dm\,g\mathbf {\hat {k}} =-\rho (\mathbf {r} )\,dV\,g\mathbf {\hat {k}},$$ wheredmis the mass at the pointr,gis the acceleration of gravity, and${\textstyle \mathbf {\hat {k}} }$is a unit vector defining the vertical direction.

Choose a reference pointRin the volume and compute theresultant forceand torque at this point, $$\mathbf {F} =\iiint _{Q}\mathbf {f} (\mathbf {r} )\,dV=\iiint _{Q}\rho (\mathbf {r} )\,dV\left(-g\mathbf {\hat {k}} \right)=-Mg\mathbf {\hat {k}},$$ and $$\mathbf {T} =\iiint _{Q}(\mathbf {r} -\mathbf {R} )\times \mathbf {f} (\mathbf {r} )\,dV=\iiint _{Q}(\mathbf {r} -\mathbf {R} )\times \left(-g\rho (\mathbf {r} )\,dV\,\mathbf {\hat {k}} \right)=\left(\iiint _{Q}\rho (\mathbf {r} )\left(\mathbf {r} -\mathbf {R} \right)dV\right)\times \left(-g\mathbf {\hat {k}} \right).$$

If the reference pointRis chosen so that it is the center of mass, then $$\iiint _{Q}\rho (\mathbf {r} )\left(\mathbf {r} -\mathbf {R} \right)dV=0,$$ which means the resultant torqueT= 0.Because the resultant torque is zero the body will move as though it is a particle with its mass concentrated at the center of mass.

By selecting the center of gravity as the reference point for a rigid body, the gravity forces will not cause the body to rotate, which means the weight of the body can be considered to be concentrated at the center of mass.

Linear and angular momentum

The linear and angular momentum of a collection of particles can be simplified by measuring the position and velocity of the particles relative to the center of mass. Let the system of particlesP_i,i= 1,...,nof massesm_ibe located at the coordinatesr_iwith velocitiesv_i.Select a reference pointRand compute the relative position and velocity vectors, $$\mathbf {r} _{i}=(\mathbf {r} _{i}-\mathbf {R} )+\mathbf {R},\quad \mathbf {v} _{i}={\frac {d}{dt}}(\mathbf {r} _{i}-\mathbf {R} )+\mathbf {v}.$$

The total linear momentum and angular momentum of the system are $$\mathbf {p} ={\frac {d}{dt}}\left(\sum _{i=1}^{n}m_{i}(\mathbf {r} _{i}-\mathbf {R} )\right)+\left(\sum _{i=1}^{n}m_{i}\right)\mathbf {v},$$ and $$\mathbf {L} =\sum _{i=1}^{n}m_{i}(\mathbf {r} _{i}-\mathbf {R} )\times {\frac {d}{dt}}(\mathbf {r} _{i}-\mathbf {R} )+\left(\sum _{i=1}^{n}m_{i}\right)\left[\mathbf {R} \times {\frac {d}{dt}}(\mathbf {r} _{i}-\mathbf {R} )+(\mathbf {r} _{i}-\mathbf {R} )\times \mathbf {v} \right]+\left(\sum _{i=1}^{n}m_{i}\right)\mathbf {R} \times \mathbf {v}$$

IfRis chosen as the center of mass these equations simplify to $$\mathbf {p} =m\mathbf {v},\quad \mathbf {L} =\sum _{i=1}^{n}m_{i}(\mathbf {r} _{i}-\mathbf {R} )\times {\frac {d}{dt}}(\mathbf {r} _{i}-\mathbf {R} )+\sum _{i=1}^{n}m_{i}\mathbf {R} \times \mathbf {v}$$ wheremis the total mass of all the particles,pis the linear momentum, andLis the angular momentum.

Thelaw of conservation of momentumpredicts that for any system not subjected to external forces the momentum of the system will remain constant, which means the center of mass will move 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 cancel in accordance withNewton's Third Law.

Determination

See also:Centroid § Determination

Plumb line method

The experimental determination of a body's center of mass makes use of gravity forces on the body and is based on the fact that the center of mass is the same as the center of gravity in the parallel gravity field near the earth's surface.

The center of mass of a body with an axis of symmetry and constant density must lie on this axis. Thus, the center of mass of a circular cylinder of constant density has its center of mass on the axis of the cylinder. In the same way, the center of mass of a spherically symmetric body of constant density is at the center of the sphere. In general, for any symmetry of a body, its center of mass will be a fixed point of that symmetry.

In two dimensions

An experimental method for locating the center of mass is to suspend the object from two locations and to dropplumb linesfrom the suspension points. The intersection of the two lines is the center of mass.

The shape of an object might already be mathematically determined, but it may be too complex to use a known formula. In this case, one can subdivide the complex shape into simpler, more elementary shapes, whose centers of mass are easy to find. If the total mass and center of mass can be determined for each area, then the center of mass of the whole is the weighted average of the centers.This method can even work for objects with holes, which can be accounted for as negative masses.

A direct development of theplanimeterknown as an integraph, or integerometer, can be used to establish the position of thecentroidor center of mass of an irregular two-dimensional 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 compare with the requireddisplacementandcenter of buoyancyof a ship, and ensure it would not capsize.

In three dimensions

An experimental method to locate the three-dimensional coordinates of the center of mass begins by supporting the object at three points and measuring the forces,F₁,F₂,andF₃that resist the weight of the object,$\mathbf {W} =-W\mathbf {\hat {k}}$($\mathbf {\hat {k}}$is the unit vector in the vertical direction). Letr₁,r₂,andr₃be the position coordinates of the support points, then the coordinatesRof the center of mass satisfy the condition that the resultant torque is zero, $$\mathbf {T} =(\mathbf {r} _{1}-\mathbf {R} )\times \mathbf {F} _{1}+(\mathbf {r} _{2}-\mathbf {R} )\times \mathbf {F} _{2}+(\mathbf {r} _{3}-\mathbf {R} )\times \mathbf {F} _{3}=0,$$ or $$\mathbf {R} \times \left(-W\mathbf {\hat {k}} \right)=\mathbf {r} _{1}\times \mathbf {F} _{1}+\mathbf {r} _{2}\times \mathbf {F} _{2}+\mathbf {r} _{3}\times \mathbf {F} _{3}.$$

This equation yields the coordinates of the center of massR* in the horizontal plane as, $$\mathbf {R} ^{*}=-{\frac {1}{W}}\mathbf {\hat {k}} \times (\mathbf {r} _{1}\times \mathbf {F} _{1}+\mathbf {r} _{2}\times \mathbf {F} _{2}+\mathbf {r} _{3}\times \mathbf {F} _{3}).$$

The center of mass lies on the vertical lineL,given by $$\mathbf {L} (t)=\mathbf {R} ^{*}+t\mathbf {\hat {k}}.$$

The three-dimensional coordinates of the center of mass are determined by performing this experiment twice with the object positioned so that these forces are measured for two different horizontal planes through the object. The center of mass will be the intersection of the two linesL₁andL₂obtained from the two experiments.

Applications

Engineering designs

Automotive applications

Engineers try to design asports carso that its center of mass is lowered to make the carhandlebetter, which is to say, maintain traction while executing relatively sharp turns.

The characteristic low profile of the U.S. militaryHumveewas designed in part to allow it to tilt farther than taller vehicles withoutrolling over,by ensuring its low center of mass stays over the space bounded by the four wheels even at angles far from thehorizontal.

Aeronautics

Main article:Center of gravity of an aircraft

The center of mass is an important point on anaircraft,which significantly affects the stability of the aircraft. To ensure the aircraft is stable enough to be safe to fly, the center of mass must fall within specified limits. If the center of mass is ahead of theforward 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 aircraft will be more maneuverable, but also less stable, and possibly unstable enough so as to be impossible to fly. The moment arm of theelevatorwill also be reduced, which makes it more difficult to recover from astalledcondition.

Forhelicoptersinhover,the center of mass is always directly below therotorhead.In forward flight, the center of mass will move forward to balance the negative pitch torque produced by applyingcycliccontrol to propel the helicopter forward; consequently a cruising helicopter flies "nose-down" in level flight.

Astronomy

Main article:Barycenter

Two bodies orbiting theirbarycenter(red cross)

The center of mass plays an important role in astronomy and astrophysics, where it is commonly referred to as thebarycenter.The barycenter is the point between two objects where they balance each other; it is the center of mass where two or more celestial bodiesorbiteach other. When amoonorbits aplanet,or a planet orbits astar,both bodies are actually orbiting a point that lies away from the center of the primary (larger) body.For example, the Moon does not orbit the exact center of theEarth,but a point on a line between the center of the Earth and the Moon, approximately 1,710 km (1,062 miles) below the surface of the Earth, where their respective masses balance. This is the point about which the Earth and Moon orbit as they travel around theSun.If the masses are more similar, e.g.,Pluto and Charon,the barycenter will fall outside both bodies.

Rigging and safety

Knowing the location of the center of gravity whenriggingis crucial, possibly resulting in severe injury or death if assumed incorrectly. A center of gravity that is at or above the lift point will most likely result in a tip-over incident. In general, the further the center of gravity below the pick point, the safer the lift. There are other things to consider, such as shifting loads, strength of the load and mass, distance between pick points, and number of pick points. Specifically, when selecting lift points, it is very important to place the center of gravity at the center and well below the lift points.

Body motion

Main article:Kinesiology

In kinesiology and biomechanics, the center of mass is an important parameter that assists people in understanding their human locomotion. Typically, a human's center of mass is detected with one of two methods: the reaction board method is a static analysis that involves the person lying down on that instrument, and use of theirstatic equilibriumequation to find their center of mass; the segmentation method relies on a mathematical solution based on thephysical principlethat thesummationof thetorquesof individual body sections,relative toa specifiedaxis,must equal the torque of the whole system that constitutes the body, measured relative to the same axis.

Optimization

TheCenter-of-gravity methodis a method for convex optimization, which uses the center-of-gravity of the feasible region.

See also

• Barycenter
• Buoyancy
• Center of percussion
• Center of pressure (fluid mechanics)
• Center of pressure (terrestrial locomotion)
• Centroid
• Circumcenter of mass
• Expected value
• Mass point geometry
• Metacentric height
• Roll center
• Weight distribution

References

• Asimov, Isaac(1988) [1966],Understanding Physics,Barnes & Noble Books,ISBN978-0-88029-251-1
• Bai, Linge; Breen, David (2008). "Calculating Center of Mass in an Unbounded 2D Environment".Journal of Graphics, GPU, and Game Tools.13(4): 53–60.doi:10.1080/2151237X.2008.10129266.S2CID40807367.
• Baron, Margaret E.(2004) [1969],The Origins of the Infinitesimal Calculus,Courier Dover Publications,ISBN978-0-486-49544-6
• Beatty, Millard F. (2006),Principles of Engineering Mechanics, Volume 2: Dynamics—The Analysis of Motion,Mathematical Concepts and Methods in Science and Engineering, vol. 33, Springer,ISBN978-0-387-23704-6
• De Silva, Clarence W. (2002),Vibration and shock handbook,CRC Press,ISBN978-0-8493-1580-0
• Federal Aviation Administration(2007),Aircraft Weight and Balance Handbook(PDF),United States Government Printing Office,archived fromthe original(PDF)on 2011-10-19,retrieved2011-10-23
• Feynman, Richard;Leighton, Robert B.;Sands, Matthew(1963),The Feynman Lectures on Physics,vol. 1 (Sixth printing, February 1977 ed.), Addison-Wesley,ISBN978-0-201-02010-6
• Frautschi, Steven C.;Olenick, Richard P.;Apostol, Tom M.;Goodstein, David L.(1986),The Mechanical Universe: Mechanics and heat, advanced edition,Cambridge University Press,ISBN978-0-521-30432-0
• Giambattista, Alan; Richardson, Betty McCarthy; Richardson, Robert Coleman (2007),College physics,vol. 1 (2nd ed.), McGraw-Hill Higher Education,ISBN978-0-07-110608-5
• Goldstein, Herbert;Poole, Charles;Safko, John (2001),Classical Mechanics(3rd ed.), Addison Wesley,ISBN978-0-201-65702-9
• Goldstein, Herbert;Poole, Charles; Safko, John (2002),Classical Mechanics(3rd ed.), Addison-Wesley,ISBN978-0-201-65702-9
• Goodman, Lawrence E.; Warner, William H. (2001) [1964],Statics,Dover,ISBN978-0-486-42005-9
• Hamill, Patrick (2009),Intermediate Dynamics,Jones & Bartlett Learning,ISBN978-0-7637-5728-1
• Jong, I. G.; Rogers, B. G. (1995),Engineering Mechanics: Statics,Saunders College Publishing,ISBN978-0-03-026309-5
• Kleppner, Daniel;Kolenkow, Robert(1973),An Introduction to Mechanics(2nd ed.), McGraw-Hill,ISBN978-0-07-035048-9
• Levi, Mark (2009),The Mathematical Mechanic: Using Physical Reasoning to Solve Problems,Princeton University Press,ISBN978-0-691-14020-9
• Mancosu, Paolo (1999),Philosophy of mathematics and mathematical practice in the seventeenth century,Oxford University Press,ISBN978-0-19-513244-1
• Millikan, Robert Andrews(1902),Mechanics, molecular physics and heat: a twelve weeks' college course,Chicago: Scott, Foresman and Company,retrieved2011-05-25
• Murray, Carl; Dermott, Stanley (1999),Solar System Dynamics,Cambridge University Press,ISBN978-0-521-57295-8
• O'Donnell, Peter J. (2015),Essential Dynamics and Relativity,CRC Press,ISBN978-1-466-58839-4
• Pollard, David D.; Fletcher, Raymond C. (2005),Fundamentals of Structural Geology,Cambridge University Press,ISBN978-0-521-83927-3
• Pytel, Andrew; Kiusalaas, Jaan (2010),Engineering Mechanics: Statics,vol. 1 (3rd ed.), Cengage Learning,ISBN978-0-495-29559-4
• Rosen, Joe; Gothard, Lisa Quinn (2009),Encyclopedia of Physical Science,Infobase Publishing,ISBN978-0-8160-7011-4
• Sangwin, Christopher J. (2006),"Locating the centre of mass by mechanical means"(PDF),Journal of the Oughtred Society,15(2), archived fromthe original(PDF)on 2011-10-05,retrieved2011-10-23
• Serway, Raymond A.; Jewett, John W. (2006),Principles of physics: a calculus-based text,vol. 1 (4th ed.), Thomson Learning,Bibcode:2006ppcb.book.....J,ISBN978-0-534-49143-7
• Shirley, James H.; Fairbridge, Rhodes Whitmore (1997),Encyclopedia of planetary sciences,Springer,ISBN978-0-412-06951-2
• Shore, Steven N. (2008),Forces in Physics: A Historical Perspective,Greenwood Press,ISBN978-0-313-33303-3
• Symon, Keith R. (1971),Mechanics(3rd ed.), Addison-Wesley,ISBN978-0-201-07392-8
• Tipler, Paul A.; Mosca, Gene (2004),Physics for Scientists and Engineers,vol. 1A (5th ed.), W. H. Freeman and Company,ISBN978-0-7167-0900-8
• Van Pelt, Michael (2005),Space Tourism: Adventures in Earth Orbit and Beyond,Springer,ISBN978-0-387-40213-0
• Vint, Peter (2003),"LAB: Center of Mass (Center of Gravity) of the Human Body"(PDF),KIN 335 - Biomechanics,retrieved2013-10-18
• Walton, William (1855),A collection of problems in illustration of the principles of theoretical mechanics(2nd ed.), Deighton, Bell & Co.

External links

• Motion of the Center of Massshows that the motion of the center of mass of an object in free fall is the same as the motion of a point object.
• The Solar System's barycenter,simulations showing the effect each planet contributes to the Solar System's barycenter.