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Center of mass

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(Redirected fromCentre of gravity)
Unique point where the weighted relative position of the distributed mass sums to zero

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

Inphysics, thecenter of mass of a distribution ofmass inspace (sometimes referred to as thebarycenter orbalance point) is the unique point at any given time where the weightedrelative position of the distributed mass sums to zero. For a rigid body containing its center of mass, this is the point to which a force may be applied to cause alinear acceleration without anangular acceleration. Calculations inmechanics are 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 the application ofNewton's laws of motion.[1]

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 forhollow or open-shaped objects, such as ahorseshoe. In the case of a distribution of separate bodies, such as theplanets of 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 in mechanics that involve masses distributed in space, such as thelinear andangular momentum of planetary bodies andrigid body dynamics. Inorbital mechanics, the equations of motion of planets are formulated aspoint masses located at the centers of mass (seeBarycenter (astronomy) for details). Thecenter of mass frame is aninertial frame in which the center of mass of a system is at rest with respect to the origin of thecoordinate system.

History

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The concept of center of gravity or center of weight was studied extensively byArchimedes of Syracuse, an ancient Greekmathematician,physicist, andengineer. 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 thetorque 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 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.[2]

Other ancient mathematicians who contributed to the theory of the center of mass includeHero of Alexandria andPappus of Alexandria. In theRenaissance andEarly Modern periods, work byGuido Ubaldi,Francesco Maurolico,[3]Federico Commandino,[4]Evangelista Torricelli,Simon Stevin,[5]Luca Valerio,[6]Jean-Charles de la Faille,Paul Guldin,[7]John Wallis,Christiaan Huygens,[8]Louis Carré,Pierre Varignon, andAlexis Clairaut expanded the concept further.[9]

Newton's second law is reformulated with respect to the center of mass inEuler's first law.[10]

Definition

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The center of mass is the unique point at the center of a distribution of mass in space that has the property that the weightedposition vectors relative 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

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In the case of a system of particlesPi,i = 1, ..., n , each with massmi that are located in space with coordinatesri,i = 1, ..., n , the coordinatesR of the center of mass satisfy∑i=1nmi(ri−R)=0.{\displaystyle \sum _{i=1}^{n}m_{i}(\mathbf {r} _{i}-\mathbf {R} )=\mathbf {0} .}

Solving this equation forR yields the formulaR=∑i=1nmiri∑i=1nmi.{\displaystyle \mathbf {R} ={\sum _{i=1}^{n}m_{i}\mathbf {r} _{i} \over \sum _{i=1}^{n}m_{i}}.}

A continuous volume

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If the mass distribution is continuous with the density ρ(r) within a solidQ, then the integral of the weighted position coordinates of the points in this volume relative to the center of massR over the volumeV is zero, that is∭Qρ(r)(r−R)dV=0.{\displaystyle \iiint _{Q}\rho (\mathbf {r} )\left(\mathbf {r} -\mathbf {R} \right)dV=\mathbf {0} .}

Solve this equation for the coordinatesR to obtainR=1M∭Qρ(r)rdV,{\displaystyle \mathbf {R} ={\frac {1}{M}}\iiint _{Q}\rho (\mathbf {r} )\mathbf {r} \,dV,}whereM is the total mass in the volume.

If a continuous mass distribution has uniform density, which means thatρ is constant, then the center of mass is the same as thecentroid of the volume.[11]

Barycentric coordinates

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Further information:Barycentric coordinate system

The coordinatesR of the center of mass of a two-particle system,P1 andP2, with massesm1 andm2 is given byR=m1r1+m2r2m1+m2.{\displaystyle \mathbf {R} ={{m_{1}\mathbf {r} _{1}+m_{2}\mathbf {r} _{2}} \over m_{1}+m_{2}}.}

Let the percentage of the total mass divided between these two particles vary from 100%P1 and 0%P2 through 50%P1 and 50%P2 to 0%P1 and 100%P2, then the center of massR moves along the line fromP1 toP2. The percentages of mass at each point can be viewed as projective coordinates of the pointR on 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

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For particles in a system withperiodic boundary conditions two particles can be neighbours even though they are on opposite sides of the system. This occurs often inmolecular dynamics simulations, 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,x andy and/orz, as if it were on a circle instead of a line.[12] The calculation takes every particle'sx coordinate and computesN possible centers of mass,

x¯n=x¯0+2πnN,n=1,2,…,N,{\bar {x}}_{n}={\bar {x}}_{0}+{\frac {2\pi n}{N}},\;\;\;n=1,2,\ldots ,N,

wherex¯0{\bar {x}}_{0} is the naïve mass-weighted average position for all particles. The true center of mass is then the value in the vectorx¯=[x¯1,x¯2,…,x¯n]{\bar {\mathbf {x} }}=[{\bar {x}}_{1},{\bar {x}}_{2},\ldots ,{\bar {x}}_{n}], with the lowest sum of squared arc distances between the points.

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 analysis to "unfold" a cluster straddling the periodic boundaries. This approach computes the intrinsic center of mass.[13] The extrinsic circular mean is a popular way to estimate the center of mass,[14] however this approach can be inaccurate.[12]

Center of gravity

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"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 the resultanttorque due to gravity forces vanishes.[15] Where a gravity field can be considered to be uniform, the center of mass 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 the parts 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.[16] 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 theresultant of the gravity forces on a continuous body. Consider a bodyQ of volumeV with densityρ(r) at each pointr in the volume. In a parallel gravity field the forcef at each pointr is given by,f(r)=−dmgk^=−ρ(r)dVgk^,{\displaystyle \mathbf {f} (\mathbf {r} )=-dm\,g\mathbf {\hat {k}} =-\rho (\mathbf {r} )\,dV\,g\mathbf {\hat {k}} ,}wheredm is the mass at the pointr,g is the acceleration of gravity, andk^{\textstyle \mathbf {\hat {k}} } is a unit vector defining the vertical direction.

Choose a reference pointR in the volume and compute theresultant force and torque at this point,F=∭Qf(r)dV=∭Qρ(r)dV(−gk^)=−Mgk^,{\displaystyle \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}} ,}andT=∭Q(r−R)×f(r)dV=∭Q(r−R)×(−gρ(r)dVk^)=(∭Qρ(r)(r−R)dV)×(−gk^).{\displaystyle \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 pointR is chosen so that it is the center of mass, then∭Qρ(r)(r−R)dV=0,{\displaystyle \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

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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 particlesPi,i = 1, ...,n of massesmi be located at the coordinatesri with velocitiesvi. Select a reference pointR and compute the relative position and velocity vectors,ri=(ri−R)+R,vi=ddt(ri−R)+v.{\displaystyle \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 arep=ddt(∑i=1nmi(ri−R))+(∑i=1nmi)v,{\displaystyle \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} ,}andL=∑i=1nmi(ri−R)×ddt(ri−R)+(∑i=1nmi)[R×ddt(ri−R)+(ri−R)×v]+(∑i=1nmi)R×v{\displaystyle \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} }

IfR is chosen as the center of mass these equations simplify top=mv,L=∑i=1nmi(ri−R)×ddt(ri−R)+∑i=1nmiR×v{\displaystyle \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} }wherem is the total mass of all the particles,p is the linear momentum, andL is the angular momentum.

Thelaw of conservation of momentum predicts 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.[17]

Determination

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See also:Centroid § Determination

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.[18]

In two dimensions

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Plumb line method

An experimental method for locating the center of mass is to suspend the object from two different points, in series, and each time to drop aplumb line from the suspension point. The intersection of the two lines is the center of mass.[19]

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.[20] This method can even work for objects with holes, which can be accounted for as negative masses.[21]

A direct development of theplanimeter known as an integraph, or integerometer, can be used to establish the position of thecentroid or 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 requireddisplacement andcenter of buoyancy of a ship, and ensure it would not capsize.[22][23]

In three dimensions

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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, andF3 that resist the weight of the object,W=−Wk^{\displaystyle \mathbf {W} =-W\mathbf {\hat {k}} } (k^{\displaystyle \mathbf {\hat {k}} } is the unit vector in the vertical direction). Letr1,r2, andr3 be the position coordinates of the support points, then the coordinatesR of the center of mass satisfy the condition that the resultant torque is zero,T=(r1−R)×F1+(r2−R)×F2+(r3−R)×F3=0,{\displaystyle \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,}orR×(−Wk^)=r1×F1+r2×F2+r3×F3.{\displaystyle \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,R∗=−1Wk^×(r1×F1+r2×F2+r3×F3).{\displaystyle \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 byL(t)=R∗+tk^.{\displaystyle \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 linesL1 andL2 obtained from the two experiments.

Applications

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Engineering designs

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Automotive applications

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

The characteristic low profile of the U.S. militaryHumvee was designed in part to allow it to tilt farther than taller vehicles without rolling over, by ensuring its low center of mass stays over the space bounded by the four wheels even at angles far from the horizontal.

Aeronautics

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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.[24] 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 theelevator will also be reduced, which makes it more difficult to recover from astalled condition.[25]

Forhelicopters inhover, 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 applyingcyclic control to propel the helicopter forward; consequently a cruising helicopter flies "nose-down" in level flight.[26]

Astronomy

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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 bodies orbit each other. When a moon orbits aplanet, or a planet orbits astar, both bodies are actually orbiting a point that lies away from the center of the primary (larger) body.[27] For example, the Moon does not orbit the exact center of the Earth, 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 the Sun. If the masses are more similar, e.g.,Pluto and Charon, the barycenter will fall outside both bodies.

Rigging and safety

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Knowing the location of the center of gravity whenrigging is 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.[28]

Body motion

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Main article:Kinesiology

The center of mass of the adult human body vertically is 10 cm above thetrochanter (where the femur joins the hip),[29] 1.4 cm forward of the knee[clarification needed] and 1.0 cm behind the trochanter.[30] 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 equilibrium equation to find their center of mass; the segmentation method relies on a mathematical solution based on the physical principle that the summation of thetorques of individual body sections, relative to a specified axis, must equal the torque of the whole system that constitutes the body, measured relative to the same axis.[31]

Optimization

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Thecenter-of-gravity method is a method for convex optimization which uses the center of gravity of the feasible region.

See also

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Notes

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  1. ^"10.3: The center of mass".Physics LibreTexts. 17 September 2019. Retrieved2024-10-15.
  2. ^Shore 2008, pp. 9–11.
  3. ^Baron 2004, pp. 91–94.
  4. ^Baron 2004, pp. 94–96.
  5. ^Baron 2004, pp. 96–101.
  6. ^Baron 2004, pp. 101–106.
  7. ^Mancosu 1999, pp. 56–61.
  8. ^Erlichson, H. (1996)."Christiaan Huygens' discovery of the center of oscillation formula".American Journal of Physics.64 (5):571–574.Bibcode:1996AmJPh..64..571E.doi:10.1119/1.18156.ISSN 0002-9505.
  9. ^Walton 1855, p. 2.
  10. ^Beatty 2006, p. 29.
  11. ^Levi 2009, p. 85.
  12. ^abRichardson, Dunn & McCluskey 2025.
  13. ^Hotz 2013.
  14. ^Bai & Breen 2008.
  15. ^Resnick, R. and Halliday, D. (1962)Physics, 9-1 “Center of Mass”, Wiley
  16. ^Resnick, R. and Halliday, D. (1962)Physics, 14-3 “Center of Gravity”, Wiley
  17. ^Kleppner & Kolenkow 1973, p. 117.
  18. ^The Feynman Lectures on Physics Vol. I Ch. 19: Center of Mass; Moment of Inertia
  19. ^Kleppner & Kolenkow 1973, pp. 119–120.
  20. ^Feynman, Leighton & Sands 1963, pp. 19.1–19.2.
  21. ^Hamill 2009, pp. 20–21.
  22. ^"The theory and design of British shipbuilding".Amos Lowrey Ayre. p. 3. Retrieved2012-08-20.
  23. ^Sangwin 2006, p. 7.
  24. ^Federal Aviation Administration 2007, p. 1.4.
  25. ^Federal Aviation Administration 2007, p. 1.3.
  26. ^"Helicopter Aerodynamics"(PDF). p. 82. Archived fromthe original(PDF) on 2012-03-24. Retrieved2013-11-23.
  27. ^Murray & Dermott 1999, pp. 45–47.
  28. ^"Structural Collapse Technician: Module 4 – Lifting and Rigging"(PDF).FEMA.gov. Retrieved2019-11-27.
  29. ^Palmer, Carroll E. (1944)."Studies of the Center of Gravity in the Human Body".Child Development.15 (2/3):99–180.doi:10.2307/1125537.JSTOR 1125537.
  30. ^Woodhull, A. M.; Maltrud, K.; Mello, B. L. (1985)."Alignment of the human body in standing".European Journal of Applied Physiology and Occupational Physiology.54 (1):109–115.doi:10.1007/BF00426309.ISSN 0301-5548.PMID 4018044.
  31. ^Vint 2003, pp. 1–11.

References

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External links

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