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2018:
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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.
3484:
2084:
1892:
3684:
1059:. 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.
4368:
4060:
2706:{\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).}
4206:
1708:
3166:
4523:, 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 the
1753:
2444:
3496:
2997:
3705:
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,
4620:
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
3752:
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
3736:
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
2103:
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
1075:
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
2099:
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
4637:
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
3159:
2095:
due 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
88:
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 application of
4445:
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 lines
4217:
3902:
4065:
1542:
4600:, 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
4621:
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.
212:, 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.
2244:
1393:
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3764:
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 required
2865:
1896:
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 using
3005:
3732:
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.
979:
620:
3479:{\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+\left(\sum _{i=1}^{n}m_{i}\right)\mathbf {R} \times \mathbf {v} }
808:
2797:
728:
2809:
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.
502:
4565:
1887:{\displaystyle {\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}}}
3679:{\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} }
1967:
4441:
4519:, 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 the
5027:
2142:
1150:
1291:
198:. 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 the
4363:{\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}).}
4055:{\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,}
2818:
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 particles
1758:
1547:
1296:
3843:
4201:{\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}.}
3753:
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.
1703:{\displaystyle {\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}}}
1213:
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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 pointâtheir center of mass. In his work
1481:
1454:
422:
4596:, 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 the
1286:
3781:
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,
2439:{\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}} ,}
2992:{\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} .}
5906:
4379:
4720:
3154:{\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} ,}
5295:
2078:
5682:
5286:
1095:
1904:
2104:
in common usage and when gravity gradient effects are negligible, center-of-gravity and mass-center are the same and are used interchangeably.
4685:
2100:
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.
5825:
5803:
5785:
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5338:
5273:
5255:
5237:
5187:
4580:. 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
3737:
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.
4715:
363:
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.
359:
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 position
2806:. 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.
6106:
5095:
2071:
1996:
is undefined. This is a correct result, because it only occurs when all particles are exactly evenly spaced. In that condition, their
5872:
5899:
2057:
346:
4520:
4510:
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5282:
2035:
324:
2039:
328:
272:
1056:
838:
6177:
5947:
5892:
3805:
2001:
1068:
4617:
4546:. In forward flight, the center of mass will move forward to balance the negative pitch torque produced by applying
2239:{\displaystyle \mathbf {f} (\mathbf {r} )=-dm\,g\mathbf {\hat {k}} =-\rho (\mathbf {r} )\,dV\,g\mathbf {\hat {k}} ,}
6187:
6101:
6083:
5427:
5404:
4547:
224:
90:
5875:
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.
4576:
The center of mass plays an important role in astronomy and astrophysics, where it is commonly referred to as the
2028:
317:
6192:
6135:
4497:, by ensuring its low center of mass stays over the space bounded by the four wheels even at angles far from the
256:
4550:
control to propel the helicopter forward; consequently a cruising helicopter flies "nose-down" in level flight.
2107:
In physics the benefits of using the center of mass to model a mass distribution can be seen by considering the
1388:{\displaystyle {\begin{aligned}\xi _{i}&=\cos(\theta _{i})\\\zeta _{i}&=\sin(\theta _{i})\end{aligned}}}
5987:
5962:
4671:
3745:
An experimental method for locating the center of mass is to suspend the object from two locations and to drop
3702:
4642:
equation to find their center of mass; the segmentation method relies on a mathematical solution based on the
5567:
5530:
4498:
2087:
Diagram of an educational toy that balances on a point: the center of mass (C) settles below its support (P)
1166:
4662:, must equal the torque of the whole system that constitutes the body, measured relative to the same axis.
1972:
1720:
1218:
1071:
two particles can be neighbours even though they are on opposite sides of the system. This occurs often in
6093:
1513:
286:
244:
156:
3848:
6130:
5942:
5177:
1486:
228:
3707:
32:
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5400:
5299:
5972:
5719:
4860:
4730:
4710:
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220:
160:
77:
6113:
5927:
4755:
4740:
4524:
3766:
1400:
240:
236:
144:
73:
72:
of the distributed mass sums to zero. This is the point to which a force may be applied to cause a
5504:
5320:
5213:
4745:
4639:
4539:
3770:
1072:
268:
216:
208:
1459:
128:, the center of mass may not correspond to the position of any individual member of the system.
5878:
3706:
and so on. More formally, this is true for any internal forces that cancel in accordance with
974:{\displaystyle \mathbf {R} ={{m_{1}\mathbf {r} _{1}+m_{2}\mathbf {r} _{2}} \over m_{1}+m_{2}}.}
615:{\displaystyle \mathbf {R} ={\sum _{i=1}^{n}m_{i}\mathbf {r} _{i} \over \sum _{i=1}^{n}m_{i}}.}
6150:
5821:
5799:
5781:
5763:
5745:
5727:
5668:
5650:
5632:
5610:
5592:
5554:
5536:
5516:
5487:
5469:
5451:
5433:
5423:
5410:
5396:
5383:
5363:
5334:
5269:
5251:
5233:
5196:
Bai, Linge; Breen, David (2008). "Calculating Center of Mass in an
Unbounded 2D Environment".
5183:
4876:
4674:
is a method for convex optimization, which uses the center-of-gravity of the feasible region.
4585:
1432:
1051:. The percentages of mass at each point can be viewed as projective coordinates of the point
639:
168:
148:
69:
6118:
5997:
5347:
5205:
4868:
4659:
4494:
1898:
140:
5835:
4979:
1901:
to "unfold" a cluster straddling the periodic boundaries. If both average values are zero,
1264:
6012:
6002:
5881:, simulations showing the effect each planet contributes to the Solar System's barycenter.
5500:
5355:
5316:
4655:
2295:
2108:
2092:
803:{\displaystyle \mathbf {R} ={\frac {1}{M}}\iiint _{Q}\rho (\mathbf {r} )\mathbf {r} \,dV,}
280:
276:
109:
65:
5377:
2792:{\displaystyle \iiint _{Q}\rho (\mathbf {r} )\left(\mathbf {r} -\mathbf {R} \right)dV=0,}
723:{\displaystyle \iiint _{Q}\rho (\mathbf {r} )\left(\mathbf {r} -\mathbf {R} \right)dV=0.}
290:
215:
Other ancient mathematicians who contributed to the theory of the center of mass include
5723:
5250:, Mathematical Concepts and Methods in Science and Engineering, vol. 33, Springer,
4864:
36:
This toy uses the principles of center of mass to keep balance when sitting on a finger.
6046:
6036:
6031:
5814:
5351:
5225:
4735:
4691:
4528:
164:
3760:
known as an integraph, or integerometer, can be used to establish the position of the
1088:, as if it were on a circle instead of a line. The calculation takes every particle's
27:
Unique point where the weighted relative position of the distributed mass sums to zero
6182:
6171:
6123:
5992:
5957:
5509:
5324:
5131:
4605:
3749:
from the suspension points. The intersection of the two lines is the center of mass.
184:
167:
in which the center of mass of a system is at rest with respect to the origin of the
5217:
6145:
6067:
5858:
A collection of problems in illustration of the principles of theoretical mechanics
5376:
Giambattista, Alan; Richardson, Betty McCarthy; Richardson, Robert
Coleman (2007),
5209:
5173:
4643:
4486:
better, which is to say, maintain traction while executing relatively sharp turns.
4483:
360:
252:
248:
232:
125:
100:, the center of mass is fixed in relation to the body, and if the body has uniform
5625:
5856:
5571:
4848:
497:{\displaystyle \sum _{i=1}^{n}m_{i}(\mathbf {r} _{i}-\mathbf {R} )=\mathbf {0} .}
6026:
6021:
5952:
4980:
The
Feynman Lectures on Physics Vol. I Ch. 19: Center of Mass; Moment of Inertia
4750:
4630:
2017:
306:
264:
260:
81:
3899:
of the center of mass satisfy the condition that the resultant torque is zero,
3724:
6140:
5551:
Philosophy of mathematics and mathematical practice in the seventeenth century
5248:
Principles of
Engineering Mechanics, Volume 2: DynamicsâThe Analysis of Motion
4700:
4681:
4569:
4559:
4535:
4493:
was designed in part to allow it to tilt farther than taller vehicles without
4479:
3757:
3746:
984:
195:
152:
97:
17:
4880:
1429:
plane, these coordinates lie on a circle of radius 1. From the collection of
6062:
5982:
5937:
4647:
4543:
188:
132:
117:
85:
1962:{\displaystyle \left({\overline {\xi }},{\overline {\zeta }}\right)=(0,0)}
5967:
4725:
4705:
4516:
3761:
3719:
988:
826:
192:
136:
105:
5333:, vol. 1 (Sixth printing, February 1977 ed.), Addison-Wesley,
3895:
be the position coordinates of the support points, then the coordinates
113:
2083:
2042: in this section. Unsourced material may be challenged and removed.
1288:
for the center of mass or given a value of 1 for the geometric center:
818:
331: in this section. Unsourced material may be challenged and removed.
121:
101:
41:
5884:
4527:
will also be reduced, which makes it more difficult to recover from a
4436:{\displaystyle \mathbf {L} (t)=\mathbf {R} ^{*}+t\mathbf {\hat {k}} .}
5573:
Mechanics, molecular physics and heat: a twelve weeks' college course
5532:
The
Mathematical Mechanic: Using Physical Reasoning to Solve Problems
4651:
4589:
4490:
643:
199:
180:
4872:
4849:"Christiaan Huygens' discovery of the center of oscillation formula"
1261:
can be generated, which can be weighted by the mass of the particle
131:
The center of mass is a useful reference point for calculations in
4597:
4581:
4564:
3723:
2082:
203:
53:
31:
3001:
The total linear momentum and angular momentum of the system are
5977:
5132:"Structural Collapse Technician: Module 4 - Lifting and Rigging"
4593:
120:. In the case of a distribution of separate bodies, such as the
49:
5888:
5075:
5063:
4601:
2011:
300:
5360:
The
Mechanical Universe: Mechanics and heat, advanced edition
5382:, vol. 1 (2nd ed.), McGraw-Hill Higher Education,
3493:
is chosen as the center of mass these equations simplify to
2111:
of the gravity forces on a continuous body. Consider a body
5836:"LAB: Center of Mass (Center of Gravity) of the Human Body"
4210:
This equation yields the coordinates of the center of mass
1145:{\displaystyle \theta _{i}={\frac {x_{i}}{x_{\max }}}2\pi }
5798:, vol. 1A (5th ed.), W. H. Freeman and Company,
630:
If the mass distribution is continuous with the density Ï(
2091:
A body's center of gravity is the point around which the
2862:
and compute the relative position and velocity vectors,
2070:"Center of gravity" redirects here. For other uses, see
825:
is constant, then the center of mass is the same as the
5740:
Shirley, James H.; Fairbridge, Rhodes
Whitmore (1997),
3164:
3003:
2449:
2300:
513:
420:
151:, the equations of motion of planets are formulated as
4482:
so that its center of mass is lowered to make the car
2264:
289:
is reformulated with respect to the center of mass in
135:
that involve masses distributed in space, such as the
4382:
4220:
4068:
3905:
3851:
3808:
3499:
3169:
3008:
2868:
2726:
2454:
2305:
2145:
2131:
in the volume. In a parallel gravity field the force
1975:
1907:
1756:
1723:
1545:
1516:
1489:
1462:
1435:
1403:
1294:
1267:
1221:
1169:
1098:
881:
742:
660:
518:
425:
5002:
4489:
The characteristic low profile of the U.S. military
3874:
is the unit vector in the vertical direction). Let
6092:
6076:
6055:
6045:
6011:
5920:
5816:
Space
Tourism: Adventures in Earth Orbit and Beyond
4616:Knowing the location of the center of gravity when
5813:
5624:
5508:
5446:Goodman, Lawrence E.; Warner, William H. (2001) ,
4542:, the center of mass is always directly below the
4435:
4362:
4200:
4054:
3866:
3838:{\displaystyle \mathbf {W} =-W\mathbf {\hat {k}} }
3837:
3678:
3478:
3153:
2991:
2791:
2705:
2438:
2287:is a unit vector defining the vertical direction.
2279:
2238:
1988:
1961:
1886:
1748:coordinate of the center of mass can be obtained:
1736:
1714:is the sum of the masses of all of the particles.
1702:
1529:
1502:
1475:
1448:
1421:
1387:
1280:
1253:
1207:
1144:
973:
802:
722:
614:
496:
64:) is the unique point at any given time where the
5683:"Locating the centre of mass by mechanical means"
4424:
4255:
4094:
3858:
3829:
2720:is chosen so that it is the center of mass, then
2689:
2592:
2427:
2398:
2271:
2227:
2184:
108:. The center of mass may be located outside the
5718:, vol. 1 (4th ed.), Thomson Learning,
5649:, vol. 1 (3rd ed.), Cengage Learning,
5623:Pollard, David D.; Fletcher, Raymond C. (2005),
4990:
4967:
4608:, the barycenter will fall outside both bodies.
1855:
1197:
1129:
847:of the center of mass of a two-particle system,
5028:"The theory and design of British shipbuilding"
4515:The center of mass is an important point on an
1717:These values are mapped back into a new angle,
2000:coordinates are mathematically identical in a
1076:periodic systems is to treat each coordinate,
817:If a continuous mass distribution has uniform
5900:
5118:
4372:The center of mass lies on the vertical line
638:, then the integral of the weighted position
183:was studied extensively by the ancient Greek
8:
5716:Principles of physics: a calculus-based text
5714:Serway, Raymond A.; Jewett, John W. (2006),
3773:of a ship, and ensure it would not capsize.
1483:values from all the particles, the averages
987:of the total mass divided between these two
5760:Forces in Physics: A Historical Perspective
4721:Center of pressure (terrestrial locomotion)
395:that are located in space with coordinates
6052:
5907:
5893:
5885:
5230:The Origins of the Infinitesimal Calculus
4419:
4418:
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2684:
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2640:
2627:
2615:
2587:
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2585:
2578:
2570:
2542:
2534:
2525:
2511:
2503:
2495:
2484:
2476:
2467:
2455:
2453:
2422:
2421:
2393:
2392:
2374:
2366:
2354:
2340:
2332:
2324:
2318:
2306:
2304:
2266:
2265:
2263:
2222:
2221:
2217:
2210:
2202:
2179:
2178:
2174:
2154:
2146:
2144:
2058:Learn how and when to remove this message
1976:
1974:
1926:
1913:
1906:
1860:
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1266:
1242:
1229:
1220:
1196:
1174:
1168:
1128:
1118:
1112:
1103:
1097:
1063:Systems with periodic boundary conditions
959:
946:
934:
929:
922:
909:
904:
897:
892:
890:
882:
880:
790:
785:
777:
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743:
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698:
690:
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659:
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579:
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562:
555:
545:
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527:
519:
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486:
475:
466:
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451:
441:
430:
424:
347:Learn how and when to remove this message
5861:(2nd ed.), Deighton, Bell & Co.
5663:Rosen, Joe; Gothard, Lisa Quinn (2009),
5296:United States Government Printing Office
5198:Journal of Graphics, GPU, and Game Tools
4929:
4604:. If the masses are more similar, e.g.,
4563:
3690:is the total mass of all the particles,
2079:Centers of gravity in non-uniform fields
732:Solve this equation for the coordinates
5645:Pytel, Andrew; Kiusalaas, Jaan (2010),
5587:Murray, Carl; Dermott, Stanley (1999),
5051:
4834:
4767:
5576:, Chicago: Scott, Foresman and Company
5426:; Poole, Charles; Safko, John (2002),
5014:
4905:
4893:
3802:that resist the weight of the object,
1208:{\displaystyle x_{i}\in [0,x_{\max })}
5794:Tipler, Paul A.; Mosca, Gene (2004),
4822:
4810:
4798:
4786:
4774:
1989:{\displaystyle {\overline {\theta }}}
1737:{\displaystyle {\overline {\theta }}}
1254:{\displaystyle (\xi _{i},\zeta _{i})}
371:In the case of a system of particles
7:
5796:Physics for Scientists and Engineers
5288:Aircraft Weight and Balance Handbook
5157:
5076:Federal Aviation Administration 2007
5064:Federal Aviation Administration 2007
4954:Resnick, R. and Halliday, D. (1962)
4941:Resnick, R. and Halliday, D. (1962)
4917:
4716:Center of pressure (fluid mechanics)
2258:is the acceleration of gravity, and
2040:adding citations to reliable sources
1530:{\displaystyle {\overline {\zeta }}}
1092:coordinate and maps it to an angle,
329:adding citations to reliable sources
179:The concept of center of gravity or
155:located at the centers of mass (see
5482:Jong, I. G.; Rogers, B. G. (1995),
4460:obtained from the two experiments.
3867:{\displaystyle \mathbf {\hat {k}} }
5742:Encyclopedia of planetary sciences
5627:Fundamentals of Structural Geology
5003:Feynman, Leighton & Sands 1963
4638:that instrument, and use of their
2072:Center of gravity (disambiguation)
1503:{\displaystyle {\overline {\xi }}}
1215:. From this angle, two new points
116:or open-shaped objects, such as a
25:
5607:Essential Dynamics and Relativity
5468:, Jones & Bartlett Learning,
4958:, 14-3 âCenter of Gravityâ, Wiley
2799:which means the resultant torque
814:is the total mass in the volume.
5780:(3rd ed.), Addison-Wesley,
5681:Sangwin, Christopher J. (2006),
5665:Encyclopedia of Physical Science
5432:(3rd ed.), Addison-Wesley,
5409:(3rd ed.), Addison Wesley,
4684:
4511:Center of gravity of an aircraft
4421:
4402:
4384:
4344:
4329:
4314:
4299:
4284:
4269:
4252:
4223:
4185:
4170:
4155:
4140:
4125:
4110:
4091:
4070:
4033:
4021:
4007:
3989:
3977:
3963:
3945:
3933:
3919:
3907:
3855:
3826:
3810:
3672:
3664:
3622:
3608:
3578:
3564:
3521:
3512:
3501:
3472:
3464:
3410:
3399:
3385:
3370:
3356:
3329:
3272:
3258:
3228:
3214:
3171:
3144:
3087:
3073:
3010:
2982:
2971:
2957:
2924:
2914:
2903:
2889:
2871:
2765:
2757:
2744:
2686:
2649:
2641:
2628:
2589:
2571:
2543:
2535:
2504:
2496:
2485:
2477:
2456:
2424:
2395:
2367:
2333:
2325:
2307:
2280:{\textstyle \mathbf {\hat {k}} }
2268:
2224:
2203:
2181:
2155:
2147:
2016:
930:
905:
883:
786:
778:
744:
699:
691:
678:
563:
520:
487:
476:
462:
418:of the center of mass satisfy
305:
5690:Journal of the Oughtred Society
5486:, Saunders College Publishing,
5330:The Feynman Lectures on Physics
5283:Federal Aviation Administration
3703:law of conservation of momentum
3519:
2921:
2027:needs additional citations for
1067:For particles in a system with
646:relative to the center of mass
316:needs additional citations for
112:, as is sometimes the case for
5647:Engineering Mechanics: Statics
5631:, Cambridge University Press,
5591:, Cambridge University Press,
5535:, Princeton University Press,
5484:Engineering Mechanics: Statics
5362:, Cambridge University Press,
5264:De Silva, Clarence W. (2002),
5232:, Courier Dover Publications,
5210:10.1080/2151237X.2008.10129266
4394:
4388:
4354:
4264:
4214:* in the horizontal plane as,
4025:
4002:
3981:
3958:
3937:
3914:
3626:
3603:
3582:
3559:
3403:
3380:
3374:
3351:
3276:
3253:
3232:
3209:
3091:
3068:
2975:
2952:
2907:
2884:
2840:be located at the coordinates
2748:
2740:
2632:
2624:
2575:
2567:
2547:
2531:
2508:
2500:
2489:
2473:
2371:
2363:
2337:
2329:
2294:in the volume and compute the
2207:
2199:
2159:
2151:
1956:
1944:
1416:
1404:
1378:
1365:
1335:
1322:
1248:
1222:
1202:
1183:
782:
774:
682:
674:
480:
457:
283:expanded the concept further.
56:(sometimes referred to as the
1:
5879:The Solar System's barycenter
5515:(2nd ed.), McGraw-Hill,
4945:, 9-1 âCenter of Massâ, Wiley
4654:of individual body sections,
3720:Centroid § Determination
1422:{\displaystyle (\xi ,\zeta )}
1055:on this line, and are termed
839:Barycentric coordinate system
5948:Electronic Stability Control
5873:Motion of the Center of Mass
5605:O'Donnell, Peter J. (2015),
5511:An Introduction to Mechanics
5266:Vibration and shock handbook
5182:, Barnes & Noble Books,
5094:. p. 82. Archived from
4991:Kleppner & Kolenkow 1973
4968:Kleppner & Kolenkow 1973
3756:A direct development of the
3694:is the linear momentum, and
2858:. Select a reference point
1981:
1931:
1918:
1866:
1813:
1797:
1766:
1729:
1630:
1555:
1522:
1495:
1069:periodic boundary conditions
104:, it will be located at the
5553:, Oxford University Press,
5246:Beatty, Millard F. (2006),
4853:American Journal of Physics
2814:Linear and angular momentum
6209:
5812:Van Pelt, Michael (2005),
4628:
4568:Two bodies orbiting their
4557:
4508:
4478:Engineers try to design a
3717:
2298:and torque at this point,
2076:
2069:
1476:{\displaystyle \zeta _{i}}
1159:is the system size in the
1037:moves along the line from
1033:, then the center of mass
836:
507:Solving this equation for
5758:Shore, Steven N. (2008),
5119:Murray & Dermott 1999
5089:"Helicopter Aerodynamics"
3698:is the angular momentum.
2290:Choose a reference point
2250:is the mass at the point
257:Jean-Charles de la Faille
5855:Walton, William (1855),
5776:Symon, Keith R. (1971),
5568:Millikan, Robert Andrews
5464:Hamill, Patrick (2009),
4672:Center-of-gravity method
1449:{\displaystyle \xi _{i}}
143:of planetary bodies and
96:In the case of a single
5667:, Infobase Publishing,
5549:Mancosu, Paolo (1999),
5350:; Olenick, Richard P.;
4592:, or a planet orbits a
4474:Automotive applications
2716:If the reference point
1057:barycentric coordinates
833:Barycentric coordinates
91:Newton's laws of motion
5843:KIN 335 - Biomechanics
5403:; Safko, John (2001),
4847:Erlichson, H. (1996).
4573:
4437:
4364:
4202:
4056:
3868:
3839:
3729:
3680:
3652:
3548:
3480:
3447:
3307:
3198:
3155:
3127:
3057:
2993:
2793:
2707:
2440:
2281:
2240:
2088:
1990:
1963:
1888:
1738:
1704:
1672:
1597:
1531:
1504:
1477:
1450:
1423:
1389:
1282:
1255:
1209:
1146:
975:
804:
724:
642:of the points in this
616:
595:
550:
498:
446:
245:Evangelista Torricelli
196:Archimedes of Syracuse
157:Barycenter (astronomy)
37:
5589:Solar System Dynamics
5466:Intermediate Dynamics
5179:Understanding Physics
5005:, pp. 19.1â19.2.
4567:
4438:
4365:
4203:
4057:
3869:
3840:
3727:
3681:
3632:
3528:
3481:
3427:
3287:
3178:
3156:
3107:
3037:
2994:
2794:
2708:
2441:
2282:
2241:
2086:
1991:
1964:
1889:
1739:
1705:
1652:
1577:
1532:
1505:
1478:
1451:
1424:
1390:
1283:
1281:{\displaystyle x_{i}}
1256:
1210:
1147:
976:
837:Further information:
805:
725:
617:
575:
530:
499:
426:
367:A system of particles
48:of a distribution of
35:
5834:Vint, Peter (2003),
5348:Frautschi, Steven C.
4930:Bai & Breen 2008
4731:Circumcenter of mass
4711:Center of percussion
4380:
4218:
4066:
3903:
3849:
3806:
3497:
3167:
3006:
2866:
2724:
2452:
2303:
2262:
2143:
2036:improve this article
1973:
1905:
1754:
1721:
1543:
1514:
1487:
1460:
1433:
1401:
1292:
1265:
1219:
1167:
1096:
879:
740:
658:
516:
423:
325:improve this article
221:Pappus of Alexandria
161:center of mass frame
78:angular acceleration
6178:Classical mechanics
5915:Automotive handling
5762:, Greenwood Press,
5724:2006ppcb.book.....J
5696:(2), archived from
5529:Levi, Mark (2009),
5429:Classical Mechanics
5406:Classical Mechanics
5356:Goodstein, David L.
5321:Leighton, Robert B.
4993:, pp. 119â120.
4865:1996AmJPh..64..571E
4825:, pp. 101â106.
4756:Weight distribution
4741:Mass point geometry
4584:each other. When a
4469:Engineering designs
3777:In three dimensions
821:, which means that
626:A continuous volume
511:yields the formula
287:Newton's second law
241:Federico Commandino
237:Francesco Maurolico
145:rigid body dynamics
74:linear acceleration
5424:Goldstein, Herbert
5397:Goldstein, Herbert
5226:Baron, Margaret E.
4813:, pp. 96â101.
4746:Metacentric height
4644:physical principle
4640:static equilibrium
4612:Rigging and safety
4574:
4433:
4360:
4198:
4052:
3864:
3835:
3771:center of buoyancy
3730:
3708:Newton's Third Law
3676:
3476:
3151:
2989:
2789:
2703:
2436:
2277:
2236:
2089:
1986:
1959:
1884:
1882:
1734:
1700:
1698:
1527:
1500:
1473:
1446:
1419:
1385:
1383:
1278:
1251:
1205:
1142:
1073:molecular dynamics
971:
800:
720:
612:
494:
414:, the coordinates
269:Christiaan Huygens
217:Hero of Alexandria
209:On Floating Bodies
159:for details). The
38:
6188:Geometric centers
6165:
6164:
6161:
6160:
6154:
6151:Semi-trailing arm
6127:
6110:
5827:978-0-387-40213-0
5805:978-0-7167-0900-8
5787:978-0-201-07392-8
5769:978-0-313-33303-3
5751:978-0-412-06951-2
5733:978-0-534-49143-7
5674:978-0-8160-7011-4
5656:978-0-495-29559-4
5638:978-0-521-83927-3
5616:978-1-466-58839-4
5598:978-0-521-57295-8
5560:978-0-19-513244-1
5542:978-0-691-14020-9
5522:978-0-07-035048-9
5493:978-0-03-026309-5
5475:978-0-7637-5728-1
5457:978-0-486-42005-9
5439:978-0-201-65702-9
5416:978-0-201-65702-9
5389:978-0-07-110608-5
5369:978-0-521-30432-0
5340:978-0-201-02010-6
5275:978-0-8493-1580-0
5257:978-0-387-23704-6
5239:978-0-486-49544-6
5189:978-0-88029-251-1
5121:, pp. 45â47.
5017:, pp. 20â21.
4837:, pp. 56â61.
4801:, pp. 94â96.
4789:, pp. 91â94.
4427:
4258:
4247:
4097:
3861:
3832:
3741:In two dimensions
3728:Plumb line method
3601:
3349:
3251:
3030:
2950:
2692:
2595:
2430:
2401:
2274:
2230:
2187:
2068:
2067:
2060:
2008:Center of gravity
1984:
1934:
1921:
1878:
1869:
1840:
1816:
1800:
1769:
1744:, from which the
1732:
1650:
1633:
1575:
1558:
1525:
1498:
1134:
966:
759:
654:is zero, that is
634:) within a solid
607:
388:, each with mass
357:
356:
349:
291:Euler's first law
231:periods, work by
169:coordinate system
149:orbital mechanics
16:(Redirected from
6200:
6193:Moment (physics)
6148:
6121:
6119:MacPherson strut
6104:
6077:Semi-independent
6053:
5998:Vehicle dynamics
5909:
5902:
5895:
5886:
5862:
5851:
5850:
5849:
5840:
5830:
5819:
5808:
5790:
5772:
5754:
5736:
5710:
5709:
5708:
5702:
5687:
5677:
5659:
5641:
5630:
5619:
5601:
5583:
5582:
5581:
5563:
5545:
5525:
5514:
5505:Kolenkow, Robert
5501:Kleppner, Daniel
5496:
5478:
5460:
5442:
5419:
5392:
5372:
5343:
5317:Feynman, Richard
5312:
5311:
5310:
5304:
5298:, archived from
5293:
5278:
5260:
5242:
5221:
5192:
5161:
5160:, pp. 1â11.
5155:
5149:
5148:
5146:
5145:
5136:
5128:
5122:
5116:
5110:
5109:
5107:
5106:
5100:
5093:
5085:
5079:
5073:
5067:
5061:
5055:
5049:
5043:
5042:
5040:
5039:
5032:Amos Lowrey Ayre
5024:
5018:
5012:
5006:
5000:
4994:
4988:
4982:
4977:
4971:
4965:
4959:
4952:
4946:
4939:
4933:
4927:
4921:
4915:
4909:
4903:
4897:
4891:
4885:
4884:
4844:
4838:
4832:
4826:
4820:
4814:
4808:
4802:
4796:
4790:
4784:
4778:
4777:, pp. 9â11.
4772:
4694:
4689:
4688:
4606:Pluto and Charon
4442:
4440:
4439:
4434:
4429:
4428:
4420:
4411:
4410:
4405:
4387:
4369:
4367:
4366:
4361:
4353:
4352:
4347:
4338:
4337:
4332:
4323:
4322:
4317:
4308:
4307:
4302:
4293:
4292:
4287:
4278:
4277:
4272:
4260:
4259:
4251:
4248:
4240:
4232:
4231:
4226:
4207:
4205:
4204:
4199:
4194:
4193:
4188:
4179:
4178:
4173:
4164:
4163:
4158:
4149:
4148:
4143:
4134:
4133:
4128:
4119:
4118:
4113:
4104:
4100:
4099:
4098:
4090:
4073:
4061:
4059:
4058:
4053:
4042:
4041:
4036:
4024:
4016:
4015:
4010:
3998:
3997:
3992:
3980:
3972:
3971:
3966:
3954:
3953:
3948:
3936:
3928:
3927:
3922:
3910:
3873:
3871:
3870:
3865:
3863:
3862:
3854:
3844:
3842:
3841:
3836:
3834:
3833:
3825:
3813:
3685:
3683:
3682:
3677:
3675:
3667:
3662:
3661:
3651:
3646:
3625:
3617:
3616:
3611:
3602:
3600:
3589:
3581:
3573:
3572:
3567:
3558:
3557:
3547:
3542:
3524:
3515:
3504:
3486:
3483:
3482:
3477:
3475:
3467:
3462:
3458:
3457:
3456:
3446:
3441:
3418:
3414:
3413:
3402:
3394:
3393:
3388:
3373:
3365:
3364:
3359:
3350:
3348:
3337:
3332:
3322:
3318:
3317:
3316:
3306:
3301:
3275:
3267:
3266:
3261:
3252:
3250:
3239:
3231:
3223:
3222:
3217:
3208:
3207:
3197:
3192:
3174:
3161:
3158:
3157:
3152:
3147:
3142:
3138:
3137:
3136:
3126:
3121:
3098:
3094:
3090:
3082:
3081:
3076:
3067:
3066:
3056:
3051:
3031:
3029:
3018:
3013:
2998:
2996:
2995:
2990:
2985:
2974:
2966:
2965:
2960:
2951:
2949:
2938:
2933:
2932:
2927:
2917:
2906:
2898:
2897:
2892:
2880:
2879:
2874:
2849:with velocities
2805:
2798:
2796:
2795:
2790:
2773:
2769:
2768:
2760:
2747:
2736:
2735:
2713:
2710:
2709:
2704:
2699:
2695:
2694:
2693:
2685:
2668:
2664:
2657:
2653:
2652:
2644:
2631:
2620:
2619:
2602:
2598:
2597:
2596:
2588:
2574:
2546:
2538:
2530:
2529:
2507:
2499:
2488:
2480:
2472:
2471:
2459:
2446:
2443:
2442:
2437:
2432:
2431:
2423:
2408:
2404:
2403:
2402:
2394:
2370:
2359:
2358:
2336:
2328:
2323:
2322:
2310:
2286:
2284:
2283:
2278:
2276:
2275:
2267:
2245:
2243:
2242:
2237:
2232:
2231:
2223:
2206:
2189:
2188:
2180:
2158:
2150:
2127:) at each point
2093:resultant torque
2063:
2056:
2052:
2049:
2043:
2020:
2012:
1995:
1993:
1992:
1987:
1985:
1977:
1968:
1966:
1965:
1960:
1940:
1936:
1935:
1927:
1922:
1914:
1899:cluster analysis
1893:
1891:
1890:
1885:
1883:
1879:
1877:
1862:
1861:
1859:
1858:
1842:
1841:
1838:
1822:
1818:
1817:
1809:
1801:
1793:
1770:
1762:
1743:
1741:
1740:
1735:
1733:
1725:
1713:
1709:
1707:
1706:
1701:
1699:
1692:
1691:
1682:
1681:
1671:
1666:
1651:
1643:
1634:
1626:
1617:
1616:
1607:
1606:
1596:
1591:
1576:
1568:
1559:
1551:
1537:are calculated.
1536:
1534:
1533:
1528:
1526:
1518:
1509:
1507:
1506:
1501:
1499:
1491:
1482:
1480:
1479:
1474:
1472:
1471:
1455:
1453:
1452:
1447:
1445:
1444:
1428:
1426:
1425:
1420:
1394:
1392:
1391:
1386:
1384:
1377:
1376:
1351:
1350:
1334:
1333:
1308:
1307:
1287:
1285:
1284:
1279:
1277:
1276:
1260:
1258:
1257:
1252:
1247:
1246:
1234:
1233:
1214:
1212:
1211:
1206:
1201:
1200:
1179:
1178:
1151:
1149:
1148:
1143:
1135:
1133:
1132:
1123:
1122:
1113:
1108:
1107:
980:
978:
977:
972:
967:
965:
964:
963:
951:
950:
940:
939:
938:
933:
927:
926:
914:
913:
908:
902:
901:
891:
886:
843:The coordinates
809:
807:
806:
801:
789:
781:
770:
769:
760:
752:
747:
729:
727:
726:
721:
707:
703:
702:
694:
681:
670:
669:
650:over the volume
622:
619:
618:
613:
608:
606:
605:
604:
594:
589:
573:
572:
571:
566:
560:
559:
549:
544:
528:
523:
504:
501:
500:
495:
490:
479:
471:
470:
465:
456:
455:
445:
440:
413:
394:
387:
352:
345:
341:
338:
332:
309:
301:
141:angular momentum
21:
6208:
6207:
6203:
6202:
6201:
6199:
6198:
6197:
6168:
6167:
6166:
6157:
6102:Double wishbone
6088:
6072:
6041:
6007:
6003:Weight transfer
5916:
5913:
5869:
5854:
5847:
5845:
5838:
5833:
5828:
5811:
5806:
5793:
5788:
5775:
5770:
5757:
5752:
5739:
5734:
5713:
5706:
5704:
5700:
5685:
5680:
5675:
5662:
5657:
5644:
5639:
5622:
5617:
5604:
5599:
5586:
5579:
5577:
5566:
5561:
5548:
5543:
5528:
5523:
5499:
5494:
5481:
5476:
5463:
5458:
5445:
5440:
5422:
5417:
5395:
5390:
5379:College physics
5375:
5370:
5352:Apostol, Tom M.
5346:
5341:
5315:
5308:
5306:
5302:
5291:
5281:
5276:
5263:
5258:
5245:
5240:
5224:
5195:
5190:
5172:
5169:
5164:
5156:
5152:
5143:
5141:
5134:
5130:
5129:
5125:
5117:
5113:
5104:
5102:
5098:
5091:
5087:
5086:
5082:
5074:
5070:
5062:
5058:
5050:
5046:
5037:
5035:
5026:
5025:
5021:
5013:
5009:
5001:
4997:
4989:
4985:
4978:
4974:
4966:
4962:
4953:
4949:
4940:
4936:
4928:
4924:
4916:
4912:
4904:
4900:
4892:
4888:
4873:10.1119/1.18156
4846:
4845:
4841:
4833:
4829:
4821:
4817:
4809:
4805:
4797:
4793:
4785:
4781:
4773:
4769:
4765:
4760:
4690:
4683:
4680:
4668:
4633:
4627:
4614:
4562:
4556:
4513:
4507:
4476:
4471:
4466:
4459:
4452:
4400:
4378:
4377:
4342:
4327:
4312:
4297:
4282:
4267:
4221:
4216:
4215:
4183:
4168:
4153:
4138:
4123:
4108:
4081:
4077:
4064:
4063:
4031:
4005:
3987:
3961:
3943:
3917:
3901:
3900:
3894:
3887:
3880:
3847:
3846:
3804:
3803:
3801:
3794:
3787:
3779:
3743:
3722:
3716:
3653:
3606:
3593:
3562:
3549:
3495:
3494:
3448:
3426:
3422:
3383:
3354:
3341:
3327:
3323:
3308:
3286:
3282:
3256:
3243:
3212:
3199:
3165:
3163:
3128:
3106:
3102:
3071:
3058:
3036:
3032:
3022:
3004:
3002:
2955:
2942:
2922:
2887:
2869:
2864:
2863:
2857:
2848:
2838:
2823:
2816:
2800:
2755:
2751:
2727:
2722:
2721:
2676:
2672:
2639:
2635:
2611:
2610:
2606:
2557:
2553:
2521:
2463:
2450:
2448:
2385:
2381:
2350:
2314:
2301:
2299:
2296:resultant force
2260:
2259:
2141:
2140:
2081:
2075:
2064:
2053:
2047:
2044:
2033:
2021:
2010:
2002:periodic system
1971:
1970:
1912:
1908:
1903:
1902:
1881:
1880:
1870:
1850:
1843:
1833:
1830:
1829:
1788:
1784:
1771:
1752:
1751:
1719:
1718:
1711:
1697:
1696:
1683:
1673:
1635:
1622:
1621:
1608:
1598:
1560:
1541:
1540:
1512:
1511:
1485:
1484:
1463:
1458:
1457:
1436:
1431:
1430:
1399:
1398:
1382:
1381:
1368:
1352:
1342:
1339:
1338:
1325:
1309:
1299:
1290:
1289:
1268:
1263:
1262:
1238:
1225:
1217:
1216:
1192:
1170:
1165:
1164:
1158:
1124:
1114:
1099:
1094:
1093:
1065:
1050:
1043:
1032:
1025:
1018:
1011:
1004:
997:
991:vary from 100%
955:
942:
941:
928:
918:
903:
893:
877:
876:
874:
867:
860:
853:
841:
835:
829:of the volume.
761:
738:
737:
689:
685:
661:
656:
655:
628:
596:
574:
561:
551:
529:
514:
512:
460:
447:
421:
419:
404:
396:
393:
389:
377:
372:
369:
353:
342:
336:
333:
322:
310:
299:
281:Alexis Clairaut
277:Pierre Varignon
177:
28:
23:
22:
15:
12:
11:
5:
6206:
6204:
6196:
6195:
6190:
6185:
6180:
6170:
6169:
6163:
6162:
6159:
6158:
6156:
6155:
6143:
6138:
6136:Sliding pillar
6133:
6128:
6116:
6111:
6098:
6096:
6090:
6089:
6087:
6086:
6080:
6078:
6074:
6073:
6071:
6070:
6065:
6059:
6057:
6050:
6043:
6042:
6040:
6039:
6034:
6029:
6024:
6018:
6016:
6009:
6008:
6006:
6005:
6000:
5995:
5990:
5985:
5980:
5975:
5970:
5965:
5960:
5955:
5950:
5945:
5940:
5935:
5933:Center of mass
5930:
5924:
5922:
5918:
5917:
5914:
5912:
5911:
5904:
5897:
5889:
5883:
5882:
5876:
5868:
5867:External links
5865:
5864:
5863:
5852:
5831:
5826:
5809:
5804:
5791:
5786:
5773:
5768:
5755:
5750:
5737:
5732:
5711:
5678:
5673:
5660:
5655:
5642:
5637:
5620:
5615:
5602:
5597:
5584:
5564:
5559:
5546:
5541:
5526:
5521:
5497:
5492:
5479:
5474:
5461:
5456:
5443:
5438:
5420:
5415:
5401:Poole, Charles
5393:
5388:
5373:
5368:
5344:
5339:
5325:Sands, Matthew
5313:
5279:
5274:
5261:
5256:
5243:
5238:
5222:
5193:
5188:
5168:
5165:
5163:
5162:
5150:
5123:
5111:
5080:
5078:, p. 1.3.
5068:
5066:, p. 1.4.
5056:
5044:
5019:
5007:
4995:
4983:
4972:
4970:, p. 117.
4960:
4947:
4934:
4922:
4910:
4898:
4886:
4859:(5): 571â574.
4839:
4827:
4815:
4803:
4791:
4779:
4766:
4764:
4761:
4759:
4758:
4753:
4748:
4743:
4738:
4736:Expected value
4733:
4728:
4723:
4718:
4713:
4708:
4703:
4697:
4696:
4695:
4692:Physics portal
4679:
4676:
4667:
4664:
4629:Main article:
4626:
4623:
4613:
4610:
4558:Main article:
4555:
4552:
4509:Main article:
4506:
4503:
4475:
4472:
4470:
4467:
4465:
4462:
4457:
4450:
4432:
4426:
4423:
4417:
4414:
4409:
4404:
4399:
4396:
4393:
4390:
4386:
4359:
4356:
4351:
4346:
4341:
4336:
4331:
4326:
4321:
4316:
4311:
4306:
4301:
4296:
4291:
4286:
4281:
4276:
4271:
4266:
4263:
4257:
4254:
4246:
4243:
4238:
4235:
4230:
4225:
4197:
4192:
4187:
4182:
4177:
4172:
4167:
4162:
4157:
4152:
4147:
4142:
4137:
4132:
4127:
4122:
4117:
4112:
4107:
4103:
4096:
4093:
4087:
4084:
4080:
4076:
4072:
4051:
4048:
4045:
4040:
4035:
4030:
4027:
4023:
4019:
4014:
4009:
4004:
4001:
3996:
3991:
3986:
3983:
3979:
3975:
3970:
3965:
3960:
3957:
3952:
3947:
3942:
3939:
3935:
3931:
3926:
3921:
3916:
3913:
3909:
3892:
3885:
3878:
3860:
3857:
3831:
3828:
3822:
3819:
3816:
3812:
3799:
3792:
3785:
3778:
3775:
3742:
3739:
3715:
3712:
3674:
3670:
3666:
3660:
3656:
3650:
3645:
3642:
3639:
3635:
3631:
3628:
3624:
3620:
3615:
3610:
3605:
3599:
3596:
3592:
3587:
3584:
3580:
3576:
3571:
3566:
3561:
3556:
3552:
3546:
3541:
3538:
3535:
3531:
3527:
3523:
3518:
3514:
3510:
3507:
3503:
3474:
3470:
3466:
3461:
3455:
3451:
3445:
3440:
3437:
3434:
3430:
3425:
3421:
3417:
3412:
3408:
3405:
3401:
3397:
3392:
3387:
3382:
3379:
3376:
3372:
3368:
3363:
3358:
3353:
3347:
3344:
3340:
3335:
3331:
3326:
3321:
3315:
3311:
3305:
3300:
3297:
3294:
3290:
3285:
3281:
3278:
3274:
3270:
3265:
3260:
3255:
3249:
3246:
3242:
3237:
3234:
3230:
3226:
3221:
3216:
3211:
3206:
3202:
3196:
3191:
3188:
3185:
3181:
3177:
3173:
3150:
3146:
3141:
3135:
3131:
3125:
3120:
3117:
3114:
3110:
3105:
3101:
3097:
3093:
3089:
3085:
3080:
3075:
3070:
3065:
3061:
3055:
3050:
3047:
3044:
3040:
3035:
3028:
3025:
3021:
3016:
3012:
2988:
2984:
2980:
2977:
2973:
2969:
2964:
2959:
2954:
2948:
2945:
2941:
2936:
2931:
2926:
2920:
2916:
2912:
2909:
2905:
2901:
2896:
2891:
2886:
2883:
2878:
2873:
2853:
2844:
2836:
2821:
2815:
2812:
2788:
2785:
2782:
2779:
2776:
2772:
2767:
2763:
2759:
2754:
2750:
2746:
2742:
2739:
2734:
2730:
2702:
2698:
2691:
2688:
2682:
2679:
2675:
2671:
2667:
2663:
2660:
2656:
2651:
2647:
2643:
2638:
2634:
2630:
2626:
2623:
2618:
2614:
2609:
2605:
2601:
2594:
2591:
2584:
2581:
2577:
2573:
2569:
2566:
2563:
2560:
2556:
2552:
2549:
2545:
2541:
2537:
2533:
2528:
2524:
2520:
2517:
2514:
2510:
2506:
2502:
2498:
2494:
2491:
2487:
2483:
2479:
2475:
2470:
2466:
2462:
2458:
2435:
2429:
2426:
2420:
2417:
2414:
2411:
2407:
2400:
2397:
2391:
2388:
2384:
2380:
2377:
2373:
2369:
2365:
2362:
2357:
2353:
2349:
2346:
2343:
2339:
2335:
2331:
2327:
2321:
2317:
2313:
2309:
2273:
2270:
2235:
2229:
2226:
2220:
2216:
2213:
2209:
2205:
2201:
2198:
2195:
2192:
2186:
2183:
2177:
2173:
2170:
2167:
2164:
2161:
2157:
2153:
2149:
2135:at each point
2077:Main article:
2066:
2065:
2048:September 2024
2024:
2022:
2015:
2009:
2006:
1983:
1980:
1958:
1955:
1952:
1949:
1946:
1943:
1939:
1933:
1930:
1925:
1920:
1917:
1911:
1876:
1873:
1868:
1865:
1857:
1853:
1849:
1846:
1844:
1836:
1832:
1831:
1828:
1825:
1821:
1815:
1812:
1807:
1804:
1799:
1796:
1791:
1787:
1783:
1780:
1777:
1774:
1772:
1768:
1765:
1760:
1759:
1731:
1728:
1695:
1690:
1686:
1680:
1676:
1670:
1665:
1662:
1659:
1655:
1649:
1646:
1641:
1638:
1636:
1632:
1629:
1624:
1623:
1620:
1615:
1611:
1605:
1601:
1595:
1590:
1587:
1584:
1580:
1574:
1571:
1566:
1563:
1561:
1557:
1554:
1549:
1548:
1524:
1521:
1497:
1494:
1470:
1466:
1443:
1439:
1418:
1415:
1412:
1409:
1406:
1380:
1375:
1371:
1367:
1364:
1361:
1358:
1355:
1353:
1349:
1345:
1341:
1340:
1337:
1332:
1328:
1324:
1321:
1318:
1315:
1312:
1310:
1306:
1302:
1298:
1297:
1275:
1271:
1250:
1245:
1241:
1237:
1232:
1228:
1224:
1204:
1199:
1195:
1191:
1188:
1185:
1182:
1177:
1173:
1163:direction and
1156:
1141:
1138:
1131:
1127:
1121:
1117:
1111:
1106:
1102:
1064:
1061:
1048:
1041:
1030:
1023:
1016:
1009:
1002:
995:
970:
962:
958:
954:
949:
945:
937:
932:
925:
921:
917:
912:
907:
900:
896:
889:
885:
872:
865:
861:, with masses
858:
851:
834:
831:
799:
796:
793:
788:
784:
780:
776:
773:
768:
764:
758:
755:
750:
746:
719:
716:
713:
710:
706:
701:
697:
693:
688:
684:
680:
676:
673:
668:
664:
627:
624:
611:
603:
599:
593:
588:
585:
582:
578:
570:
565:
558:
554:
548:
543:
540:
537:
533:
526:
522:
493:
489:
485:
482:
478:
474:
469:
464:
459:
454:
450:
444:
439:
436:
433:
429:
400:
391:
375:
368:
365:
355:
354:
337:September 2024
313:
311:
304:
298:
295:
176:
173:
165:inertial frame
46:center of mass
26:
24:
18:Centre of mass
14:
13:
10:
9:
6:
4:
3:
2:
6205:
6194:
6191:
6189:
6186:
6184:
6181:
6179:
6176:
6175:
6173:
6152:
6147:
6144:
6142:
6139:
6137:
6134:
6132:
6129:
6125:
6124:Chapman strut
6120:
6117:
6115:
6112:
6108:
6103:
6100:
6099:
6097:
6095:
6091:
6085:
6082:
6081:
6079:
6075:
6069:
6066:
6064:
6061:
6060:
6058:
6054:
6051:
6048:
6044:
6038:
6035:
6033:
6030:
6028:
6025:
6023:
6020:
6019:
6017:
6014:
6010:
6004:
6001:
5999:
5996:
5994:
5993:Unsprung mass
5991:
5989:
5986:
5984:
5981:
5979:
5976:
5974:
5971:
5969:
5966:
5964:
5961:
5959:
5958:Inboard brake
5956:
5954:
5951:
5949:
5946:
5944:
5941:
5939:
5936:
5934:
5931:
5929:
5926:
5925:
5923:
5919:
5910:
5905:
5903:
5898:
5896:
5891:
5890:
5887:
5880:
5877:
5874:
5871:
5870:
5866:
5860:
5859:
5853:
5844:
5837:
5832:
5829:
5823:
5818:
5817:
5810:
5807:
5801:
5797:
5792:
5789:
5783:
5779:
5774:
5771:
5765:
5761:
5756:
5753:
5747:
5743:
5738:
5735:
5729:
5725:
5721:
5717:
5712:
5703:on 2011-10-05
5699:
5695:
5691:
5684:
5679:
5676:
5670:
5666:
5661:
5658:
5652:
5648:
5643:
5640:
5634:
5629:
5628:
5621:
5618:
5612:
5609:, CRC Press,
5608:
5603:
5600:
5594:
5590:
5585:
5575:
5574:
5569:
5565:
5562:
5556:
5552:
5547:
5544:
5538:
5534:
5533:
5527:
5524:
5518:
5513:
5512:
5506:
5502:
5498:
5495:
5489:
5485:
5480:
5477:
5471:
5467:
5462:
5459:
5453:
5449:
5444:
5441:
5435:
5431:
5430:
5425:
5421:
5418:
5412:
5408:
5407:
5402:
5398:
5394:
5391:
5385:
5381:
5380:
5374:
5371:
5365:
5361:
5357:
5353:
5349:
5345:
5342:
5336:
5332:
5331:
5326:
5322:
5318:
5314:
5305:on 2011-10-19
5301:
5297:
5290:
5289:
5284:
5280:
5277:
5271:
5268:, CRC Press,
5267:
5262:
5259:
5253:
5249:
5244:
5241:
5235:
5231:
5227:
5223:
5219:
5215:
5211:
5207:
5203:
5199:
5194:
5191:
5185:
5181:
5180:
5175:
5174:Asimov, Isaac
5171:
5170:
5166:
5159:
5154:
5151:
5140:
5133:
5127:
5124:
5120:
5115:
5112:
5101:on 2012-03-24
5097:
5090:
5084:
5081:
5077:
5072:
5069:
5065:
5060:
5057:
5053:
5048:
5045:
5033:
5029:
5023:
5020:
5016:
5011:
5008:
5004:
4999:
4996:
4992:
4987:
4984:
4981:
4976:
4973:
4969:
4964:
4961:
4957:
4951:
4948:
4944:
4938:
4935:
4931:
4926:
4923:
4920:, p. 85.
4919:
4914:
4911:
4908:, p. 29.
4907:
4902:
4899:
4895:
4890:
4887:
4882:
4878:
4874:
4870:
4866:
4862:
4858:
4854:
4850:
4843:
4840:
4836:
4831:
4828:
4824:
4819:
4816:
4812:
4807:
4804:
4800:
4795:
4792:
4788:
4783:
4780:
4776:
4771:
4768:
4762:
4757:
4754:
4752:
4749:
4747:
4744:
4742:
4739:
4737:
4734:
4732:
4729:
4727:
4724:
4722:
4719:
4717:
4714:
4712:
4709:
4707:
4704:
4702:
4699:
4698:
4693:
4687:
4682:
4677:
4675:
4673:
4665:
4663:
4661:
4657:
4653:
4649:
4645:
4641:
4635:
4632:
4624:
4622:
4619:
4611:
4609:
4607:
4603:
4599:
4595:
4591:
4587:
4583:
4579:
4571:
4566:
4561:
4553:
4551:
4549:
4545:
4541:
4537:
4532:
4530:
4526:
4522:
4521:forward limit
4518:
4512:
4504:
4502:
4500:
4496:
4492:
4487:
4485:
4481:
4473:
4468:
4463:
4461:
4456:
4449:
4443:
4430:
4415:
4412:
4407:
4397:
4391:
4375:
4370:
4357:
4349:
4339:
4334:
4324:
4319:
4309:
4304:
4294:
4289:
4279:
4274:
4261:
4244:
4241:
4236:
4233:
4228:
4213:
4208:
4195:
4190:
4180:
4175:
4165:
4160:
4150:
4145:
4135:
4130:
4120:
4115:
4105:
4101:
4085:
4082:
4078:
4074:
4049:
4046:
4043:
4038:
4028:
4017:
4012:
3999:
3994:
3984:
3973:
3968:
3955:
3950:
3940:
3929:
3924:
3911:
3898:
3891:
3884:
3877:
3820:
3817:
3814:
3798:
3791:
3784:
3776:
3774:
3772:
3768:
3763:
3759:
3754:
3750:
3748:
3740:
3738:
3734:
3726:
3721:
3714:Determination
3713:
3711:
3709:
3704:
3699:
3697:
3693:
3689:
3668:
3658:
3654:
3648:
3643:
3640:
3637:
3633:
3629:
3618:
3613:
3597:
3594:
3590:
3585:
3574:
3569:
3554:
3550:
3544:
3539:
3536:
3533:
3529:
3525:
3516:
3508:
3505:
3492:
3487:
3485:
3468:
3459:
3453:
3449:
3443:
3438:
3435:
3432:
3428:
3423:
3419:
3415:
3406:
3395:
3390:
3377:
3366:
3361:
3345:
3342:
3338:
3333:
3324:
3319:
3313:
3309:
3303:
3298:
3295:
3292:
3288:
3283:
3279:
3268:
3263:
3247:
3244:
3240:
3235:
3224:
3219:
3204:
3200:
3194:
3189:
3186:
3183:
3179:
3175:
3160:
3148:
3139:
3133:
3129:
3123:
3118:
3115:
3112:
3108:
3103:
3099:
3095:
3083:
3078:
3063:
3059:
3053:
3048:
3045:
3042:
3038:
3033:
3026:
3023:
3019:
3014:
2999:
2986:
2978:
2967:
2962:
2946:
2943:
2939:
2934:
2929:
2918:
2910:
2899:
2894:
2881:
2876:
2861:
2856:
2852:
2847:
2843:
2839:
2832:
2828:
2824:
2813:
2811:
2807:
2803:
2786:
2783:
2780:
2777:
2774:
2770:
2761:
2752:
2737:
2732:
2728:
2719:
2714:
2712:
2700:
2696:
2680:
2677:
2673:
2669:
2665:
2661:
2658:
2654:
2645:
2636:
2621:
2616:
2612:
2607:
2603:
2599:
2582:
2579:
2564:
2561:
2558:
2554:
2550:
2539:
2526:
2522:
2518:
2515:
2512:
2492:
2481:
2468:
2464:
2460:
2445:
2433:
2418:
2415:
2412:
2409:
2405:
2389:
2386:
2382:
2378:
2375:
2360:
2355:
2351:
2347:
2344:
2341:
2319:
2315:
2311:
2297:
2293:
2288:
2257:
2253:
2249:
2233:
2218:
2214:
2211:
2196:
2193:
2190:
2175:
2171:
2168:
2165:
2162:
2139:is given by,
2138:
2134:
2130:
2126:
2122:
2119:with density
2118:
2114:
2110:
2105:
2101:
2097:
2094:
2085:
2080:
2073:
2062:
2059:
2051:
2041:
2037:
2031:
2030:
2025:This section
2023:
2019:
2014:
2013:
2007:
2005:
2003:
1999:
1978:
1953:
1950:
1947:
1941:
1937:
1928:
1923:
1915:
1909:
1900:
1894:
1874:
1871:
1863:
1851:
1847:
1845:
1834:
1826:
1823:
1819:
1810:
1805:
1802:
1794:
1789:
1785:
1781:
1778:
1775:
1773:
1763:
1749:
1747:
1726:
1715:
1693:
1688:
1684:
1678:
1674:
1668:
1663:
1660:
1657:
1653:
1647:
1644:
1639:
1637:
1627:
1618:
1613:
1609:
1603:
1599:
1593:
1588:
1585:
1582:
1578:
1572:
1569:
1564:
1562:
1552:
1538:
1519:
1492:
1468:
1464:
1441:
1437:
1413:
1410:
1407:
1395:
1373:
1369:
1362:
1359:
1356:
1354:
1347:
1343:
1330:
1326:
1319:
1316:
1313:
1311:
1304:
1300:
1273:
1269:
1243:
1239:
1235:
1230:
1226:
1193:
1189:
1186:
1180:
1175:
1171:
1162:
1155:
1139:
1136:
1125:
1119:
1115:
1109:
1104:
1100:
1091:
1087:
1083:
1079:
1074:
1070:
1062:
1060:
1058:
1054:
1047:
1040:
1036:
1029:
1022:
1015:
1008:
1001:
994:
990:
986:
981:
968:
960:
956:
952:
947:
943:
935:
923:
919:
915:
910:
898:
894:
887:
871:
864:
857:
850:
846:
840:
832:
830:
828:
824:
820:
815:
813:
797:
794:
791:
771:
766:
762:
756:
753:
748:
735:
730:
717:
714:
711:
708:
704:
695:
686:
671:
666:
662:
653:
649:
645:
641:
637:
633:
625:
623:
621:
609:
601:
597:
591:
586:
583:
580:
576:
568:
556:
552:
546:
541:
538:
535:
531:
524:
510:
505:
503:
491:
483:
472:
467:
452:
448:
442:
437:
434:
431:
427:
417:
412:
408:
403:
399:
386:
382:
378:
366:
364:
362:
351:
348:
340:
330:
326:
320:
319:
314:This section
312:
308:
303:
302:
296:
294:
292:
288:
284:
282:
278:
274:
270:
266:
262:
258:
254:
250:
246:
242:
238:
234:
230:
226:
222:
218:
213:
211:
210:
205:
202:exerted on a
201:
197:
194:
190:
186:
185:mathematician
182:
174:
172:
170:
166:
162:
158:
154:
150:
146:
142:
138:
134:
129:
127:
123:
119:
115:
111:
110:physical body
107:
103:
99:
94:
92:
87:
83:
79:
75:
71:
67:
63:
62:balance point
59:
55:
51:
47:
43:
34:
30:
19:
6146:Trailing arm
6068:De Dion tube
5932:
5928:Car handling
5857:
5846:, retrieved
5842:
5820:, Springer,
5815:
5795:
5777:
5759:
5744:, Springer,
5741:
5715:
5705:, retrieved
5698:the original
5693:
5689:
5664:
5646:
5626:
5606:
5588:
5578:, retrieved
5572:
5550:
5531:
5510:
5483:
5465:
5447:
5428:
5405:
5378:
5359:
5329:
5307:, retrieved
5300:the original
5287:
5265:
5247:
5229:
5204:(4): 53â60.
5201:
5197:
5178:
5153:
5142:. Retrieved
5138:
5126:
5114:
5103:. Retrieved
5096:the original
5083:
5071:
5059:
5054:, p. 7.
5052:Sangwin 2006
5047:
5036:. Retrieved
5031:
5022:
5010:
4998:
4986:
4975:
4963:
4955:
4950:
4942:
4937:
4925:
4913:
4901:
4896:, p. 2.
4889:
4856:
4852:
4842:
4835:Mancosu 1999
4830:
4818:
4806:
4794:
4782:
4770:
4669:
4666:Optimization
4658:a specified
4636:
4634:
4615:
4577:
4575:
4533:
4514:
4495:rolling over
4488:
4477:
4464:Applications
4454:
4447:
4444:
4373:
4371:
4211:
4209:
3896:
3889:
3882:
3875:
3796:
3789:
3782:
3780:
3767:displacement
3755:
3751:
3744:
3735:
3731:
3700:
3695:
3691:
3687:
3490:
3488:
3000:
2859:
2854:
2850:
2845:
2841:
2834:
2830:
2826:
2819:
2817:
2808:
2801:
2717:
2715:
2291:
2289:
2255:
2251:
2247:
2136:
2132:
2128:
2124:
2120:
2116:
2112:
2106:
2102:
2098:
2090:
2054:
2045:
2034:Please help
2029:verification
2026:
1997:
1895:
1750:
1745:
1716:
1539:
1396:
1160:
1153:
1089:
1085:
1081:
1077:
1066:
1052:
1045:
1038:
1034:
1027:
1020:
1013:
1006:
1005:through 50%
999:
992:
982:
875:is given by
869:
862:
855:
848:
844:
842:
822:
816:
811:
733:
731:
651:
647:
635:
631:
629:
508:
506:
415:
410:
406:
401:
397:
384:
380:
373:
370:
358:
343:
334:
323:Please help
318:verification
315:
285:
253:Luca Valerio
249:Simon Stevin
233:Guido Ubaldi
229:Early Modern
214:
207:
178:
153:point masses
130:
126:Solar System
95:
82:Calculations
61:
57:
45:
39:
29:
6094:Independent
5978:Tire / Tyre
5953:Fishtailing
5921:Main topics
5034:. p. 3
5015:Hamill 2009
4906:Beatty 2006
4894:Walton 1855
4751:Roll center
4656:relative to
4631:Kinesiology
4625:Body motion
4572:(red cross)
4536:helicopters
4531:condition.
4505:Aeronautics
4376:, given by
3747:plumb lines
640:coordinates
273:Louis Carré
265:John Wallis
261:Paul Guldin
225:Renaissance
76:without an
6172:Categories
6141:Swing axle
6131:Multi-link
6107:Jaguar IRS
6084:Twist beam
6047:Suspension
5988:Understeer
5973:Suspension
5848:2013-10-18
5707:2011-10-23
5580:2011-05-25
5309:2011-10-23
5167:References
5144:2019-11-27
5105:2013-11-23
5038:2012-08-20
4823:Baron 2004
4811:Baron 2004
4799:Baron 2004
4787:Baron 2004
4775:Shore 2008
4701:Barycenter
4578:barycenter
4570:barycenter
4560:Barycenter
4499:horizontal
4480:sports car
3758:planimeter
3718:See also:
2833:of masses
2829:= 1, ...,
2115:of volume
985:percentage
736:to obtain
409:= 1,â...,
383:= 1,â...,
297:Definition
98:rigid body
58:barycenter
6063:Beam axle
6056:Dependent
6032:Pneumatic
5983:Transaxle
5963:Oversteer
5938:Downforce
5778:Mechanics
5450:, Dover,
5228:(2004) ,
5176:(1988) ,
5158:Vint 2003
4918:Levi 2009
4881:0002-9505
4648:summation
4646:that the
4588:orbits a
4554:Astronomy
4544:rotorhead
4425:^
4408:∗
4340:×
4310:×
4280:×
4262:×
4256:^
4237:−
4229:∗
4181:×
4151:×
4121:×
4095:^
4083:−
4075:×
4029:×
4018:−
3985:×
3974:−
3941:×
3930:−
3859:^
3830:^
3818:−
3669:×
3634:∑
3619:−
3586:×
3575:−
3530:∑
3469:×
3429:∑
3407:×
3396:−
3367:−
3334:×
3289:∑
3269:−
3236:×
3225:−
3180:∑
3109:∑
3084:−
3039:∑
2968:−
2900:−
2762:−
2738:ρ
2729:∭
2690:^
2678:−
2670:×
2646:−
2622:ρ
2613:∭
2593:^
2565:ρ
2559:−
2551:×
2540:−
2523:∭
2493:×
2482:−
2465:∭
2428:^
2413:−
2399:^
2387:−
2361:ρ
2352:∭
2316:∭
2272:^
2228:^
2197:ρ
2194:−
2185:^
2166:−
2109:resultant
1982:¯
1979:θ
1932:¯
1929:ζ
1919:¯
1916:ξ
1875:π
1867:¯
1864:θ
1827:π
1814:¯
1811:ξ
1806:−
1798:¯
1795:ζ
1790:−
1782:
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1764:θ
1730:¯
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1685:ζ
1654:∑
1631:¯
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1610:ξ
1579:∑
1556:¯
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1523:¯
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1496:¯
1493:ξ
1465:ζ
1438:ξ
1414:ζ
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1370:θ
1363:
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1327:θ
1320:
1301:ξ
1240:ζ
1227:ξ
1181:∈
1140:π
1101:θ
1026:and 100%
989:particles
772:ρ
763:∭
696:−
672:ρ
663:∭
577:∑
532:∑
473:−
428:∑
223:. In the
189:physicist
133:mechanics
118:horseshoe
86:mechanics
68:relative
6114:Dubonnet
5968:Steering
5943:Drifting
5570:(1902),
5507:(1973),
5358:(1986),
5327:(1963),
5285:(2007),
5218:40807367
5139:FEMA.gov
4726:Centroid
4706:Buoyancy
4678:See also
4525:elevator
4517:aircraft
3762:centroid
1012:and 50%
983:Let the
827:centroid
193:engineer
106:centroid
70:position
66:weighted
6037:Torsion
5720:Bibcode
5448:Statics
4956:Physics
4943:Physics
4861:Bibcode
4652:torques
4650:of the
4618:rigging
4529:stalled
1969:, then
1397:In the
1084:and/or
998:and 0%
819:density
361:vectors
175:History
124:of the
122:planets
102:density
42:physics
6013:Spring
5824:
5802:
5784:
5766:
5748:
5730:
5671:
5653:
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4548:cyclic
4491:Humvee
4484:handle
3888:, and
3795:, and
3686:where
2246:where
1710:where
1152:where
1019:to 0%
810:Where
644:volume
279:, and
200:torque
191:, and
181:weight
163:is an
137:linear
114:hollow
44:, the
6049:types
6015:types
5839:(PDF)
5701:(PDF)
5686:(PDF)
5303:(PDF)
5292:(PDF)
5214:S2CID
5135:(PDF)
5099:(PDF)
5092:(PDF)
4763:Notes
4598:Earth
4582:orbit
4540:hover
1779:atan2
204:lever
147:. In
54:space
6183:Mass
6027:Leaf
6022:Coil
5822:ISBN
5800:ISBN
5782:ISBN
5764:ISBN
5746:ISBN
5728:ISBN
5669:ISBN
5651:ISBN
5633:ISBN
5611:ISBN
5593:ISBN
5555:ISBN
5537:ISBN
5517:ISBN
5488:ISBN
5470:ISBN
5452:ISBN
5434:ISBN
5411:ISBN
5384:ISBN
5364:ISBN
5335:ISBN
5270:ISBN
5252:ISBN
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4877:ISSN
4670:The
4660:axis
4594:star
4586:moon
4534:For
4453:and
3769:and
3701:The
3162:and
2447:and
1510:and
1456:and
1080:and
868:and
854:and
227:and
219:and
139:and
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5206:doi
4869:doi
4602:Sun
4538:in
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2004:.
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