Center of mass

center of gravitycentre of gravitycentre of massbarycentercenter of balancecenter-of-masscenters of gravitycenters of massc.g.center-of-gravity
In physics, the center of mass of a distribution of mass in space is the unique point where the weighted relative position of the distributed mass sums to zero.wikipedia
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Centroid

centroidsgeographic centerbarycenter
In the case of a single rigid body, the center of mass is fixed in relation to the body, and if the body has uniform density, it will be located at the centroid. A direct development of the planimeter known as an integraph, or integerometer, can be used to establish the position of the centroid or center of mass of an irregular two-dimensional shape.
While in geometry the word barycenter is a synonym for centroid, in astrophysics and astronomy, the barycenter is the center of mass of two or more bodies that orbit each other.

Angular momentum

conservation of angular momentumangular momentamomentum
The center of mass is a useful reference point for calculations in mechanics that involve masses distributed in space, such as the linear and angular momentum of planetary bodies and rigid body dynamics.
The spin angular momentum of an object is defined as the angular momentum about its centre of mass coordinate.

Rigid body dynamics

rigid-body dynamicsRigidrigid body mechanics
The center of mass is a useful reference point for calculations in mechanics that involve masses distributed in space, such as the linear and angular momentum of planetary bodies and rigid body dynamics.
Use the center of mass C as the reference point, so these equations for Newton's laws simplify to become

Rigid body

rigid bodiesrigidrigid-body
In the case of a single rigid body, the center of mass is fixed in relation to the body, and if the body has uniform density, it will be located at the centroid.
The linear position can be represented by a vector with its tail at an arbitrary reference point in space (the origin of a chosen coordinate system) and its tip at an arbitrary point of interest on the rigid body, typically coinciding with its center of mass or centroid.

Center-of-momentum frame

center of momentum framecenter of mass framecenter-of-mass frame
The center of mass frame is an inertial frame in which the center of mass of a system is at rest with respect to the origin of the coordinate system.
A special case of the center-of-momentum frame is the center-of-mass frame: an inertial frame in which the center of mass (which is a physical point) remains at the origin.

Point particle

point chargepoint massparticle
In orbital mechanics, the equations of motion of planets are formulated as point masses located at the centers of mass.
For example, spherical objects interacting in 3-dimensional space whose interactions are described by the inverse square law behave in such a way as if all their matter were concentrated in their centers of mass.

Luca Valerio

ValerioValerio, Luca
Later mathematicians who developed the theory of the center of mass include Pappus of Alexandria, Guido Ubaldi, Francesco Maurolico, Federico Commandino, Simon Stevin, Luca Valerio, Jean-Charles de la Faille, Paul Guldin, John Wallis, Louis Carré, Pierre Varignon, and Alexis Clairaut.
He developed ways to find volumes and centers of gravity of solid bodies using the methods of Archimedes.

Weight function

weighted sumweightedweights
In physics, the center of mass of a distribution of mass in space is the unique point where the weighted relative position of the distributed mass sums to zero.
The terminology weight function arises from mechanics: if one has a collection of n objects on a lever, with weights (where weight is now interpreted in the physical sense) and locations :, then the lever will be in balance if the fulcrum of the lever is at the center of mass

Automobile handling

handlingcar handlingvehicle handling
Engineers try to design a sports car so that its center of mass is lowered to make the car handle better, that is maintaining traction while executing relatively sharp turns.
The centre of mass height, also known as the centre of gravity height, or CGZ, relative to the track, determines load transfer (related to, but not exactly weight transfer) from side to side and causes body lean.

Momentum

conservation of momentumlinear momentummomenta
The center of mass is a useful reference point for calculations in mechanics that involve masses distributed in space, such as the linear and angular momentum of planetary bodies and rigid body dynamics.
A system of particles has a center of mass, a point determined by the weighted sum of their positions:

Torque

moment armmomentlever arm
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 torque exerted on a lever by weights resting at various points along the lever is the same as what it would be if all of the weights were moved to a single point—their center of mass. In work on floating bodies he demonstrated that the orientation of a floating object is the one that makes its center of mass as low as possible.
Mathematically, for rotation about a fixed axis through the center of mass, the work W can be expressed as

Euler's laws of motion

Euler's equations of motionEuler's first lawEuler's laws
Newton's second law is reformulated with respect to the center of mass in Euler's first law.
and the velocity of its center of mass

Sports car

sports carssportssportscar
Engineers try to design a sports car so that its center of mass is lowered to make the car handle better, that is maintaining traction while executing relatively sharp turns.
The Mercedes included pioneering features such as a pressed-steel chassis, a gated 4-speed transmission, pushrod-actuated overhead inlet valves, a honeycomb radiator, low-tension magneto ignition, a long wheelbase, a low centre of mass and a very effective suspension system.

Archimedes

Archimedes of SyracuseArchimedeanArchimedes Heat Ray
The concept of "center of mass" in the form of the center of gravity was first introduced by the great ancient Greek physicist, mathematician, and engineer Archimedes of Syracuse.

Rollover

rolled overrollover accidentroll over
The characteristic low profile of the U. S. military Humvee was designed in part to allow it tilt farther than taller vehicles, without a rollover, because its low center of mass would stay over the space bounded the four wheels even at angles far from the horizontal.
Generally, rollover tendency increases with the height of the center of mass, narrowness of the axle track, steering sensitivity, and increased speed.

Center of gravity of an aircraft

center of gravitycentre of gravityweight and balance
If the center of mass is ahead of the forward limit, the aircraft will be less maneuverable, possibly to the point of being unable to rotate for takeoff or flare for landing.
To ensure the aircraft is safe to fly, the center of gravity must fall within specified limits established by the aircraft manufacturer.

Elevator (aeronautics)

elevatorelevatorsbalanced elevator
The moment arm of the elevator will also be reduced, which makes it more difficult to recover from a stalled condition.
The horizontal stabilizer usually creates a downward force which balances the nose down moment created by the wing lift force, which typically applies at a point (the wing center of lift) situated aft of the airplane's center of gravity.

Alexis Clairaut

Alexis Claude ClairautClairautAlexis Claude Clairault
Later mathematicians who developed the theory of the center of mass include Pappus of Alexandria, Guido Ubaldi, Francesco Maurolico, Federico Commandino, Simon Stevin, Luca Valerio, Jean-Charles de la Faille, Paul Guldin, John Wallis, Louis Carré, Pierre Varignon, and Alexis Clairaut.
This hydrostatic model of the shape of the Earth was founded on a paper by Colin Maclaurin, which had shown that a mass of homogeneous fluid set in rotation about a line through its centre of mass would, under the mutual attraction of its particles, take the form of an ellipsoid.

Humvee

HMMWVHumveesHigh Mobility Multipurpose Wheeled Vehicle
The characteristic low profile of the U. S. military Humvee was designed in part to allow it tilt farther than taller vehicles, without a rollover, because its low center of mass would stay over the space bounded the four wheels even at angles far from the horizontal.
The MaxxPro Line has been shown to have the highest rate of vehicle rollover accidents to its very high center of gravity and immense weight.

Planimeter

planometerPlatometer
A direct development of the planimeter known as an integraph, or integerometer, can be used to establish the position of the centroid or center of mass of an irregular two-dimensional shape.
Developments of the planimeter can establish the position of the first moment of area (center of mass), and even the second moment of area.

Orbit

orbitsorbital motionplanetary motion
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.
Newton showed that, for a pair of bodies, the orbits' sizes are in inverse proportion to their masses, and that those bodies orbit their common center of mass.

Barycenter

barycentrebarycentricbarycentric coordinates
In astronomy, the barycenter (or barycentre; from the Ancient Greek βαρύς heavy + κέντρον center ) is the center of mass of two or more bodies that orbit one another and is the point about which the bodies orbit.

Jean-Charles della Faille

Jean-Charles de la Failledella Faille, Jean-Charles
Later mathematicians who developed the theory of the center of mass include Pappus of Alexandria, Guido Ubaldi, Francesco Maurolico, Federico Commandino, Simon Stevin, Luca Valerio, Jean-Charles de la Faille, Paul Guldin, John Wallis, Louis Carré, Pierre Varignon, and Alexis Clairaut.
His most famous book is Theoremata de centro gravitatis partium circuli et ellipsis (1632) in which he determined the centre of gravity of the sector of a circle, for the first time.

Center of mass (relativistic)

Relativistic center of mass
In physics, relativistic center of mass refers to the mathematical and physical concepts that define the center of mass of a system of particles in relativistic mechanics and relativistic quantum mechanics.

Earth

Earth's surfaceterrestrialworld
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.
The point on the surface farthest from Earth's center of mass is the summit of the equatorial Chimborazo volcano in Ecuador (6384.4 km).