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In physics, the center of mass or barycenter is the weighted average location of all the mass in a body or group of bodies. Various important calculations in mechanics become simplified when quantities are referenced to the center of mass, or when the entire mass of a body is treated as if it is concentrated at the center of mass.

In the case of a rigid body, the center of mass is fixed in relation to the body, and it does not necessarily coincide with the geometric center. Nor does the center of mass necessarily coincide with any point on the body, as is often the case for hollow or open-shaped objects, like a horseshoe. In the case of a loose distribution of particles or bodies, such as the planets of the Solar System, the center of mass of the entire group may not correspond to the position of any individual member.

The mass center often obeys simple equations of motion, and it is a convenient reference point for many other calculations in mechanics, such as angular momentum and moment of inertia. In many applications, such as orbital mechanics, objects can be replaced by point masses located at their mass centers for the purposes of several types of analysis. The center of mass frame is an inertial frame in which the center of mass of a system is at rest at the origin of the coordinate system.

Gravity
Diagram of an educational toy that balances on a point: the CM (C) settles below its support (P)
The suspending chair trick makes use of the human body's surprisingly high center of mass

The center of mass is often called the center of gravity because any uniform gravitational field g acts on a system as if the mass M of the system were concentrated at the center of mass R. Specifically, the gravitational potential energy is equal to the potential energy of a point mass M at R,[1] and the gravitational torque is equal to the torque of a force Mg acting at R.[2] In a uniform gravitational field, the center of mass is a center of gravity, and in common usage, the two phrases are used as synonyms.

In a non-uniform field, gravitational effects such as potential energy, force, and torque can no longer be calculated using the center of mass alone. In particular, a non-uniform gravitational field can produce a torque on an object, causing it to rotate. The center of gravity, an application point of the resultant gravitational force, may not exist or not be unique; see centers of gravity in non-uniform fields.
Definition

The center of mass $$\mathbf{R}$$ of a system of particles of total mass M is defined as the average of their positions, $$\mathbf{r}_i$$, weighted by their masses, $$m_i$$:[3]

$$\mathbf{R} = \frac{1}{M} \sum m_i \mathbf{r}_i.$$

For a continuous distribution with mass density \rho(\mathbf{r}), the sum becomes an integral:[4]

$$\mathbf R =\frac 1M \int \mathbf{r} \; dm = \frac 1M \int\rho(\mathbf{r})\, \mathbf{r} \ dV.$$

If an object has uniform density then its center of mass is the same as the centroid of its shape.[5]
Examples

The center of mass of a two-particle system lies on the line connecting the particles (or, more precisely, their individual centers of mass). The center of mass is closer to the more massive object; for details, see below.
The center of mass of a uniform ring is at the center of the ring; outside the material that makes up the ring.
The center of mass of a uniform solid triangle lies on all three medians and therefore at the centroid, which is also the average of the three vertices.
The center of mass of a uniform rectangle is at the intersection of the two diagonals.
In a spherically symmetric body, the center of mass is at the geometric center.[6] This approximately applies to the Earth: the density varies considerably, but it mainly depends on depth and less on the latitude and longitude coordinates.

More generally, for any symmetry of a body, its center of mass will be a fixed point of that symmetry.[2]
Properties
Momentum
Further information: Derivation of the center of mass, Center of mass frame

For any system with no external forces, the center of mass moves 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 satisfy Newton's Third Law.[3]

The total momentum for any system of particles is given by

$$\mathbf{p}=M\mathbf{v}_\mathrm{cm},$$

where M indicates the total mass, and vcm is the velocity of the center of mass.[7] This velocity can be computed by taking the time derivative of the position of the center of mass. An analogue to Newton's Second Law is

$$\mathbf{F} = M\mathbf{a}_\mathrm{cm},$$

where F indicates the sum of all external forces on the system, and acm indicates the acceleration of the center of mass. It is this principle that gives precise expression to the intuitive notion that the system as a whole behaves like a mass of M placed at R.[3]

The angular momentum vector for a system is equal to the angular momentum of all the particles around the center of mass, plus the angular momentum of the center of mass, as if it were a single particle of mass M:[8]

$$\mathbf{L}_\mathrm{sys} = \mathbf{L}_\mathrm{cm} + \mathbf{L}_\mathrm{around\,cm}.$$

This is a corollary of the parallel axis theorem.[9]

History

The concept of a center of gravity was first introduced by the ancient Greek physicist, mathematician, and engineer Archimedes of Syracuse. He worked with simplified assumptions about gravity that amount to a uniform field, thus arriving at the mathematical properties of what we now call the center of mass. Archimedes showed that 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. He developed mathematical techniques for finding the centers of mass of objects of uniform density of various well-defined shapes.[10]

Later mathematicians who developed the theory of the center of mass include Pappus of Alexandria, Guido Ubaldi, Francesco Maurolico,[11] Federico Commandino,[12] Simon Stevin,[13] Luca Valerio,[14] Jean-Charles de la Faille, Paul Guldin,[15] John Wallis, Louis Carré, Pierre Varignon, and Alexis Clairaut.[16]

Newton's second law is reformulated with respect to the center of mass in Euler's first law.[17]
Locating the center of mass
Main article: Locating the center of mass
Plumb line method

An experimental method for locating the center of mass is to suspend the object from two locations and to drop plumb lines from the suspension points. The intersection of the two lines is the center of mass.[18]

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

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. 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 ensure the ship would not capsize.[21]
Applications
Estimated center of mass/gravity of a gynmast at the end of performing a cartwheel. Notice center is outside the body in this position.

Engineers try to design a sports car's center of mass as low as possible to make the car handle better. When high jumpers perform a "Fosbury Flop", they bend their body in such a way that it clears the bar while its center of mass does not.[22]
Aeronautics
Main article: Center of gravity of an aircraft

The center of mass is an important point on an aircraft, 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 forward limit, the aircraft will be less maneuverable, possibly to the point of being unable to rotate for takeoff or flare for landing.[23] If the center of mass is behind the aft limit, the aircraft will be more maneuverable, but also less stable, and possibly so unstable that it is impossible to fly. The moment arm of the elevator will also be reduced, which makes it more difficult to recover from a stalled condition.[24]

For helicopters in hover, the center of mass is always directly below the rotorhead. In forward flight, the center of mass will move aft to balance the negative pitch torque produced by applying cyclic control to propel the helicopter forward; consequently a cruising helicopter flies "nose-down" in level flight.
Astronomy

Two bodies orbiting a barycenter internal to one body
Main article: Barycentric coordinates (astronomy)

The center of mass plays an important role in astronomy and astrophysics, where it is commonly referred to as the barycenter. 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 a planet, or a planet orbits a star, both bodies are actually orbiting around a point that lies away from the center of the primary (larger) body.[25] 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 (1062 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 difference in mass is smaller, e.g. Pluto and Charon, the barycenter will fall outside both bodies.

Center of percussion
Center of pressure (fluid mechanics)
Center of pressure (terrestrial locomotion)
Mass point geometry
Metacentric height
Roll center
Weight distribution
Expected value

Notes

^ Goldstein, Poole & Safko 2001, p. 185.
^ a b Feynman, Leighton & Sands 1963, p. 19.3.
^ a b c Kleppner & Kolenkow 1973, p. 117.
^ Kleppner & Kolenkow 1973, p. 119.
^ Levi 2009, p. 85.
^ Giambattista, Richardson & Richardson 2007, p. 235.
^ Kleppner & Kolenkow 1973, p. 116.
^ Kleppner & Kolenkow 1973, p. 262.
^ Kleppner & Kolenkow 1973, p. 252.
^ Shore 2008, pp. 9–11.
^ Baron 2004, pp. 91–94.
^ Baron 2004, pp. 94–96.
^ Baron 2004, pp. 96–101.
^ Baron 2004, pp. 101–106.
^ Mancosu 1999, pp. 56–61.
^ Walton 1855, p. 2.
^ Beatty 2006, p. 29.
^ Kleppner & Kolenkow 1973, pp. 119–120.
^ Feynman, Leighton & Sands 1963, pp. 19.1–19.2.
^ Hamill 2009, pp. 20–21.
^ Sangwin 2006, p. 7.
^ Van Pelt 2005, p. 185.
^ Federal Aviation Administration 2007, p. 1.4.
^ Federal Aviation Administration 2007, p. 1.3.
^ Murray & Dermott 1999, pp. 45–47.

References

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Federal Aviation Administration (2007), Aircraft Weight and Balance Handbook, United States Government Printing Office, retrieved 23 October 2011
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