Center of mass

# Center of mass

Overview
In physics
Physics
Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...

, the center of mass or barycenter of a system is the average
Mean
In statistics, mean has two related meanings:* the arithmetic mean .* the expected value of a random variable, which is also called the population mean....

location of all of its mass
Mass
Mass can be defined as a quantitive measure of the resistance an object has to change in its velocity.In physics, mass commonly refers to any of the following three properties of matter, which have been shown experimentally to be equivalent:...

. In the case of a rigid body
Rigid body
In physics, a rigid body is an idealization of a solid body of finite size in which deformation is neglected. In other words, the distance between any two given points of a rigid body remains constant in time regardless of external forces exerted on it...

, the position of the center of mass is fixed in relation to the body. In the case of a loose distribution of masses in free space, such as shot
Lead shot is a collective term for small balls of lead. These were the original projectiles for muskets and early rifles, but today lead shot is fired primarily from shotguns. It is also used for a variety of other purposes...

from a shotgun
Shotgun
A shotgun is a firearm that is usually designed to be fired from the shoulder, which uses the energy of a fixed shell to fire a number of small spherical pellets called shot, or a solid projectile called a slug...

or the planets of the Solar System
Solar System
The Solar System consists of the Sun and the astronomical objects gravitationally bound in orbit around it, all of which formed from the collapse of a giant molecular cloud approximately 4.6 billion years ago. The vast majority of the system's mass is in the Sun...

, the position of the center of mass is a point in space
Space
Space is the boundless, three-dimensional extent in which objects and events occur and have relative position and direction. Physical space is often conceived in three linear dimensions, although modern physicists usually consider it, with time, to be part of a boundless four-dimensional continuum...

among them that may not correspond to the position of any individual mass.

The use of the mass center often allows the use of simplified equations of motion, and it is a convenient reference point for many other calculations in physics, such as angular momentum
Angular momentum
In physics, angular momentum, moment of momentum, or rotational momentum is a conserved vector quantity that can be used to describe the overall state of a physical system...

or the moment of inertia
Moment of inertia
In classical mechanics, moment of inertia, also called mass moment of inertia, rotational inertia, polar moment of inertia of mass, or the angular mass, is a measure of an object's resistance to changes to its rotation. It is the inertia of a rotating body with respect to its rotation...

.
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Encyclopedia
In physics
Physics
Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...

, the center of mass or barycenter of a system is the average
Mean
In statistics, mean has two related meanings:* the arithmetic mean .* the expected value of a random variable, which is also called the population mean....

location of all of its mass
Mass
Mass can be defined as a quantitive measure of the resistance an object has to change in its velocity.In physics, mass commonly refers to any of the following three properties of matter, which have been shown experimentally to be equivalent:...

. In the case of a rigid body
Rigid body
In physics, a rigid body is an idealization of a solid body of finite size in which deformation is neglected. In other words, the distance between any two given points of a rigid body remains constant in time regardless of external forces exerted on it...

, the position of the center of mass is fixed in relation to the body. In the case of a loose distribution of masses in free space, such as shot
Lead shot is a collective term for small balls of lead. These were the original projectiles for muskets and early rifles, but today lead shot is fired primarily from shotguns. It is also used for a variety of other purposes...

from a shotgun
Shotgun
A shotgun is a firearm that is usually designed to be fired from the shoulder, which uses the energy of a fixed shell to fire a number of small spherical pellets called shot, or a solid projectile called a slug...

or the planets of the Solar System
Solar System
The Solar System consists of the Sun and the astronomical objects gravitationally bound in orbit around it, all of which formed from the collapse of a giant molecular cloud approximately 4.6 billion years ago. The vast majority of the system's mass is in the Sun...

, the position of the center of mass is a point in space
Space
Space is the boundless, three-dimensional extent in which objects and events occur and have relative position and direction. Physical space is often conceived in three linear dimensions, although modern physicists usually consider it, with time, to be part of a boundless four-dimensional continuum...

among them that may not correspond to the position of any individual mass.

The use of the mass center often allows the use of simplified equations of motion, and it is a convenient reference point for many other calculations in physics, such as angular momentum
Angular momentum
In physics, angular momentum, moment of momentum, or rotational momentum is a conserved vector quantity that can be used to describe the overall state of a physical system...

or the moment of inertia
Moment of inertia
In classical mechanics, moment of inertia, also called mass moment of inertia, rotational inertia, polar moment of inertia of mass, or the angular mass, is a measure of an object's resistance to changes to its rotation. It is the inertia of a rotating body with respect to its rotation...

. In many applications, such as orbital mechanics, objects can be replaced by point masses located at their mass centers for the purposes 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.

In a uniform gravitational field, such as for small bodies near the surface of Earth, the weight of a body acts as if it were concentrated at the center of mass. For this reason, the center of mass is also called the center of gravity. Where gravity is not uniform, there can be a gravitational torque acting on objects. In some situations, this torque can be explained as arising from a generalized center of gravity that is distinct from the center of mass.

The center of mass of a body does not generally coincide with its geometric center
Centre (geometry)
In geometry, the centre of an object is a point in some sense in the middle of the object. If geometry is regarded as the study of isometry groups then the centre is a fixed point of the isometries.-Circles:...

, and this property can be exploited. Engineers try to design a sports car
Sports car
A sports car is a small, usually two seat, two door automobile designed for high speed driving and maneuverability....

's center of mass as low as possible to make the car handle
Car handling
Automobile handling and vehicle handling are descriptions of the way wheeled vehicles perform transverse to their direction of motion, particularly during cornering and swerving. It also includes their stability when moving at rest. Handling and braking are the major components of a vehicle's...

better. When high jump
High jump
The high jump is a track and field athletics event in which competitors must jump over a horizontal bar placed at measured heights without the aid of certain devices in its modern most practiced format; auxiliary weights and mounds have been used for assistance; rules have changed over the years....

ers perform a "Fosbury Flop
Fosbury Flop
The Fosbury Flop is a style used in the athletics event of high jump. It was popularized and perfected by American athlete Dick Fosbury, whose gold medal in the 1968 Summer Olympics brought it to the world's attention...

", they bend their body in such a way that it clears the bar while its center of mass does not.

## Definition

The center of mass of a system of particles of total mass is defined as the average
Average
In mathematics, an average, or central tendency of a data set is a measure of the "middle" value of the data set. Average is one form of central tendency. Not all central tendencies should be considered definitions of average....

of their positions, , weighted
Weight function
A weight function is a mathematical device used when performing a sum, integral, or average in order to give some elements more "weight" or influence on the result than other elements in the same set. They occur frequently in statistics and analysis, and are closely related to the concept of a...

by their mass
Mass
Mass can be defined as a quantitive measure of the resistance an object has to change in its velocity.In physics, mass commonly refers to any of the following three properties of matter, which have been shown experimentally to be equivalent:...

es, :

For a continuous distribution
Continuum mechanics
Continuum mechanics is a branch of mechanics that deals with the analysis of the kinematics and the mechanical behavior of materials modelled as a continuous mass rather than as discrete particles...

with mass density , the sum becomes an integral:

If an object has uniform density
Density
The mass density or density of a material is defined as its mass per unit volume. The symbol most often used for density is ρ . In some cases , density is also defined as its weight per unit volume; although, this quantity is more properly called specific weight...

then its center of mass is the same as the centroid
Centroid
In geometry, the centroid, geometric center, or barycenter of a plane figure or two-dimensional shape X is the intersection of all straight lines that divide X into two parts of equal moment about the line. Informally, it is the "average" of all points of X...

of its shape.

### 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 median
Median (geometry)
In geometry, a median of a triangle is a line segment joining a vertex to the midpoint of the opposing side. Every triangle has exactly three medians; one running from each vertex to the opposite side...

s and therefore at the centroid
Centroid
In geometry, the centroid, geometric center, or barycenter of a plane figure or two-dimensional shape X is the intersection of all straight lines that divide X into two parts of equal moment about the line. Informally, it is the "average" of all points of X...

, 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. This approximately applies to the Earth
Earth
Earth is the third planet from the Sun, and the densest and fifth-largest of the eight planets in the Solar System. It is also the largest of the Solar System's four terrestrial planets...

: the density varies considerably, but it mainly depends on depth and less on the latitude
Latitude
In geography, the latitude of a location on the Earth is the angular distance of that location south or north of the Equator. The latitude is an angle, and is usually measured in degrees . The equator has a latitude of 0°, the North pole has a latitude of 90° north , and the South pole has a...

and longitude
Longitude
Longitude is a geographic coordinate that specifies the east-west position of a point on the Earth's surface. It is an angular measurement, usually expressed in degrees, minutes and seconds, and denoted by the Greek letter lambda ....

coordinates.

More generally, for any symmetry of a body, its center of mass will be a fixed point of that symmetry.

### Momentum

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.

The total momentum for any system of particles is given by

where M indicates the total mass, and vcm is the velocity of the center of mass. 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
Newton's laws of motion
Newton's laws of motion are three physical laws that form the basis for classical mechanics. They describe the relationship between the forces acting on a body and its motion due to those forces...

is

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.

The angular momentum
Angular momentum
In physics, angular momentum, moment of momentum, or rotational momentum is a conserved vector quantity that can be used to describe the overall state of a physical system...

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 :

This is a corollary of the parallel axis theorem
Parallel axis theorem
In physics, the parallel axis theorem or Huygens-Steiner theorem can be used to determine the second moment of area or the mass moment of inertia of a rigid body about any axis, given the body's moment of inertia about a parallel axis through the object's centre of mass and the perpendicular...

.

### Gravity

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, and the gravitational torque
Torque
Torque, moment or moment of force , is the tendency of a force to rotate an object about an axis, fulcrum, or pivot. Just as a force is a push or a pull, a torque can be thought of as a twist....

is equal to the torque of a force Mg acting at R.

If the gravitational field acting on a body is not uniform, then the center of mass does not necessarily exhibit these convenient properties concerning gravity. A non-uniform gravitational field can produce a torque on an object about its center of mass, causing it to rotate. The center of gravity seeks to model the gravitational torque as a resultant force at a point. Such a point may not exist, and if it exists, it is not unique. When a unique center of gravity can be defined, its location depends on the external field, so its motion is harder to determine than the motion of the center of mass; this problem limits its usefulness in applications.

## History

The concept of center of mass was first introduced by the ancient Greek physicist, mathematician, and engineer Archimedes of Syracuse
Archimedes
Archimedes of Syracuse was a Greek mathematician, physicist, engineer, inventor, and astronomer. Although few details of his life are known, he is regarded as one of the leading scientists in classical antiquity. Among his advances in physics are the foundations of hydrostatics, statics and an...

. Archimedes showed that the torque
Torque
Torque, moment or moment of force , is the tendency of a force to rotate an object about an axis, fulcrum, or pivot. Just as a force is a push or a pull, a torque can be thought of as a twist....

exerted on a lever
Lever
In physics, a lever is a rigid object that is used with an appropriate fulcrum or pivot point to either multiply the mechanical force that can be applied to another object or resistance force , or multiply the distance and speed at which the opposite end of the rigid object travels.This leverage...

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.

Later mathematicians who developed the theory of the center of mass include Pappus of Alexandria
Pappus of Alexandria
Pappus of Alexandria was one of the last great Greek mathematicians of Antiquity, known for his Synagoge or Collection , and for Pappus's Theorem in projective geometry...

, Guido Ubaldi,
Francesco Maurolico
Francesco Maurolico
Francesco Maurolico was a Greek mathematician and astronomer of Sicily. Throughout his lifetime, he made contributions to the fields of geometry, optics, conics, mechanics, music, and astronomy...

,
Federico Commandino
Federico Commandino
Federico Commandino was an Italian humanist and mathematician.Born in Urbino, he studied at Padua and at Ferrara, where he received his doctorate in medicine. He translated the works of ancient mathematicians and was responsible for the publication of the works of Archimedes...

,
Simon Stevin
Simon Stevin
Simon Stevin was a Flemish mathematician and military engineer. He was active in a great many areas of science and engineering, both theoretical and practical...

,
Luca Valerio
Luca Valerio
Luca Valerio was an Italian mathematician. He developed ways to find volumes and centers of gravity of solid bodies using the methods of Archimedes. He corresponded with Galileo Galilei and was a member of the Accademia dei Lincei.-Biography:...

, Jean-Charles de la Faille
Jean-Charles de la Faille
Jean-Charles de la Faille or Jan-Karel della Faille was a Flemish Jesuit mathematician....

, Paul Guldin
Paul Guldin
Paul Guldin was a Swiss Jesuit mathematician and astronomer. He discovered the Guldinus theorem to determine the surface and the volume of a solid of revolution. This theorem is also known as Pappus–Guldinus theorem and Pappus's centroid theorem, attributed to Pappus of Alexandria...

, John Wallis, Louis Carré
Louis Carré (mathematician)
Louis Carré was a French mathematician and member of the French Academy of Sciences. He was the author of one of the first books on integral calculus.-References:...

, Pierre Varignon
Pierre Varignon
Pierre Varignon was a French mathematician. He was educated at the Jesuit College and the University in Caen, where he received his M.A. in 1682. He took Holy Orders the following year....

, and Alexis Clairaut.

Newton's second law is reformulated with respect to the center of mass in Euler's first law.

## Locating the center of mass

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.

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

A direct development of the planimeter
Planimeter
A planimeter is a measuring instrument used to determine the area of an arbitrary two-dimensional shape.-Construction:There are several kinds of planimeters, but all operate in a similar way. The precise way in which they are constructed varies, with the main types of mechanical planimeter being...

known as an integraph, or integerometer, can be used to establish the position of the centroid
Centroid
In geometry, the centroid, geometric center, or barycenter of a plane figure or two-dimensional shape X is the intersection of all straight lines that divide X into two parts of equal moment about the line. Informally, it is the "average" of all points of X...

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.

### Aeronautics

The center of mass is an important point on an aircraft
Aircraft
An aircraft is a vehicle that is able to fly by gaining support from the air, or, in general, the atmosphere of a planet. An aircraft counters the force of gravity by using either static lift or by using the dynamic lift of an airfoil, or in a few cases the downward thrust from jet engines.Although...

, 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. 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
Elevator (aircraft)
Elevators are flight control surfaces, usually at the rear of an aircraft, which control the aircraft's orientation by changing the pitch of the aircraft, and so also the angle of attack of the wing. In simplified terms, they make the aircraft nose-up or nose-down...

will also be reduced, which makes it more difficult to recover from a stalled
Stall (flight)
In fluid dynamics, a stall is a reduction in the lift coefficient generated by a foil as angle of attack increases. This occurs when the critical angle of attack of the foil is exceeded...

condition.

For helicopter
Helicopter
A helicopter is a type of rotorcraft in which lift and thrust are supplied by one or more engine-driven rotors. This allows the helicopter to take off and land vertically, to hover, and to fly forwards, backwards, and laterally...

s in hover
Hover
Hover may refer to:*Hovering , the process by which an object is suspended by a physical force against gravity, in a stable position without solid physical contactIn transport* Hover , nearly stationary flight in a helicopter...

, the center of mass is always directly below the rotorhead
In helicopters the rotorhead is the part of the rotor assembly that joins the blades to the shaft, cyclic and collective mechanisms. It is sometimes referred to as the rotor "hub"...

. 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

The center of mass plays an important role in astronomy and astrophysics, where it is common 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
Orbit
In physics, an orbit is the gravitationally curved path of an object around a point in space, for example the orbit of a planet around the center of a star system, such as the Solar System...

each other. When a moon
Natural satellite
A natural satellite or moon is a celestial body that orbits a planet or smaller body, which is called its primary. The two terms are used synonymously for non-artificial satellites of planets, of dwarf planets, and of minor planets....

orbits a planet
Planet
A planet is a celestial body orbiting a star or stellar remnant that is massive enough to be rounded by its own gravity, is not massive enough to cause thermonuclear fusion, and has cleared its neighbouring region of planetesimals.The term planet is ancient, with ties to history, science,...

, or a planet orbits a star
Star
A star is a massive, luminous sphere of plasma held together by gravity. At the end of its lifetime, a star can also contain a proportion of degenerate matter. The nearest star to Earth is the Sun, which is the source of most of the energy on Earth...

, both bodies are actually orbiting around a point that lies outside the center of the primary (the larger body). For example, the moon does not orbit the exact center of the Earth
Earth
Earth is the third planet from the Sun, and the densest and fifth-largest of the eight planets in the Solar System. It is also the largest of the Solar System's four terrestrial planets...

, 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
Sun
The Sun is the star at the center of the Solar System. It is almost perfectly spherical and consists of hot plasma interwoven with magnetic fields...

.

• Center of percussion
Center of percussion
The center of percussion is the point on an object where a perpendicular impact will produce translational and rotational forces which perfectly cancel each other out at some given pivot point, so that the pivot will not be moving momentarily after the impulse....

• Center of pressure
Center of pressure
The center of pressure is the point on a body where the total sum of a pressure field acts, causing a force and no moment about that point. The total force vector acting at the center of pressure is the value of the integrated vectorial pressure field. The resultant force and center of pressure...

• Mass point geometry
Mass point geometry
Mass point geometry, colloquially known as mass points, is a geometry problem-solving technique which applies the physical principle of the center of mass to geometry problems involving triangles and intersecting cevians...

• Metacentric height
Metacentric height
The metacentric height is a measurement of the static stability of a floating body. It is calculated as the distance between the centre of gravity of a ship and its metacentre . A larger metacentric height implies greater stability against overturning...

• Roll center
Roll center
The roll center of a vehicle is the notional point at which the cornering forces in the suspension are reacted to the vehicle body.-Theory:There are two definitions of roll center...

• Weight distribution
Weight distribution
Weight distribution is the apportioning of weight within a vehicle, especially cars, airplanes, and trains.In a vehicle which relies on gravity in some way, weight distribution directly affects a variety of vehicle characteristics, including handling, acceleration, traction, and component life...