Center of mass

In physics, the center of mass of a distribution of mass in space is the unique point at any given time where the weighted relative position of the distributed mass sums to zero. For a rigid body containing its center of mass, this is the point to which a force may be applied to cause a linear acceleration without an angular acceleration. Calculations in mechanics are often simplified when formulated with respect to the center of mass. It is a hypothetical point where the entire mass of an object may be assumed to be concentrated to visualise its motion. In other words, the center of mass is the particle equivalent of a given object for the application of Newton's laws of motion.
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 center of mass may be located outside the physical body, as is sometimes the case for hollow or open-shaped objects, such as a horseshoe. In the case of a distribution of separate bodies, such as the planets of the Solar System, the center of mass may not correspond to the position of any individual member of the system.
The center of mass is a useful reference point for calculations in mechanics that involve masses distributed in space, such as the linear and angular momentum of planetary bodies and rigid body dynamics. In orbital mechanics, the equations of motion of planets are formulated as point masses located at the centers of mass. 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.

History

The origins of the concept of the center of gravity are unknown, but it appears to have been a well-known philosophical concept in the ancient Greek world. It is discussed in the works of Hero of Alexandria and Pappus of Alexandria in detail in the 2nd and 3rd century CE,
but makes its first known appearance in the 3rd century BCE in the writings of
Archimedes of Syracuse, an ancient Greek mathematician, physicist, and engineer. 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 his work On Floating Bodies, Archimedes demonstrated that the orientation of a floating object is the one that makes its center of mass as low as possible. He developed mathematical techniques for finding the centers of mass of objects of uniform density of various well-defined shapes. However, the surviving writings of Archimedes treat some form of the center of gravity itself as a preexisting idea, implying that it has earlier origins, either in his own preceding but lost studies, or in unknown works of his contemporaries or predecessors
In the Renaissance and Early Modern periods, work by Guido Ubaldi, Francesco Maurolico, Federico Commandino, Evangelista Torricelli, Simon Stevin, Luca Valerio, Jean-Charles de la Faille, Paul Guldin, John Wallis, Christiaan Huygens, Louis Carré, Pierre Varignon, and Alexis Clairaut expanded the concept further.
Newton's second law is reformulated with respect to the center of mass in Euler's first law.

Definition

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 vectors relative to this point sum to zero. In analogy to statistics, the center of mass is the mean location of a distribution of mass in space.

A system of particles

In the case of a system of particles, each with mass that are located in space with coordinates, the coordinates R of the center of mass satisfy
Solving this equation for R yields the formula

A continuous volume

If the mass distribution is continuous with the density ρ within a solid Q, then the integral of the weighted position coordinates of the points in this volume relative to the center of mass R over the volume V is zero, that is
Solve this equation for the coordinates R to obtain
where M is the total mass in the volume.
If a continuous mass distribution has uniform density, which means that ρ is constant, then the center of mass is the same as the centroid of the volume.

Barycentric coordinates

The coordinates R of the center of mass of a two-particle system, P₁ and P₂, with masses m₁ and m₂ is given by
Let the percentage of the total mass divided between these two particles vary from 100% P₁ and 0% P₂ through 50% P₁ and 50% P₂ to 0% P₁ and 100% P₂, then the center of mass R moves along the line from P₁ to P₂. The percentages of mass at each point can be viewed as projective coordinates of the point R on this line, and are termed barycentric coordinates. Another way of interpreting the process here is the mechanical balancing of moments about an arbitrary point. The numerator gives the total moment that is then balanced by an equivalent total force at the center of mass. This can be generalized to three points and four points to define projective coordinates in the plane, and in space, respectively.

Systems with periodic boundary conditions

For particles in a system with periodic boundary conditions two particles can be neighbours even though they are on opposite sides of the system. This occurs often in molecular dynamics simulations, for example, in which clusters form at random locations and sometimes neighbouring atoms cross the periodic boundary. When a cluster straddles the periodic boundary, a naive calculation of the center of mass will be incorrect. A generalized method for calculating the center of mass for periodic systems is to treat each coordinate, x and y and/or z, as if it were on a circle instead of a line. The calculation takes every particle's x coordinate and computes N possible centers of mass,
where is the naïve mass-weighted average position for all particles. The true center of mass is then the value in the vector, with the lowest sum of squared arc distances between the points.
The process can be repeated for all dimensions of the system to determine the complete center of mass. The utility of the algorithm is that it allows the mathematics to determine where the "best" center of mass is, instead of guessing or using cluster analysis to "unfold" a cluster straddling the periodic boundaries. This approach computes the intrinsic center of mass. The extrinsic circular mean is a popular way to estimate the center of mass, however this approach can be inaccurate.

Center of gravity

A body's center of gravity is the point around which the resultant torque due to gravity forces vanishes. Where a gravity field can be considered to be uniform, the center of mass and the center of gravity will be the same. However, for satellites in orbit around a planet, in the absence of other torques being applied to a satellite, the slight variation in gravitational field between the parts closer to and further from the planet 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.
The mass center is a fixed property for a given rigid body, whereas the center of gravity may, in addition, depend upon its orientation in a non-uniform gravitational field. In the latter case, the center of gravity will always be located somewhat closer to the main attractive body as compared to the mass center, and thus will change its position in the body of interest as its orientation is changed.
In the study of the dynamics of aircraft, vehicles and vessels, forces and moments need to be resolved relative to the mass center. That is true independent of whether gravity itself is a consideration. Referring to the mass center as the center of gravity is something of a colloquialism, but it is in common usage and when gravity gradient effects are negligible, center of gravity and mass center are the same and are used interchangeably.
In physics the benefits of using the center of mass to model a mass distribution can be seen by considering the resultant of the gravity forces on a continuous body. Consider a body Q of volume V with density ρ at each point r in the volume. In a parallel gravity field the force f at each point r is given by,
where dm is the mass at the point r, g is the acceleration of gravity, and is a unit vector defining the vertical direction.
Choose a reference point R in the volume and compute the resultant force and torque at this point,
and
If the reference point R is chosen so that it is the center of mass, then
which means the resultant torque. Because the resultant torque is zero the body will move as though it is a particle with its mass concentrated at the center of mass.
By selecting the center of gravity as the reference point for a rigid body, the gravity forces will not cause the body to rotate, which means the weight of the body can be considered to be concentrated at the center of mass.

Linear and angular momentum

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 P_i, i = 1,..., n of masses m_i be located at the coordinates r_i with velocities v_i. Select a reference point R and compute the relative position and velocity vectors,
The total linear momentum and angular momentum of the system are
and
If R is chosen as the center of mass these equations simplify to
where m is the total mass of all the particles, p is the linear momentum, and L is the angular momentum.
The law of conservation of momentum predicts that for any system not subjected to external forces the momentum of the system will remain constant, which means the center of mass will move with constant velocity. This applies for all systems with classical internal forces, including magnetic fields, electric fields, chemical reactions, and so on. More formally, this is true for any internal forces that cancel in accordance with Newton's third law.