In physics, the center of mass of a distribution of mass in space (sometimes referred to as the barycenter or balance point) is the unique point at any given time where the weighted relative position of the distributed mass sums to zero. 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 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 (see Barycenter (astronomy) for details). 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 concept of center of gravity or weight was studied extensively by the ancient Greek mathematician, physicist, 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 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.^[1]

Other ancient mathematicians who contributed to the theory of the center of mass include Hero of Alexandria and Pappus of Alexandria. In the Renaissance and Early Modern periods, work by Guido Ubaldi, Francesco Maurolico,^[2] Federico Commandino,^[3] Evangelista Torricelli, Simon Stevin,^[4] Luca Valerio,^[5] Jean-Charles de la Faille, Paul Guldin,^[6] John Wallis, Christiaan Huygens,^[7] Louis Carré, Pierre Varignon, and Alexis Clairaut expanded the concept further.^[8]

Newton's second law is reformulated with respect to the center of mass in Euler's first law.^[9]

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 P_i, i = 1, ..., n , each with mass m_i that are located in space with coordinates r_i, i = 1, ..., n , the coordinates R of the center of mass satisfy the condition

Solving this equation for R yields the formula

A continuous volume

If the mass distribution is continuous with the density ρ(r) 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.^[10]

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.^[11] The calculation takes every particle's x coordinate and maps it to an angle,

where x_max is the system size in the x direction and ${\displaystyle x_{i}\in 0,x_{\max })}$. From this angle, two new points ${\displaystyle (\xi _{i},\zeta _{i})}$ can be generated, which can be weighted by the mass of the particle ${\displaystyle x_{i}}$ for the center of mass or given a value of 1 for the geometric center:

In the ${\displaystyle (\xi ,\zeta )}$ plane, these coordinates lie on a circle of radius 1. From the collection of ${\displaystyle \xi _{i}}$ and ${\displaystyle \zeta _{i}}$ values from all the particles, the averages ${\displaystyle {\overline {\xi }}}$ and ${\displaystyle {\overline {\zeta }}}$ are calculated.

where M is the sum of the masses of all of the particles.

These values are mapped back into a new angle, ${\displaystyle {\overline {\theta }}}$, from which the x coordinate of the center of mass can be obtained: