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.

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.

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 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.

He developed mathematical techniques for finding the centers of mass of objects of uniform density of various well-defined shapes.

In the Renaissance and Early Modern periods, work by Guido Ubaldi, Francesco Maurolico,[3] Federico Commandino,[4] Evangelista Torricelli, Simon Stevin,[5] Luca Valerio,[6] Jean-Charles de la Faille, Paul Guldin,[7] John Wallis, Christiaan Huygens,[8] Louis Carré, Pierre Varignon, and Alexis Clairaut expanded the concept further.

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 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 process can be repeated for all dimensions of the system to determine the complete center of mass.

The mass-center is a fixed property for a given rigid body (e.g. with no slosh or articulation), whereas the center-of-gravity may, in addition, depend upon its orientation in a non-uniform gravitational field.

In the study of the dynamics of aircraft, vehicles and vessels, forces and moments need to be resolved relative to the mass center.

Let the system of particles Pi, i = 1, ..., n of masses mi be located at the coordinates ri with velocities vi.

Select a reference point R and compute the relative position and velocity vectors,

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.

More formally, this is true for any internal forces that cancel in accordance with Newton's Third Law.

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

[19] 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.

The characteristic low profile of the U.S. military Humvee was designed in part to allow it to tilt farther than taller vehicles without rolling over, by ensuring its low center of mass stays over the space bounded by the four wheels even at angles far from the horizontal.

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.

[22] If the center of mass is behind the aft limit, the aircraft will be more maneuverable, but also less stable, and possibly unstable enough so as to be 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] 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.

Knowing the location of the center of gravity when rigging is crucial, possibly resulting in severe injury or death if assumed incorrectly.

[28] In kinesiology and biomechanics, the center of mass is an important parameter that assists people in understanding their human locomotion.

Typically, a human's center of mass is detected with one of two methods: the reaction board method is a static analysis that involves the person lying down on that instrument, and use of their static equilibrium equation to find their center of mass; the segmentation method relies on a mathematical solution based on the physical principle that the summation of the torques of individual body sections, relative to a specified axis, must equal the torque of the whole system that constitutes the body, measured relative to the same axis.