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In physics, the center of mass of a system of particles is a specific point at which, for many purposes, the system's mass behaves as if it were concentrated. The center of mass is a function only of the positions and masses of the particles that comprise the system. In the case of a rigid body, the position of its center of mass is fixed in relation to the object (but not necessarily in contact with it). In the case of a loose distribution of masses in free space, such as shot from a shotgun, the position of the center of mass is a point in space among them that may not correspond to the position of any individual mass. In the context of a uniform gravitational field, the center of mass is sometimes called the center of gravity. The center of mass of a body does not always coincide with its intuitive geometric center, and one can exploit this freedom. Engineers try hard to make a sport car as light as possible, and then add weight on the bottom; this way, the center of mass is nearer to the street, and the car handles better. When high jumpers perform a "Fosbury Flop", they bend their body in such a way that it is possible for the jumper to clear the bar while his or her center of mass does not. Definition The center of mass of a system of particles is defined as the average of their positions , weighted by their masses : where is the total mass of the system, equal to the sum of the particle masses. For a continuous distribution with mass density , the sum becomes an integral: ho(mathbf), mathbf dV. If an object has uniform density then its center of mass is the same as the centroid of its shape. Examples History The concept of center of gravity was first introduced by the ancient Greek mathematician, physicist, and engineer Archimedes of Syracuse. 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 gravity. In work on floating bodies he demonstrated that the orientation of a floating object is the one that makes its center of gravity as low as possible. He developed mathematical techniques for finding the centers of gravity of objects of uniform density of various well-defined shapes, in particular a triangle, a hemisphere, and a frustum of a circular paraboloid. Locating the center of mass of an arbitrary 2D physical shape This method is useful when you wish to find the center of gravity of a complex planar object with unknown dimensions. Locating the center of mass of a composite shape
Motion The following equations of motion assume that there is a system of particles governed by internal and external forces. An internal force is a force caused by the interaction of the particles within the system. An external force is a force that originates from outside the system, and acts on one or more particles within the system. The external force need not be due to a uniform field. 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 the weak form of 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 the famous Newton's Second Law is Where F indicates the sum of all external forces on the system, and acm indicates the acceleration of the center of mass. Rotation and centers of gravity
CM frame 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 : This is a corollary of the Parallel Axis Theorem. Engineering Aeronautical significance The center of mass is an important point on an aircraft, which significantly affects the stability of the aircraft. To ensure the aircraft is safe to fly, it is critical that the center of gravity fall within specified limits. This range varies by aircraft, but as a rule of thumb it is centered about a point one quarter of the way from the wing leading edge to the wing trailing edge (the quarter chord point). 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 moment arm of the elevator is reduced, which makes it more difficult to recover from a stalled condition. The aircraft will be more maneuverable, but also less stable, and possibly so unstable that it is impossible to fly. Barycenter The barycenter (or barycentre; from the Greek βαρύκεντρον) is the point between two objects where they balance each other. In other words, the center of gravity 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 which lies outside the center of the greater body. For example, it's not the case that the moon orbits the exact center of the earth, but--like how the center of a see-saw would have to be moved closer to the larger of an adult or a child playing on it in order for them to balance each other--the moon orbits a point outside the earth's center where their respective masses balance each other. The barycenter is one of the foci of the elliptical orbit of each body. This is an important concept in the fields of astronomy, astrophysics, and the like (see two-body problem). In a simple two-body case, r1, the distance from the center of the first body to the barycenter is given by: where: a is the shortest distance between the two bodies; m1 and m2 are the masses of the two bodies. r1 is essentially the semi-major axis of the first body's orbit around the barycenter - and r2 = a - r1 the semi-major axis of the second body's orbit. Where the barycenter is located within the more massive body, that body will appear to "wobble" rather than following a discernable orbit. The following table sets out some examples from our solar system. Figures are given rounded to three significant figures. The last two columns show R1, the radius of the first (more massive) body, and r1/R1, the ratio of the distance to the barycenter and that radius: a value less than one shows that the barycenter lies inside the first body. If m1 >> m2 - which is true for the Sun and any planet - then the ratio r1/R1 approximates to: Hence, the barycenter of the Sun-planet system will lie outside the Sun only if: That is, where the planet is heavy and far from the Sun. If Jupiter had Mercury's orbit (57,900,000 km, 0.387 AU), the Sun-Jupiter barycenter would be only 5,500 km from the center of the Sun (r1/R1 ~ 0.08). But even if the Earth had Eris' orbit (68 AU), the Sun-Earth barycenter would still be within the Sun (just over 30,000 km from the center). To calculate the actual motion of the Sun, you would need to sum all the influences from all the planets, comets, asteroids, etc. of the solar system (see many-body problem). If all the planets were aligned on the same side of the Sun, the combined center of mass would lie about 500,000 km above the Sun's surface. The calculations above are based on the mean distance between the bodies and yield the mean value r1. But all celestial orbits are eliptical, and the distance between the bodies varies between the apses, depending on the eccentricity, e. Hence, the position of the barycenter varies too, and it is possible in some systems for the barycenter to be sometimes inside and sometimes outside the more massive body. This occurs where: Note that the Sun-Jupiter system, with eJupiter = 0.0484, just fails to qualify: 1.05 ≯ 1.07 > 0.954. Animations
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