The first theorem states that the surface area A of a surface of revolution generated by rotating a plane curve C about an axis external to C and on the same plane is equal to product of the arc length s of C and the distance d1 traveled by its centroid. For example, the surface area of the torus with minor radius r and major radius R is ...

See also:

Pappus's centroid theorem, Pappus's centroid theorem - The first theorem, Pappus's centroid theorem - The second theorem

Read more here: » Pappus's centroid theorem: Encyclopedia II - Pappus's centroid theorem - The first theorem

In geometry, the centroid or barycenter of an object X in n-dimensional space is the intersection of all hyperplanes that divide X into two parts of equal moment about the hyperplane. Informally, it is the "average" of all points of X. In physics, the words centroid and barycenter may mean either the center of mass or the center of gravity of an object, depending on ...

Including:

• Centroid - Centroid of triangle and tetrahedon
• Centroid - Centroids of cones and pyramids
• Centroid - Centroid and convexity
• Centroid - Integral formula
• Centroid - Center of symmetry
• Centroid - Physical centroids

Read more here: » Centroid: Encyclopedia - Centroid

The center of mass of an object is a point at which the object's mass can be assumed, for many purposes, to be concentrated. Center of mass - Example. For example, an object can balance on a point only if its center of mass is directly above the point. Alternatively, if you hang an object from a string, the object's center of mass will be directly below the string. Center of gravity, Centroid, Pappus's centroid theorem, Center of pressure Center ...

Including:

• Center of mass - Example
• Center of mass - Comparison with center of gravity
• Center of mass - Definition
• Center of mass - More formulas
• Center of mass - Aeronautical significance
• Center of mass - Motion of the center of mass
• Center of mass - Examples
• Center of mass - Barycenter
• Center of mass - Animations

Read more here: » Center of mass: Encyclopedia - Center of mass

The centroid of a triangle is the point of intersection of its medians (the lines joining each vertex with the midpoint of the opposite side). The centroid divides each of the medians in the ratio 2:1. The centroid is the triangle's center of mass if the triangle is made from a uniform sheet of material. Its Cartesian coordinates are the means of the coordinates of the three vertices. A similar result holds for a tetrahedron: its centroid is the intersection of all line segments that connect each vertex to the centroid of the o ...

See also:

Centroid, Centroid - Centroid of triangle and tetrahedon, Centroid - Centroids of cones and pyramids, Centroid - Centroid and convexity, Centroid - Integral formula, Centroid - Center of symmetry, Centroid - Physical centroids

Read more here: » Centroid: Encyclopedia II - Centroid - Centroid of triangle and tetrahedon

When talking about celestial bodies, the center of mass has a special relevance: when a moon orbits around planet, or a planet orbits around a star, both of them are actually orbiting around their center of mass, called the barycenter, see two-body problem. The barycenter (from the Greek βαρύκεντρον) is the center of mass of two or more bodies which are orbiting each other, and is the point around which both of them orbit. It is an important concept in the fie ...

See also:

Center of mass, Center of mass - Example, Center of mass - Comparison with center of gravity, Center of mass - Definition, Center of mass - More formulas, Center of mass - Aeronautical significance, Center of mass - Motion of the center of mass, Center of mass - Examples, Center of mass - Barycenter, Center of mass - Animations

Read more here: » Center of mass: Encyclopedia II - Center of mass - Barycenter

The abscissa of the centroid of a plane figure can be given as the integral , where f(x) is the vertical extent of the object at abscissa x. The same formula yields the first coordinate of the centroid of an object in , for any dimension n, provided that f(x) is the (n − 1)-dimensional measure of the object's cross-section at coordinate See also:

Centroid, Centroid - Centroid of triangle and tetrahedon, Centroid - Centroids of cones and pyramids, Centroid - Centroid and convexity, Centroid - Integral formula, Centroid - Center of symmetry, Centroid - Physical centroids

Read more here: » Centroid: Encyclopedia II - Centroid - Integral formula

The center of mass is defined as the weighted average of position, with in the discrete case the masses as weights, and in the continuous case the density function as the weight function. Thus the center of mass of an object is the position vector given by: . ...

See also:

Center of mass, Center of mass - Example, Center of mass - Comparison with center of gravity, Center of mass - Definition, Center of mass - More formulas, Center of mass - Aeronautical significance, Center of mass - Motion of the center of mass, Center of mass - Examples, Center of mass - Barycenter, Center of mass - Animations

Read more here: » Center of mass: Encyclopedia II - Center of mass - Definition

The center of mass is an important point on an aircraft, as it defines the amount of mass forward or behind the center of gravity that needs to be moved in order to pitch the plane up or down without applying any external forces. In conventional designs the center of mass is often located very near the line 1/3rd back from the front of the wing. That is the line where most wings generate their lift, known as the center of pressure, so by balancing the plane at that point, the lift and weight balance out with no net torque. The center ...

See also:

Center of mass, Center of mass - Example, Center of mass - Comparison with center of gravity, Center of mass - Definition, Center of mass - More formulas, Center of mass - Aeronautical significance, Center of mass - Motion of the center of mass, Center of mass - Examples, Center of mass - Barycenter, Center of mass - Animations

Read more here: » Center of mass: Encyclopedia II - Center of mass - Aeronautical significance

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, in ...

See also:

Center of mass, Center of mass - Example, Center of mass - Comparison with center of gravity, Center of mass - Definition, Center of mass - More formulas, Center of mass - Aeronautical significance, Center of mass - Motion of the center of mass, Center of mass - Examples, Center of mass - Barycenter, Center of mass - Animations

Read more here: » Center of mass: Encyclopedia II - Center of mass - Motion of the center of mass

In the discrete case: where n is the number of mass particles. Or in the continuous case: where ρ(s) is the scalar mass density as a function of the position vector. If an object has uniform density then the center of mass is the same thing as the centroid. ...

See also:

Center of mass, Center of mass - Example, Center of mass - Comparison with center of gravity, Center of mass - Definition, Center of mass - More formulas, Center of mass - Aeronautical significance, Center of mass - Motion of the center of mass, Center of mass - Examples, Center of mass - Barycenter, Center of mass - Animations

Read more here: » Center of mass: Encyclopedia II - Center of mass - More formulas

Greek mathematics has origins that are presumed to go back to the 7th century BC, but are not easily documented. It is generally believed that it built on the computational methods of earlier Babylonian and Egyptian mathematics, and it may well have had Phoenician influences. Some of the most well-known figures in Greek mathematics are Pythagoras, a shadowy figure from the isle of Samos associated partly with number mysticism and numerology, but more commonly with his theorem, and Euclid, who is known for his Elements, a canon of geom ...

See also:

Greek mathematics, Greek mathematics - Origins, Greek mathematics - Famous Greek mathematicians

Read more here: » Greek mathematics: Encyclopedia II - Greek mathematics - Origins

Pappus flourished about the end of the 3rd century A.D. In a period of general stagnation in mathematical studies, he stands out as a remarkable exception. How far he was above his contemporaries, how little appreciated or understood by them, is shown by the absence of references to him in other Greek writers, and by the fact that his work had no effect in arresting the decay of mathematical science. In this respect the fate of Pappus strikingly resembles that of Diophantus. In his Collection, Pappus gives no indication of the date of ...

See also:

Pappus of Alexandria, Pappus of Alexandria - His Work, Pappus of Alexandria - Theorems, Pappus of Alexandria - Reference

Read more here: » Pappus of Alexandria: Encyclopedia II - Pappus of Alexandria - His Work

In geometry, there are several theorems that are known by the generic name Pappus's Theorem, attributing them to Pappus of Alexandria. They include: Pappus's centroid theorem, the Pappus chain, Pappus's harmonic theorem, and Pappus's hexagon theorem. ...

See also:

Pappus of Alexandria, Pappus of Alexandria - His Work, Pappus of Alexandria - Theorems, Pappus of Alexandria - Reference

Read more here: » Pappus of Alexandria: Encyclopedia II - Pappus of Alexandria - Theorems