In uniform gravitational field the center of gravity coincides with the center of masses. In geometry of the concept "centre of gravity" and "center of masses" are also equivalent as existence of gravitational field is not considered. The center of masses is called still the center of inertia and a barycentre (from Greek barus − heavy, kentron − the center). It characterizes the movement of a body or the system of particles. So, at free fall the body rotates around the center of inertia.
Instruction
1. Let the system consist of two identical points. Then the center of gravity, obviously, is located in the middle between them. If with coordinates of x1 and x2 the different mass of m1 and m2, then coordinate of the center of masses have points x (c)= (m1 · x1+m2 · x2) / (m1+m2). Depending on chosen "zero" reference system, coordinates can be and negative.
2. Points on the plane have two coordinates: x and y. At a task in space still the third coordinate of z is added. Not to paint each coordinate separately, it is convenient to consider point radius vector: r=x · i+y · j+z · k, where i, j, k − Horta of coordinate axes.
3. Let now the system consist of three points with a mass of m1, m2 and m3. Their radius vectors, respectively, r1, r2 and r3. Then radius vector of their center of gravity of r(c)= (m1 · r1+m2 · r2+m3 · r3) / (m1+m2+m3).
4. If the system consists of any number of points, then radius vector, by definition, is on a formula: r(c)= ∑ m(i) · r(i)/∑m(i). Summation is made by index i (registers from below from the sign of the sum ∑). Here m(i) − mass of some i-go of an element of a system, r(i) − its radius vector.
5. If the body is uniform in weight, the sum passes into integral. Hurt mentally a body into infinitely small pieces the mass of DM. As the body is uniform, the mass of each piece can be written down as dm=ρ\· dV, where dV − the elementary volume of this piece, ρ − density (it is identical on all volume of a uniform body).
6. Integrated summation of mass of all pieces will give the mass of all body: ∑m(i)= ∫ dm=M. So, r(c) = 1/M turns out · ∫ ρ\· dV · dr. Density, a constant, it is possible to take out from under a sign of integration: r (c)=ρ/M·∫dV·dr. For direct integration it is required to establish concrete function between dV and dr which depends on figure parameters.
7. For example, the piece center of gravity (long uniform core) is in the middle. The center of mass of the sphere and sphere is located in the center. The barycentre of a cone is on a quarter of height of an axial piece, considering from the basis.
8. It is easy to define a barycentre of some simple figures on the plane geometrically. For example, for a flat triangle it will be a point of intersection of medians. For a parallelogram − a point of intersection of diagonals.
9. The center of gravity of a figure can be defined and by practical consideration. Cut out from the sheet of dense paper or cardboard any figure (for example, the same triangle). Try to establish it on a tip of vertically extended finger. That place on a figure for which it will turn out to make it and will be the center of inertia of a body.