How to determine the coordinates of the center of gravity
In a uniform gravitational field, the center of gravity coincides with the center of mass. In geometry, the concept of "center of gravity" and "center of mass" as equivalent, since the existence of the gravitational field is not considered. The center of mass is also called the center of inertia and barycenter (from the Greek barus -. Heavy, kentron - center). He describes the motion of a body or a system of particles. Thus, when the free fall body rotates around its center of mass.
Instruction how to determine the coordinates of the center of gravity
Let the system consists of two identical points. Then the center of gravity, obviously located midway between them. If the points with coordinates x1 and x2 have different masses m1 and m2, the center of mass coordinate of x (c) = (m1 · x1 + m2 · x2) / (m1 + m2). Depending on the "zero" reference system, the coordinates may be negative.
The points on the two planes have coordinates: x and y. When you set in space added a third coordinate z. To paint each coordinate separately, it is convenient to consider the radius vector of the point: r = x · i + y · j + z · k, where i, j, k - the unit vectors of the coordinate axes.
Suppose now that the system consists of three points with masses m1, m2 and m3. Their radius vectors, respectively, r1, r2 and r3. Then the radius vector r of the center of gravity (c) = (m1 · r1 + m2 · r2 + m3 · r3) / (m1 + m2 + m3).
If the system consists of an arbitrary number of points, then the radius vector, by definition, is given by: r (c) = Σm (i) · r (i) / Σm (i). The summation is over the index i (written below the summation sign Σ). Where m (i) - a mass of i-th element of the system, r (i) - the radius vector.
If the body is homogeneous by weight, the amount transferred to the integral. Break the mental body by an infinitely small pieces of mass dm. Since the body is homogeneous, each piece weight can be written as dm = ρ · dV, where dV - elementary volume of the piece, ρ - density (the same uniform throughout the volume of the body).
Integral sum weight of all pieces will give the entire body weight: Σm (i) = ∫dm = M. Thus obtained r (c) = 1 / M · ∫ρ · dV · dr. Density, constant value, can be taken out from under the integral sign: r (c) = ρ / M · ∫dV · dr. For direct integration will need to install a specific function between dV and dr, which depends on the shape parameters.
For example, the length of the center of gravity (long uniform rod) is in the middle. The center of mass of the ball and the sphere is located in the center. Barycenter of the cone is at a quarter of the height of the axial length, measured from the base.
Barycenter some simple figures on a plane is easy to determine geometrical. For example, for this will be a flat triangle intersection of medians. For parallelogram - the point of intersection of the diagonals.
The center of gravity and shape can be determined empirically. Cut a sheet of heavy paper or cardboard any figure (for example, the same triangle). Try to install it on the tip of a vertically elongated finger. That place in the figure for which will do this, and will be the center of inertia of the body.