You will need
- the formula for calculating the center of mass.
Note that the position of the center of mass depends on how distributed the volume of his body weight. The center of mass may not even be in the body, an example of such object is a homogeneous ring whose centre of mass is its geometric centre. That is – in the void. When calculating the center of mass can be considered a mathematical point, which focuses the whole body weight.
The concept of center of mass and center of gravity very close, so in the calculations, in most cases, they can be considered synonyms. The only difference is that the concept of center of gravity requires gravity and center of mass is present in the absence of gravity. The body is falling freely without rotation, moves under the force of gravity, applied to all its points, while its center of mass coincides with the center of gravity. To determine the center of mass in the classical mechanics uses the following formula.
Here R. TS.m. – the radius-vector of center of mass, mi – mass of i-th point, ri is the radius-vector of the i-th point of the system. In practice, in many cases, it is easy to find the center of mass if the object has a strict geometric form. For example, a homogeneous rod it is located exactly in the middle. The parallelogram at the intersection of diagonals of a triangle is the intersection point of the medians, and of a regular polygon center of mass is at the center of rotational symmetry.
For more complex bodies the task of calculation becomes complicated in this case, you must break the object into homogeneous volumes. For each of them separately calculated centers of mass, then the values are substituted in the appropriate formula is the final value.
In practice, the need to determine the center of mass (center of gravity) is typically associated with design. For example, when designing a vessel it is important to ensure its stability. If the center of gravity will be very high, the ship may capsize. How to calculate the right setting for such complex object like a ship? To do this, there are centers of gravity of the individual elements and units, and then the values are added based on their location. When designing the center of gravity generally placed as low as possible, so the most heavy units have at the bottom.
Advice 2 : How to find the center of gravity of the body
The centre of gravity of any body is considered a geometric point, where all forces of gravity acting on the body when it is any turn. Sometimes it does not coincide with any of the points of the body.
You will need
- - the body
- - line
If the body's center of gravity which you want to define homogeneous and has a simple rectangular shape, round, spherical, cylindrical, square, and has a center of symmetry, in such a case, the center of gravity coincides with the center of symmetry.
For a homogeneous rod the center of gravity is in its middle, that is, in its geometric center. The same result is obtained for the uniform circular disk. Its centre of gravity lies at the point of intersection of the diameters of the circle. Therefore, the center of gravity of the Hoop will be in the center, outside points of the Hoop. Find the center of gravity of a homogeneous sphere is at the geometric center of the sphere. The center of gravity of a homogeneous rectangular parallelepiped will be at the intersection of its diagonals.
If the body has arbitrary shape if it is patchy, for example, has recesses to calculate the position of the center of gravity difficult. See where such a body is the point of intersection of all forces of gravity that act on this figure when it is turning. To find this point is the easiest experimentally using the method of free suspension of the body on the thread.
Consistently attach the body to the filament for different points. At equilibrium the centre of gravity of body must lie on a line coinciding with the line of thread, otherwise gravity brings the body in motion.
Using a ruler and pencil draw a vertical straight coinciding with the direction of the filaments, which were attached at different points. Depending on the complexity of body shape will need to spend two to three lines. They must all intersect in one point. This point will be the center of gravity of the body, because the center of gravity must remain on all such lines.
Determine by the method of hanging the center of gravity as flat shapes, and more complex body, the form of which may vary. For example, two bars connected by a hinge, in the unfolded state have the center of gravity at the geometric center, and in a bent – their center of gravity is outside these bars.