Mechanics

# Center of mass

When launched, the iron bar has rotation and translation movement.

Let’s see the figure above: in it we can see an extended body, that is, a body that has translational movement and rotational movement. In the figure we see that a series of photos were taken, at very small intervals, of the moving iron bar. Still from the figure we can see that this bar describes a parabolic trajectory. It is as if the entire mass of the iron bar were concentrated at a single point and all the forces acting on each particle of the iron bar were also applied at that point.

This particular point is called the center of mass . An interesting fact is that the center of mass can be outside the body. Therefore, the existence of the center of mass is not limited to cases of rigid objects, as mentioned above. It also exists for systems formed by separate bodies. The Solar System, for example, has a center of mass and it is around this center of mass that the planets revolve, not around the center of the Sun, although the center of mass of the Solar System is very close to the center of the Sun.

Based on these examples, we can already see the importance of the center of mass in the analysis of the movement of a system of particles. Another important fact that we have to mention about the center of mass is that:

– internal forces do not affect the movement of the center of mass of a system.

So, for example, if during motion a body changes only as a result of internal forces, this will not change the movement of the center of mass. To illustrate this, let’s consider the center of mass of the human body. When a person is stretched out, their center of mass (C) is just below the navel. But if she raises her arms or legs, or bends her body, or her arms or legs, the center of mass will go to another position.

Location of center of mass

Let’s now get the position of the center of mass. Let us consider initially the case of a system formed by n particles of masses: 1 , m 2 , m 3 , … mn and that are in the same plane. We adopt an orthogonal Cartesian coordinate system, contained in this plane. These particles will have:

abscissa x 1 ,x 2 ,x 3 … x n
ordinates y 1 ,y 2 ,y 3 … y n

In this way, we can determine the center of mass in the x-plane and in the y-plane, as follows:

It is obvious that if we use another coordinate system, we will have other values ​​for x1, x2, etc. and y1, y2, etc. However, using the equations above, it is possible to show that this does not change the position of the center of mass in relation to the particles of the system, that is, for any coordinate system adopted, we will obtain the center of mass always in the same position in relation to the particles of the system. particles.