How do you determine the centroid of an object?

This is the centroid formula for any dimension, ri-pointer.

1-dimensional, the integral formula that the landlord wants to know, can be written.

xm=f(pdr*r) m total (f refers to the integral symbol, in fact, the horizontal in the middle of f should be removed, I won't play it much; p refers to the density of the line, because it is 1 dimensional, only the density of ** remains; r refers to the vector position of each point on the object, m always refers to the total mass of the object, and m always can also be expressed as an integral as f(pdr)).

For example, a rod of uniform mass with equal density everywhere is pxm=f(pdr*r) (pr).

1 2*PR Squared (PR).

1 2*r is in the center of the pole.

If the pole is not uniform, it belongs to the top-heavy type, and the density everywhere is p=kr, that is, the density of the feet is small, and the density of the head is large, and there is a proportional relationship.

Then xm=f(krdr*r) f(krdr)(1 3*kr cubic) (1 2*kr squared)2 3*r, which is two-thirds of the way from the foot.

The center of mass - the center of mass of an object (or system of things) is an important reference point for studying the mechanical motion of an object (or system of things). When a force (or resultant force) passes through the point, the object only translates and does not rotate; Otherwise, the object will rotate around that point while the movement occurs. When studying the motion of the center of mass, the mass of the object can be considered as concentrated in the center of mass.

Theoretically, the centroid is the mean center of the mass distribution of an object using the "weighted average method".

1. For the curve l, let the density formula be f(x,y), then the centroid.

The formula is: <>

This is the x-coordinate of the centroid, similar to finding another coordinate. At the same time, this formula can be generalized to the multivariate noisy function to find the integral, and the principle is still that the required coordinates are multiplied by the density formula integral divided by the density formula to do the integral;

2. For the enclosed region d, the density formula is f(x,y), and the formula for finding the centroid is as follows.

This is the x-coordinate of the centroid, similar to finding another coordinate. At the same time, this formula can be generalized to the integration of multivariate functions, and the principle is still that the required coordinates are multiplied by the density formula integral divided by the density formula to do the integration.

<> centroid coordinate formula:

The centroid coordinates are equal to the weighted average of all the points with respect to each coordinate.

Centroid: The center of mass is referred to as the centroid, which refers to an imaginary point on the material system where mass is considered to be concentrated. Unlike the center of gravity, the center of mass does not have to be in a system with a gravitational field.

It is important to note that the centroid and center of gravity of the same matter system are usually not at the same imaginary point unless the gravitational field is homogeneous.

If you choose different coordinate systems, the specific values of the centroid coordinates will be different, but the relative position of the centroid relative to each particle in the particle system has nothing to do with the selection of the coordinate system. The centroid of the particle system is only related to the mass of each particle and the relative position of the distribution.

The formula is dxdxdy=area of the center of gravity abscissa d.

The centroid calculation formula is dxdxdy=area of the abscissa d of the center of gravity, dydxdy=area of the ordinate d of the center of gravity. The centroid is the geometric center of the cross-sectional figure, the centroid is for the solid object, and the centroid is for the abstract geometry, and for the solid object with uniform density, the centroid and the centroid coincide.

Quality. The geometric center of a convex object is always inside it. The geometric center of a non-convex object may be outside, such as a ring or bowl with a geometric center that is not internal.

The center of gravity of the triangle is connected to the three vertices, and the six triangles formed are equal in area, and the distance from the vertex to the center of gravity is the midline.

The center of gravity, the outer center, the vertical center, the nine-point circle center and the four-point collinear. The center of gravity, the heart, the Nigel point, and the four points similar to the center of gravity are collinear. The center of gravity of the triangle is also the center of gravity of the midpoint triangle.

The centroid is the geometric center of the triangle, often referred to as the center of gravity, and the intersection of the three midlines (the line connecting the vertices and the midpoints of the opposite side) of the triangle is the center of gravity.

The center of gravity and the center of mass generally coincide.

The position of the center of gravity of an object, an object with a uniform distribution of mass (a homogeneous object), and the position of the center of gravity is only related to the shape of the object. For example, the center of gravity of a uniform thin straight rod is at the midpoint of the rod, the center of a uniform sphere is at the center of the sphere, and the center of gravity of a uniform cylinder is at the midpoint of the axis. The center of gravity of an irregular object can be determined by the suspension method, and the center of gravity of the object is not necessarily on the object.

In the case of an object with uneven mass distribution, the position of the center of gravity is not only related to the shape of the object, but also to the distribution of mass within the object. The center of gravity of a truck changes with how much is loaded and where it is loaded, and the center of gravity of a crane changes with the weight and height of the object being lifted.

The junior high school textbook talks about the center of gravity, which is the point at which the resultant force of gravity on an object is evenly distributed over the mass.

, the center of gravity is at the center of symmetry of the object

Here is the introduction of the centroid.

The concept that for an object, the center of mass and the center of gravity are the same point So why distinguish between the center of mass and the center of gravity? Because only near the ground, gravitational acceleration.

When there is no concept of gravity, there is no concept of center of gravity, such as in a spaceship.

, the object is in a weightless state, there is no gravity, of course, there is no center of gravity, but the point on the object still exists, that is, the center of mass The center of mass is a broader concept than the center of gravity For example, when we study the motion of microscopic particles, microscopic particles also have internal structures, and generally do not need to consider the role of gravity, but the center of mass of microscopic particles is still important, and the motion of the center of mass of the particle can represent the overall motion of the particle

There is a very important theorem in physics, which is called the "centroid motion theorem" in China For any object, regardless of its size, shape, or whether it will be deformed, the centroid motion theorem can be applied Let the mass of the object be m, the center of mass is point C, the acceleration of the center of mass is ac, and the "external force" exerted by other objects is f1(e), f2(e), 、...The centroid motion theorem is.

mac＝f1(e)＋f2(e)＋…

Two explanations of the centroid motion theorem:

1. The theorem of centroid motion only involves the external force on the object, and the complex interaction force (internal force) inside the object does not appear in the theorem

2. The idea of the centroid motion theorem is to "pseudo-particleize" complex real orange hail objects, its mathematical form and Newton's second law of particles.

Similarly, in American textbooks, it is simply called Newton's second law, and the Chinese are probably too rigorous! Although the name of the centroid motion theorem has not been seen by readers, in the middle school curriculum, when the size and shape of an object cannot be ignored, Newton's second definite law is actually the centroid motion theorem

It may be difficult for the reader to understand this way, so let's take a look at it through an example

Since the frictional force on the rear wheels is forward, the whole "man and bicycle" will gain forward acceleration under the action of friction In fact, the centroid motion theorem is used here