The center of mass is a geometric point located inside a body that determines the distribution of the mass of this body. Any body can be represented as the sum of a certain number of material points. In this case, the position of the center of mass determines the radius vector.
Formula 1 - Radius of the center of mass vector.
mi is the mass of this point.
ri is the radius vector of the point.
If you sum up the masses of all material points, you get the mass of the entire body. The position of the center of mass is affected by the uniformity of mass distribution over the volume of the body. The center of mass can be located both inside the body and outside it. Let's say for a ring, the center of mass is at the center of the circle. Where there is no substance. In general, for symmetrical bodies with a uniform distribution of mass, the center of mass is always located at the center of symmetry or on its axis.
Figure 1 - Centers of mass of symmetrical bodies.
If some force is applied to the body, it will begin to move. Imagine a ring lying on the surface of a table. If you apply force to it, and simply start pushing, then it will slide along the surface of the table. But the direction of movement will depend on the place where the force is applied.
If the force is directed from the outer edge to the center, perpendicular to the outer surface, then the ring will begin to move rectilinearly along the table surface in the direction of application of the force. If a force is applied tangentially to the outer radius of the ring, then it will begin to rotate relative to its center of mass. Thus, we can conclude that the motion of a body consists of the sum of translational and rotational motion relative to the center of mass. That is, the movement of any body can be described by the movement of a material point located at the center of mass and having the mass of the entire body.
Figure 2 - Translational and rotational motion of the ring.
There is also the concept of center of gravity. In general, this is not the same thing as the center of mass. The center of gravity is the point relative to which the total moment of gravity is zero. If you imagine a rod, say 1 meter long, 1 cm in diameter, and uniform in cross-section. Metal balls of equal mass are fixed at the ends of the rod. Then the center of mass of this rod will be in the middle. If this rod is placed in a non-uniform gravitational field, then the center of gravity will be shifted towards greater field strength.
Figure 3 - Body in a non-uniform and uniform gravitational field.
On the surface of the earth, where the force of gravity is uniform, the center of mass practically coincides with the center of gravity. For any constant uniform gravitational field, the center of gravity will always coincide with the center of mass.
In this section we will consider in detail a special case of a system of actually parallel forces. Namely, any material body or system of material points (discrete particles) located on Earth is subject to the action of gravity. Therefore, each particle of such mechanical systems is affected by its gravity force. Strictly speaking, all these forces are directed to one point towards the center of the Earth. But since the dimensions of earthly bodies are very small compared to the radius of the Earth (we assume that the volumes in which discrete particles are contained are also small), then with a high degree of accuracy these forces can be considered parallel. The paragraph is devoted to bringing this system of forces into account.
Specific gravity
Let us select an elementary particle in the body with a volume so small that its position can be determined by one radius vector. Let the weight of this particle be Quantity
is called specific gravity, and the quantity
Body density.
In the SI system of units, specific weight has the dimension
and density
In general, specific gravity and density are functions of the coordinates of body points. If they are the same for all points, then the body is called homogeneous.
The resultant of all elementary forces of gravity is equal to their sum and represents the weight of the body. The center of these parallel forces is called the center of gravity of the body.
Obviously, the position of the center of gravity in a body does not depend on the orientation of the body in space. This statement follows from the earlier remark that the center of parallel forces does not change its position when all forces rotate through the same angle around their points of application.
Formulas that determine the centers of gravity of a body and a system of discrete particles
To determine the center of gravity of a body, we divide it into fairly small particles with a volume of . To each of them we apply a force of gravity equal to
The resultant of these parallel forces is equal to the weight of the body, which we denote by
The radius vector of the center of gravity of the body, which we denote by , is determined by the formulas of the previous paragraph as the center of parallel forces. Thus, we will have
If the center of gravity of a system of discrete particles is determined, then the specific gravity of the particle, V, is its volume - the radius vector that determines the position of the particle. The last formula determines in this case the center of mass of a system of discrete particles.
If the mechanical system is a body formed by a continuous collection of particles, then in the limit of the sum of the last formulas they turn into integrals and the radius vector of the center of gravity of the body can be calculated using the formula:
where the integrals extend over the entire volume of the body.
If the body is homogeneous, then the last formula has the form:
where V is the volume of the entire body.
Thus, when the body is homogeneous, determining its center of gravity is reduced to a purely geometric problem. In this case, we talk about the center of gravity of the volume.
Body center of mass
The introduced concept of the center of gravity makes sense only for bodies (small compared to the size of the Earth) located near the Earth's surface. At the same time, the method of calculating the coordinates of the center of gravity allows it to be used to calculate the coordinates of a point characterizing the distribution of matter in the body. To do this, we should consider not the weight of the particles, but their mass. Each particle of a body with a volume has a mass
and replacing in the previously obtained formula with we arrive at the equality:
which defines a point called the center of mass or center of inertia of the body.
If the system consists of material points whose masses then the center of mass of the system is found by the formula:
where is the mass of the entire system. The radius vector of the center of mass of the body depends on the choice of the origin of coordinates O. If the center of inertia itself is chosen as the origin of coordinates, it will be equal to zero:
The concept of the center of mass can be introduced independently of the concept of the center of gravity. Thanks to this, it applies to any mechanical systems.
Static moments
The expressions are called, respectively, the static moments of weight, volume and mass of the body relative to point O. If the center of mass of the body is chosen as the point (origin of coordinates), then the static moments of the body relative to the center of mass will be equal to zero, which will be used repeatedly in the future.
Methods for calculating the center of mass
In the case of a body of complex shape, determining the coordinates of the center of mass using the general formulas given above usually involves painstaking calculations. In some cases, they can be significantly simplified if you use the following methods.
1) Symmetry method. Let the body have a center of material symmetry. This means that each particle with mass and radius vector drawn from this center corresponds to a particle with the same mass and radius vector. In this case, the static moment of the body mass will vanish and
Consequently, the center of mass will coincide in this case with the center of material symmetry of the body. For homogeneous bodies, this means that the center of mass coincides with the geometric center of the body's volume. If a body has a plane of material symmetry, then the center of mass is in this plane. If the body is symmetrical about an axis, then the center of mass is on this axis.
2) Method of splitting into parts. If a body can be divided into a finite number of parts, the masses and positions of the centers of mass of which are known, then we will find the center of mass of the entire body as follows: imagine that the masses of these parts are concentrated at their centers of mass, then the body is reduced to a finite number of material points. The center of mass of a system of material points is simply calculated using the given formulas.
3) Negative mass method. Let a homogeneous body of mass have holes and its center of mass is determined by the radius vector. If these holes of the bodies are filled with the substance of which the body consists, then they will have certain masses and centers of mass. The masses of these filled holes will be equal and the radii are the vectors of their centers of mass. Then the center of mass of the body with the filled holes will be determined by the radius vector
where M is the mass of the body with filled holes. From here
But therefore
The resulting formula indicates the following method for determining the center of mass of a body with holes. Mentally fill the holes with the substance that makes up the body. Then they find the mass and center of mass of the body obtained in this way, as well as the masses and centers of mass of the substance filling the holes, and assign a minus sign to these masses. After this, the center of mass of the body in question can be calculated using the partitioning method.
Any mechanical system, just like any body, has such a remarkable point as the center of mass. A person, a car, the Earth, the Universe, i.e., any object has it. Very often this point is confused with the center of gravity. Although they often coincide with each other, they have certain differences. We can say that the center of mass of a mechanical system is a broader concept compared to its center of gravity. What is it and how to find its location in the system or in a separate object? This is exactly what will be discussed in our article.
Concept and formula of definition
The center of mass is a certain point of intersection of straight lines, parallel to which external forces act, causing translational motion of a given object. This statement is true both for an individual body and for a group of elements that have a certain connection with each other. The center of mass always coincides with the center of gravity and is one of the most important geometric characteristics of the distribution of all masses in the system under study. Let us denote by m i the mass of each point of the system (i = 1,…,n). The position of any of them can be described by three coordinates: x i, y i, z i. Then it is obvious that the mass of the body (the entire system) will be equal to the sum of the masses of its particles: M=∑m i. And the center of mass (O) itself can be determined through the following relationships:
X o = ∑m i *x i /M;
Y o = ∑m i *y i /M;
Z o = ∑m i *z i /M.
Why is this point interesting? One of its main advantages is that it characterizes the movement of an object as a whole. This property allows the use of the center of mass in cases where the body has large dimensions or irregular geometric shape.
What you need to know to find this point
Practical Application
The concept under consideration is widely used in various fields of mechanics. Typically the center of mass is used as the center of gravity. The latter represents such a point, hanging an object, from which it will be possible to observe the invariance of its position. The center of mass of a system is often calculated when designing various parts in mechanical engineering. It also plays a big role in ensuring balance, which can be applied, for example, when creating alternative versions of furniture, vehicles, construction, warehousing, etc. Without knowledge of the basic principles by which the center of gravity is determined, it would be difficult organize the safety of work with massive loads and any large objects. We hope that our article was useful and answered all questions on this topic.
Draw a diagram of the system and mark the center of gravity on it. If the found center of gravity is outside the object system, you received an incorrect answer. You may have measured distances from different reference points. Repeat the measurements.
- For example, if children are sitting on a swing, the center of gravity will be somewhere between the children, and not to the right or left of the swing. Also, the center of gravity will never coincide with the point where the child is sitting.
- These arguments are valid in two-dimensional space. Draw a square that will contain all the objects of the system. The center of gravity should be inside this square.
Check your math if you get a small result. If the reference point is at one end of the system, a small result places the center of gravity near the end of the system. This may be the correct answer, but in the vast majority of cases this result indicates an error. When you calculated the moments, did you multiply the corresponding weights and distances? If instead of multiplying you added the weights and distances, you would get a much smaller result.
Correct the error if you found multiple centers of gravity. Each system has only one center of gravity. If you found multiple centers of gravity, you most likely did not add up all the moments. The center of gravity is equal to the ratio of the “total” moment to the “total” weight. There is no need to divide “every” moment by “every” weight: this way you will find the position of each object.
Check the reference point if the answer differs by some integer value. In our example, the answer is 3.4 m. Let's say you got the answer 0.4 m or 1.4 m, or another number ending in ".4". This is because you did not choose the left end of the board as your starting point, but a point that is located a whole amount to the right. In fact, your answer is correct no matter what reference point you choose! Just remember: the reference point is always at position x = 0. Here's an example:
- In our example, the reference point was at the left end of the board and we found that the center of gravity was 3.4 m from this reference point.
- If you choose as a reference point a point that is located 1 m to the right from the left end of the board, you will get the answer 2.4 m. That is, the center of gravity is 2.4 m from the new reference point, which, in turn, is located 1 m from the left end of the board. Thus, the center of gravity is at a distance of 2.4 + 1 = 3.4 m from the left end of the board. It turned out to be an old answer!
- Note: when measuring distances, remember that the distances to the “left” reference point are negative, and to the “right” reference point are positive.
Measure distances in straight lines. Suppose there are two children on a swing, but one child is much taller than the other, or one child is hanging under the board rather than sitting on it. Ignore this difference and measure the distances along the straight line of the board. Measuring distances at angles will give close but not entirely accurate results.
- For the see-saw board problem, remember that the center of gravity is between the right and left ends of the board. Later, you will learn to calculate the center of gravity of more complex two-dimensional systems.
The term “center of mass” is used not only in mechanics and in calculations of motion, but also in everyday life. It’s just that people don’t always think about what laws of nature are manifested in a given situation. For example, figure skaters in pair skating actively use the center of mass of the system when they spin holding hands.
The concept of center of mass is also used in ship design. It is necessary to take into account not just two bodies, but a huge number of them and bring everything to a single denominator. Errors in calculations mean a lack of stability of the ship: in one case, it will be excessively submerged in water, risking sinking with the slightest waves; and in another they are too elevated above sea level, creating the danger of turning over on their side. By the way, this is why every thing on board must be in its place, as specified by the calculations: the most massive ones are at the very bottom.
The center of mass is used not only in relation to celestial bodies and the design of mechanisms, but also in the study of the “behavior” of particles of the microworld. For example, many of them are born in pairs (electron-positron). Possessing initial rotation and obeying the laws of attraction/repulsion, they can be considered as a system with a common center of mass.
