Definition
In Newtonian mechanics, body mass is a scalar physical quantity, which is a measure of its inertial properties and a source of gravitational interaction. In classical physics, mass is always a positive quantity.
Weight- an additive quantity, which means: the mass of each set of material points (m) is equal to the sum of the masses of all individual parts of the system (m i):
In classical mechanics, one considers:
- body mass is not dependent on the movement of the body, on the impact of other bodies, the location of the body;
- the law of conservation of mass is fulfilled: the mass of a closed mechanical system of bodies is constant in time.
inertial mass
The property of the inertia of a material point is that if an external force acts on the point, then it has a finite acceleration in absolute value. If there are no external influences, then in the inertial frame of reference the body is at rest or moves uniformly and rectilinearly. Mass is included in Newton's second law:
where the mass determines the inertial properties of the material point (inertial mass).
gravitational mass
The mass of a material point is included in the law of universal gravitation, while it determines the gravitational properties of a given point. At the same time, it is called the gravitational (heavy) mass.
It has been empirically obtained that for all bodies the ratios of inertial masses to gravitational masses are the same. Therefore, if we correctly choose the value of the constant gravitation, then we can get that for any body the inertial and gravitational masses are the same and are associated with the force of gravity (F t) of the chosen body:
where g is the free fall acceleration. If observations are made at the same point, then the accelerations of free fall are the same.
Formula for calculating mass through body density
Body weight can be calculated as:
where is the density of the body substance, where the integration is carried out over the volume of the body. If the body is homogeneous (), then the mass can be calculated as:
Mass in special relativity
In SRT, mass is invariant, but not additive. It is defined here as:
where E is the total energy of a free body, p is the momentum of the body, c is the speed of light.
The relativistic mass of a particle is determined by the formula:
where m 0 is the rest mass of the particle, v is the velocity of the particle.
The basic unit of mass in the SI system is: [m]=kg.
In GHS: [m]=gr.
Examples of problem solving
Example
Exercise. Two particles fly towards each other with velocities equal to v (velocity is close to the speed of light). When they collide, a completely inelastic impact occurs. What is the mass of the particle that formed after the collision? The masses of the particles before the collision are equal to m.
Solution. With an absolutely inelastic collision of particles that had the same masses and velocities before the impact, one particle at rest is formed (Fig. 1), the rest energy of which is equal to:
In our case, the law of conservation of mechanical energy is fulfilled. Particles have only kinetic energy. According to the condition of the problem, the speed of particles is close to the speed of light, therefore? we operate with the concepts of relativistic mechanics:
where E 1 is the energy of the first particle before impact, E 2 is the energy of the second particle before impact.
We write the law of conservation of energy in the form:
From expression (1.3) it follows that the mass of the particle obtained as a result of the merger is equal to:
Example
Exercise. What is the mass of 2m 3 copper?
Moreover, if the substance (copper) is known, then it is possible to find its density using a reference book. The density of copper will be considered equal to Cu =8900 kg/m 3 . For the calculation, all quantities are known. Let's do the calculations.
Mass (physical value) Weight, a physical quantity, one of the main characteristics of matter, which determines its inertial and gravitational properties. Accordingly, M. is inert and M. gravitational (heavy, gravitating).
The concept of M. was introduced into the mechanics of I. Newton. In Newton's classical mechanics, M. is included in the definition of momentum ( momentum) body: the momentum p is proportional to the speed of the body v,
p = m.v.
The coefficient of proportionality - a constant value m for a given body - is the M. of the body. An equivalent definition of M. is obtained from the equation of motion of classical mechanics
f = ma.
Here M. is the coefficient of proportionality between the force acting on the body f and the acceleration of the body caused by it a. The mass defined by relations (1) and (2) is called inertial mass, or inertial mass; it characterizes the dynamic properties of the body, is a measure of the inertia of the body: at a constant force, the greater the M. of the body, the less acceleration it acquires, that is, the slower the state of its movement changes (the greater its inertia).
Acting on different bodies with the same force and measuring their accelerations, one can determine the ratios of M. of these bodies: m 1
:m 2
:m 3
... = a 1
: a 2
: a 3
...; if one of the M. is taken as a unit of measurement, one can find the M. of the remaining bodies.
In Newton's theory of gravitation, magnetism appears in a different form - as a source of the gravitational field. Each body creates a gravitational field proportional to the M. of the body (and is affected by the gravitational field created by other bodies, the strength of which is also proportional to the M. bodies). This field causes the attraction of any other body to this body with a force determined by Newton's law of gravity:
where r is the distance between the bodies, G is the universal gravitational constant, a m 1
and m 2
‒ M. attracting bodies. From formula (3) it is easy to obtain a formula for weightР bodies of mass m in the Earth's gravitational field:
P \u003d m g.
Here g = G M / r 2
is the free fall acceleration in the Earth's gravitational field, and r » R is the Earth's radius. The mass determined by relations (3) and (4) is called the gravitational mass of the body.
In principle, it does not follow from anywhere that magnetism, which creates a gravitational field, also determines the inertia of the same body. However, experience has shown that inertial magnetism and gravitational magnetism are proportional to each other (and with the usual choice of units of measurement, they are numerically equal). This fundamental law of nature is called the principle of equivalence. Its discovery is associated with the name of G. Galilee, who established that all bodies on Earth fall with the same acceleration. A. Einstein put this principle (which he formulated for the first time) at the basis of the general theory of relativity (cf. gravity). The principle of equivalence has been established experimentally with very high accuracy. For the first time (1890‒1906), a precision check of the equality of inert and gravitational magnetism was carried out by L. Eötvös, who found that M. matched with an error of ~ 10-8 . In 1959–64 the American physicists R. Dicke, R. Krotkov, and P. Roll reduced the error to 10-11 , and in 1971 the Soviet physicists V. B. Braginsky and V. I. Panov reduced the error to 10-12 .
The principle of equivalence makes it possible to most naturally determine the M. of a body weighing.
Initially, mass was considered (for example, by Newton) as a measure of the amount of matter. Such a definition has a clear meaning only for comparing homogeneous bodies built from the same material. It emphasizes the additivity of the M. ‒ The M. of a body is equal to the sum of the M. of its parts. The mass of a homogeneous body is proportional to its volume, so we can introduce the concept density‒ M. units of body volume.
In classical physics, it was believed that the M. of a body does not change in any processes. This corresponded to the law of conservation of matter (substance), discovered by M. V. Lomonosov and A. L. Lavoisier. In particular, this law stated that in any chemical reaction the sum of M. of initial components is equal to the sum of M. of final components.
The concept of M. has acquired a deeper meaning in the mechanics of special. A. Einstein's theory of relativity (see. Relativity theory), which considers the movement of bodies (or particles) with very high speeds - comparable to the speed of light with » 3×1010 cm/sec. In the new mechanics - it is called relativistic mechanics - the relationship between the momentum and the velocity of a particle is given by the relation:
At low speeds (v<< с
) это соотношение переходит в Ньютоново соотношение р
= mv
. Поэтому величину m
0
называют массой покоя, а М. движущейся частицы m определяют как зависящий от скорости коэфф. пропорциональности между р
и v
:
With this formula in mind, in particular, they say that the momentum of a particle (body) increases with an increase in its velocity. Such a relativistic increase in the momentum of a particle as its velocity increases must be taken into account when designing particle accelerators high energies. M. rest m 0 (M. in the reference frame associated with the particle) is the most important internal characteristic of the particle. All elementary particles have strictly defined values of m 0 inherent in this kind of particles.
It should be noted that in relativistic mechanics the definition of M. from the equation of motion (2) is not equivalent to the definition of M. as a proportionality factor between the momentum and velocity of a particle, since acceleration ceases to be parallel to the force that caused it, and M. turns out to depend on the direction of the particle's velocity.
According to the theory of relativity, the momentum of a particle m is related to its energy E by the relation:
M. rest determines the internal energy of the particle - the so-called rest energy E 0 \u003d m 0 c 2
. Thus, energy is always associated with M. (and vice versa). Therefore, there is no separate (as in classical physics) law of conservation of M. and the law of conservation of energy - they are merged into a single law of conservation of total (that is, including the rest energy of particles) energy. An approximate division into the law of conservation of energy and the law of conservation of magnetism is possible only in classical physics, when the particle velocities are small (v<< с
) и не происходят процессы превращения частиц.
In relativistic mechanics, magnetism is not an additive characteristic of a body. When two particles combine to form one compound stable state, an excess of energy (equal to binding energy) DE , which corresponds to M. Dm = DE / s 2
. Therefore, the M. of a composite particle is less than the sum of the M. of the particles that form it by the value DE / s 2
(so-called mass defect). This effect is especially pronounced in nuclear reactions. For example, the M. of a deuteron (d) is less than the sum of the M. of a proton (p) and a neutron (n); defect M. Dm is associated with the energy E g of the gamma quantum (g) produced during the formation of a deuteron: p + n ® d + g, E g \u003d Dm c 2
. M.'s defect, which occurs during the formation of a composite particle, reflects the organic connection of M. and energy.
The unit of M. in the CGS system of units is gram, and in International system of units SI - kilogram. The mass of atoms and molecules is usually measured in atomic mass units. It is customary to express the mass of elementary particles either in units of the mass of the electron m e , or in energy units, indicating the rest energy of the corresponding particle. So, the M. of an electron is 0.511 MeV, the M. of a proton is 1836.1 meV, or 938.2 MeV, etc.
The nature of mathematics is one of the most important unsolved problems of modern physics. It is generally accepted that the magnetism of an elementary particle is determined by the fields associated with it (electromagnetic, nuclear, and others). However, the quantitative theory of M. has not yet been created. There is also no theory explaining why the M. of elementary particles form a discrete spectrum of values, and even more so allowing to determine this spectrum.
In astrophysics, the magnetism of a body that creates a gravitational field determines the so-called gravity radius bodies R gr = 2GM/c 2
. Due to gravitational attraction, no radiation, including light, can go outside, beyond the surface of a body with a radius R £ R gr . Stars of this size would be invisible; so they were called black holes". Such celestial bodies must play an important role in the universe.
Lit .: Jammer M., The concept of mass in classical and modern physics, translated from English, M., 1967; Khaikin S. E., physical foundations of mechanics, M., 1963; Elementary textbook of physics, edited by G. S. Landsberg, 7th ed., vol. 1, M., 1971.
Ya. A. Smorodinsky.
Great Soviet Encyclopedia. - M.: Soviet Encyclopedia. 1969-1978 .
See what "Mass (physical quantity)" is in other dictionaries:
- (lat. massa, lit. lump, lump, piece), physical. value, one of the har to matter, which determines its inertial and gravitational forces. sv. The concept of "M." was introduced into mechanics by I. Newton in the definition of the momentum (number of motion) of the body momentum p proportional. ... ... Physical Encyclopedia
- (lat. massa). 1) the amount of substance in the object, regardless of the form; body, matter. 2) in the hostel: a significant amount of something. Dictionary of foreign words included in the Russian language. Chudinov A.N., 1910. MASS 1) in physics, quantity ... ... Dictionary of foreign words of the Russian language
- - 1) in the natural scientific sense, the amount of matter contained in the body; the resistance of a body to a change in its motion (inertia) is called inertial mass; the physical unit of mass is the inert mass of 1 cm3 of water, which is 1 g (gram ... ... Philosophical Encyclopedia
WEIGHT- (in the ordinary view), the amount of substance contained in a given body; the exact definition follows from the basic laws of mechanics. According to Newton's second law, "the change in motion is proportional to the acting force and has ... ... Big Medical Encyclopedia
Phys. the value characterizing the dynamic. sv va tepa. I. m. is included in Newton's second law (and, thus, is a measure of the body's inertia). Equal to gravity. mass (see MASS). Physical Encyclopedic Dictionary. Moscow: Soviet Encyclopedia. Editor-in-Chief A... Physical Encyclopedia
- (heavy mass), physical. a value that characterizes the body's power as a source of gravity; equal to the inertial mass. (see MASS). Physical Encyclopedic Dictionary. Moscow: Soviet Encyclopedia. Editor-in-Chief A. M. Prokhorov. 1983... Physical Encyclopedia
Phys. a value equal to the ratio of mass to count in VA. Unit M. m. (in SI) kg / mol. M \u003d m / n, where M M. m. in kg / mol, m is the mass in va in kg, n is the number in va in moles. Numerical value M. m., vyraz. in kg / mol, equally refers. molecular weight divided by... Big encyclopedic polytechnical dictionary - size, character ka physical. objects or phenomena of the material world, common to many objects or phenomena as qualities. relation, but individual in quantities. relationship for each of them. For example, mass, length, area, volume, electric power. current F ... Big encyclopedic polytechnic dictionary
ON THE PHYSICAL ESSENCE OF MASS
Brusin S.D., Brusin L.D.
annotation. The physical essence of the mass, given by Newton, is explained, and it is shown that the physical essence of the mass is distorted in modern textbooks.
Parameter weight first introduced by Newton and formulated as follows: "The amount of matter (mass) is a measure of that, established in proportion to its density and volume". The amount of a substance was previously determined by weighing it. However, it is known, for example, that the same piece of gold weighs more at the pole than at the equator. Therefore, the introduction of a simple parameter that clearly determines the amount of matter (substance) in the body is the greatest merit of Newton's genius. It allowed formulate the laws of motion and interaction of bodies.
First, Newton defines the momentum of a body as proportional to the amount of matter (mass) of the body, and then defines the inertia of the body (indicating its proportionality to the mass of the body) in the following formulation: “ Innate force of matter is the ability of resistance inherent in it, according to which every single body, since it is left to itself, maintains its state of rest or uniform rectilinear motion. This definition formed the basis of Newton's first law. We will pay attention that the inertia of a body is a property of matter, characterized by the mass of the body.
In accordance with Newton's II law, the amount of matter (mass) of the body affects the acceleration received by the body with the same force, and in accordance with Newton's law of universal gravitation, all bodies are attracted to each other with a force that is directly proportional to the product of the masses (amount of matter) bodies; these forces are called gravitational forces. Experimentally, this law for any bodies was shown by Cavendish. Thus, the same body mass has gravitational and inertial properties (according to Newton, this is due to vborn by the force of matter).
In modern science, the following definition of mass is given: “The mass of a body is a physical quantity that is a measure of its inertial and gravitational properties.” We do not know who and why needed to pervert the deep and simple physical meaning of the concept of mass given by Newton (not mass is a measure of the inertial properties of a body, but the inertial properties of a body are determined by its mass). Historians of science need to sort out this important question. The distortion of the physical essence of the mass led to the following:
1. Concepts appeared inertial mass and gravitational mass, and it took considerable effort and numerous experiments by Eotvos to prove the equality of inertial and gravitational masses, although the definition of mass given by Newton clearly shows that mass is one, but has inertial and gravitational properties.
2. To a misunderstanding of the physical nature of the parameters associated with a misunderstanding of the mass. For example, the essence of the density of a body is not the amount of inertia per unit volume, but the amount of matter (substance) per unit volume.
An erroneous understanding of the physical essence of mass is given in all textbooks, including school textbooks, and the rising generation misperceives the physical essence of the masses. So it is necessary to correct this situation by introducing into all textbooks the above definition of mass given by Newton
Literature:
1. Newton, I. "Mathematical Principles of Natural Philosophy",
M., "Nauka", 1989, p. 22
2. Ibid., p. 25
3. A. A. Detlaf and B. M. Yavorsky, Handbook of Physics, M. Nauka, 1974, p. 36
The concept with which we are familiar from early childhood is the mass. And yet, in the course of physics, some difficulties are associated with its study. Therefore, it is necessary to clearly define how it can be recognized? And why is it not equal to weight?
Determination of mass
The natural scientific meaning of this quantity is that it determines the amount of matter that is contained in the body. For its designation, it is customary to use the Latin letter m. The unit of measurement in the standard system is the kilogram. In tasks and everyday life, off-system ones are also often used: grams and tons.
In a school physics course, the answer to the question: “What is mass?” given in the study of the phenomenon of inertia. Then it is defined as the ability of a body to resist a change in the speed of its movement. Therefore, the mass is also called inert.
What is weight?
First, it is a force, that is, a vector. Mass, on the other hand, is a scalar weight always attached to a support or suspension and directed in the same direction as gravity, that is, vertically downwards.
The formula for calculating the weight depends on whether this support (suspension) is moving. When the system is at rest, the following expression is used:
P \u003d m * g, where P (in English sources the letter W is used) is the weight of the body, g is the acceleration of free fall. For the earth, g is usually taken equal to 9.8 m / s 2.
The mass formula can be derived from it: m = P / g.
When moving down, that is, in the direction of the weight, its value decreases. So the formula takes the form:
P \u003d m (g - a). Here "a" is the acceleration of the system.
That is, when these two accelerations are equal, a state of weightlessness is observed when the weight of the body is zero.
When the body begins to move upwards, they speak of an increase in weight. In this situation, an overload condition occurs. Because body weight increases, and its formula will look like this:
P \u003d m (g + a).
How is mass related to density?
Solution. 800 kg/m 3 . In order to use the already known formula, you need to know the volume of the spot. It is easy to calculate if we take the spot for a cylinder. Then the volume formula will be:
V = π * r 2 * h.
Moreover, r is the radius, and h is the height of the cylinder. Then the volume will be equal to 668794.88 m 3. Now you can calculate the mass. It will turn out like this: 535034904 kg.
Answer: the mass of oil is approximately equal to 535036 tons.
Task number 5. Condition: The length of the longest telephone cable is 15151 km. What is the mass of copper that went into its manufacture, if the cross section of the wires is 7.3 cm 2?
Solution. The density of copper is 8900 kg/m 3 . The volume is found by a formula that contains the product of the area of the base and the height (here, the length of the cable) of the cylinder. But first you need to convert this area into square meters. That is, divide this number by 10000. After calculations, it turns out that the volume of the entire cable is approximately equal to 11000 m 3.
Now we need to multiply the density and volume values to find out what the mass is equal to. The result is the number 97900000 kg.
Answer: the mass of copper is 97900 tons.
Another issue related to mass
Task number 6. Condition: The largest candle weighing 89867 kg was 2.59 m in diameter. What was its height?
Solution. Wax density - 700 kg / m 3. The height will need to be found from That is, V must be divided by the product of π and the square of the radius.
And the volume itself is calculated by mass and density. It turns out to be equal to 128.38 m 3. The height was 24.38 m.
Answer: the height of the candle is 24.38 m.
The main questions of the curriculum in physics (1 semester)
1. Modeling in physics and technology. Physical and mathematical models. The problem of accuracy in modeling.
To describe the motion of bodies, depending on the conditions of specific tasks, different physical models are used. No physical problem can be solved absolutely exactly. Always get an approximate value.
2. mechanical movement. Types of mechanical movement. Material point. Reference system. Average speed. Instant speed. Average acceleration. Instant acceleration. Velocity and acceleration of a material point as derivatives of the radius vector with respect to time.
Mechanical movement - change in the position of bodies (or body parts) relative to each other in space over time.
Types of mechanical movement: translational and rotational.
Material point - a body whose dimensions can be neglected under given conditions.
Reference system - set of coordinate system and clock.
Average speed -
Instant speed - 
Average and instant acceleration - 
3. Curvature and radius of curvature of the trajectory. Normal and tangential accelerations. Angular velocity and angular acceleration as a vector. Connection of angular velocity and angular acceleration with linear velocities and accelerations of points of a rotating body.
Curvature - degree of curvature of a flat curve. The reciprocal of the curvature - radius of curvature.
Normal acceleration: 
Tangential acceleration:
Angular velocity:
Angular acceleration: 
Connection:
4. The concept of mass and force. Newton's laws. Inertial reference systems. Forces during the motion of a material point along a curvilinear trajectory.
Weight - physical quantity, which is one of the main characteristics of matter, which determines its inertial and gravitational properties.
Power - a vector physical quantity, which is a measure of the intensity of the impact on a given body of other bodies, as well as fields.
Newton's laws:
1. There are such frames of reference, relative to which progressively moving bodies keep their speed constant if no other bodies act on them or the action of these bodies is compensated. Such COs are inertial.
2. The acceleration that the body acquires is directly proportional to the resultant of all forces acting on the body, and inversely proportional to the mass of the body: 
3. The forces with which the bodies act on each other are of the same nature, equal in magnitude and direction along one straight line in the opposite direction: 
5. The center of mass of a mechanical system and the law of its motion.
Center of mass - imaginary point C, the position of which characterizes the mass distribution of this system. 
6. Impulse. isolated system. External and internal forces. The law of conservation of momentum and its connection with the homogeneity of space.
Impulse - amount of movement, which is
Isolated system - a mechanical system of bodies that is not acted upon by external forces.
Forces interactions between the material points of a mechanical system are called internal.
forces, with which external bodies act on the material points of the system are called external.
The momentum does not change with time: 
7. Movement of a body with a variable mass. Jet propulsion. Meshchersky equation. Tsiolkovsky equation.
The movement of some bodies is accompanied by a change in their mass, for example, the mass of a rocket decreases due to the outflow of gases formed during the combustion of fuel.
Reactive force - force that arises as a result of the action on a given body of an attached (or separated) mass.
Meshchersky equation: 
Tsiolkovsky equation: 
,where and - the speed of the outflow of gases relative to the rocket.
8. Energy. Types of energy. The work of a force and its expression through a curvilinear integral. Kinetic energy of a mechanical system and its connection with the work of external and internal forces applied to the system. Power. Units of work and power.
Energy- a universal measure of various forms of movement and interaction. Various forms of energy are associated with various forms of motion of matter: mechanical, thermal, electromagnetic, nuclear, etc.
Force work:
Power:
Unit of work- joule (J): 1 J is the work done by a force of 1 N on a path of 1 m (1 J = 1 N m).
Power unit -watt (W): 1 W is the power at which 1 J of work is done in 1 s (1 W = 1 J/s).
9. Conservative and non-conservative forces. Potential energy in a homogeneous and central gravitational field. Potential energy of an elastically deformed spring.
The conservative forces all forces that act on the particle from the side of the central field: elastic, gravitational, and others. All forces that are not conservative non-conservative: friction forces.
10. The law of conservation of energy and its connection with the homogeneity of time. The law of conservation of mechanical energy. Energy dissipation. dissipative forces.
The law of conservation of mechanical energy: v system of bodies between which only conservative forces, the total mechanical energy is conserved, i.e. does not change with time.
The law of conservation of mechanical energy is related to uniformity of time. The homogeneity of time is manifested in the fact that physical laws are invariant with respect to the choice of the origin of time.
Energy dissipation - mechanical energy gradually decreases due to conversion to other (non-mechanical) forms of energy.
Dissipative forces- forces under the action of which on a mechanical system its total mechanical energy decreases.
