What is the physical meaning of body weight. Body mass formula
The concept with which we are familiar from the earliest childhood is 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 the weight?
Determination of mass
The natural scientific meaning of this value 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 kilogram. In tasks and Everyday life often used and off-system: gram and ton.
In a school physics course, the answer to the question: "What is mass?" is given when studying the phenomenon of inertia. Then it is defined as the ability of the body to resist a change in the speed of its movement. Therefore, the mass is also called inert.
What is weight?
Firstly, it is the force, that is, the vector. Mass is a scalar weight is always applied to a support or suspension and is directed in the same direction as the force of gravity, that is, vertically downward.
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 = m * g, where P (in English sources the letter W is used) is the weight of the body, g is the acceleration of gravity. For land g, it is customary to take 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. Therefore, the formula takes the form:
P = m (g - a). Here "a" is the acceleration of the movement of the system.
That is, when these two accelerations are equal, a state of weightlessness is observed when the body weight is zero.
When the body begins to move upward, then they talk about an increase in weight. In this situation, an overload condition occurs. Because the body weight is increasing, and its formula will look like this:
P = m (g + a).
How is mass related to density?
Solution. 800 kg / m 3. To use already known formula, you need to know the volume of the spot. It is easy to calculate if you take a spot for a cylinder. Then the formula for the volume will be as follows:
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 count the mass. It will turn out like this: 535034904 kg.
Answer: the mass of oil is approximately 535,036 tons.
Problem 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 using a formula that contains the product of the base area and the height (here the cable length) of the cylinder. But first you need to translate 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 you need to multiply the density and volume values to find out what is the mass. The result is the number 97,900,000 kg.
Answer: the mass of copper is 97,900 tons.
Another task related to mass
Problem number 6. Condition: The largest candle weighing 89867 kg was 2.59 m in diameter. What was its height?
Solution. The density of the wax is 700 kg / m 3. The height will need to be found from That is, V needs to 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 candle height is 24.38 m.
Essential questions of the curriculum in physics (1 semester)
1. Modeling in physics and technology. Physical and mathematical models. The problem of accuracy in modeling.
Different physical models are used to describe the motion of bodies, depending on the conditions of specific problems. 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. Speed and acceleration material point as derivatives of the vector radius 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 these conditions.
Reference system - a set of coordinate systems and clocks.
Average speed -
Instantaneous speed - 
Average and instant acceleration -
3. Curvature and radius of curvature of the trajectory. Normal and tangential acceleration. 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 - the degree of curvature of a flat curve. The reciprocal of the curvature is radius of curvature.
Normal acceleration: 
Tangential acceleration:
Angular velocity:
Angular acceleration: 
Connection:
4. The concept of mass and force. Newton's laws. Inertial frames of reference. Forces when a material point moves along a curved trajectory.
Weight - physical quantity, which is one of the main characteristics of matter, which determines its inertial and gravitational properties.
Power - 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 the translationally moving bodies keep their speed constant, if other bodies do not act on them or the action of these bodies is compensated. Such COs - 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. Center of mass of a mechanical system and the law of its motion.
Center of mass - an imaginary point C, the position of which characterizes the distribution of the mass 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, which is not acted upon by external forces.
Forces interactions between 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 over time: 
7. The movement of a body with variable mass. Jet propulsion. Meshchersky's 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 - the force that arises as a result of the action of the attached (or detached) mass on a given body.
Meshchersky's equation: 
Tsiolkovsky equation: 
,where and - the velocity of the outflow of gases relative to the rocket.
8. Energy. Types of energy. The work of the force and its expression through the curvilinear integral. Kinetic energy of a mechanical system and its relationship 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.
Work of force:
Power:
Unit of work- joule (J): 1 J - 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 performed during 1 s (1 W = 1 J / s).
9. Conservative and non-conservative forces. Potential energy in a uniform and central gravitational field. Potential energy of an elastically deformed spring.
Conservative forces - all forces that act on a particle from the side of the central field: elastic, gravitational and others. All non-conservative forces - non-conservative: frictional forces.
10. The law of conservation of energy and its relationship with the homogeneity of time. The law of conservation of mechanical energy. Energy dissipation. Dissipative forces.
Mechanical energy conservation law: v system of bodies between which only conservative forces, the total mechanical energy is conserved, that is, it does not change with time.
The law of conservation of mechanical energy is associated with 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 into other (non-mechanical) forms of energy.
Dissipative forces- forces, under the action of which on a mechanical system, its total mechanical energy decreases.
ABOUT THE PHYSICAL ESSENCE OF MASS
Brusin S.D., Brusin L.D.
annotation. The physical essence of mass, given by Newton, is explained, and it is shown that in modern textbooks it is distorted physical entity masses.
Parameter weight first introduced by Newton and formulated as follows: "The amount of matter (mass) is a measure of that, set in proportion to its density and volume"... The amount of the substance was previously determined by weighing it. However, it is known, for example, that the same piece of gold at the pole weighs more than at the equator. Therefore, the introduction of a simple parameter that clearly determines the amount of matter (substance) in the body - greatest merit genius of Newton. It allowed formulate the laws of motion and interaction of bodies.
First, Newton defines the amount of motion 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: The innate power of matter there is an inherent ability of resistance, according to which any separately taken 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'll 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 a body affects the acceleration received by the body at 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 masses (amount of matter) bodies; these forces are called gravitational forces. Cavendish showed this law experimentally for any body. Thus, the same body mass has gravitational and inertial properties (according to Newton's expression, this is due to vborn by the power of matter).
In modern science, it is given following definition mass: "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 it took to pervert the deep and simple physical meaning 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 must understand this important issue. The distortion of the physical essence of mass has led to the following:
1. There were concepts inert mass and gravitational mass, and it took considerable effort and numerous experiments by Eotvos to prove the equality of inertial and gravitational masses, although Newton's definition of mass clearly shows that there is one mass, but has inertial and gravitational properties.
2. To a misunderstanding of the physical essence of the parameters associated with misunderstanding of mass. For example, the essence of the density of a body consists not in the amount of inertia per unit volume, but in 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 younger generation misunderstands the physical essence of the mass... That's why 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., "Science", 1989, p. 22
2. Ibid, p. 25
3. Detlaf AA, Yavorskiy BM Handbook of physics, M. "Science", 1974, p. 36
Definition
In Newtonian mechanics, the mass of a body is called a scalar physical quantity, which is a measure of its inertial properties and a source of gravitational interaction. In classical physics, mass is always positive.
Weight- an additive value, which means: the mass of each set of material points (m) is equal to the sum of the masses of all separate parts of the system (m i):
In classical mechanics, it is considered:
- body weight is not dependent on body movement, on the impact of other bodies, body position;
- the law of conservation of mass is fulfilled: the mass of a closed mechanical system of bodies is unchanged in time.
Inert mass
The property of inertness of a material point is that if an external force acts on the point, then it has an acceleration of finite magnitude. If there are no external influences, then in the inertial frame of reference the body is at rest or moves uniformly and rectilinearly. Mass enters into Newton's second law:
where mass determines the inert properties of a material point (inert 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, while it is called gravitational (heavy) mass.
It was empirically obtained that for all bodies the ratios of inert to gravitational masses are the same. Therefore, if we correctly choose the value of the constant gravity, then we can get that for any body the inert and gravitational masses are the same and are associated with the force of gravity (F t) of the selected body:
where g is the acceleration due to gravity. If we carry out observations at the same point, then the accelerations of gravity are the same.
Formula for calculating mass through body density
Body weight can be calculated as:
where is the density of the substance of the body, 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 masses of the particle, v is the speed of the particle.
The basic SI unit of mass is: [m] = kg.
In the SGS: [m] = gr.
Examples of problem solving
Example
Exercise. Two particles fly towards each other with speeds equal to v (the speed is close to the speed of light). When they collide, an absolutely 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 absolutely inelastic collision of particles, which before the impact had the same masses and velocities, 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. The particles have only kinetic energy. By the condition of the problem, the speed of the particles is close to the speed of light, hence? 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 fusion is equal to:
Example
Exercise. What is the mass of 2m 3 of copper?
Moreover, if the substance (copper) is known, then you can find its density using the reference book. The density of copper will be considered equal to Cu = 8900 kg / m 3. All quantities are known for the calculation. Let's carry out the calculations.
Mass (physical quantity) Weight, a physical quantity, one of the main characteristics of matter, which determines its inertial and gravitational properties. Accordingly, a distinction is made between inert and gravitational (heavy, gravitational) magnetic fields.
The concept of M. was introduced into mechanics by I. Newton. In classical Newtonian mechanics, M. is included in the definition of momentum ( amount of motion) of the body: momentum p is proportional to the speed of motion of the body v,
p = mv.
The coefficient of proportionality - constant for a given body value m - 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 f acting on the body and the acceleration of the body a caused by it. Determined by relations (1) and (2) M. is called inertial mass, or inertial mass; it characterizes the dynamic properties of a body, is a measure of the body's inertia: with a constant force, the greater the body's M., the less acceleration it acquires, that is, the slower the state of its motion changes (the greater its inertia).
Acting on different bodies with the same force and measuring their accelerations, it is possible to determine the M. ratio 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, you can find the M. of the rest of the bodies.
In Newton's theory of gravity, M. acts in a different form - as a source of the gravitational field. Each body creates a gravitational field proportional to the body's M. (and experiences the influence of the gravitational field created by other bodies, the force of which is also proportional to the body's M.). This field causes the attraction of any other body to this body with a force determined Newton's law of gravitation:
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 weights P of a body of mass m in the Earth's gravitational field:
P = m g.
Here g = G M / r 2
- acceleration of free fall in the gravitational field of the Earth, and r »R - the radius of the Earth. M., defined by relations (3) and (4), is called the gravitational mass of the body.
In principle, it does not follow from anywhere that the magnetic field, which creates the gravitational field, also determines the inertia of the same body. However, experience has shown that inert magnetic field and gravitational magnetic field 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 (formulated by him for the first time) as the basis general theory relativity (see. Gravitation). The principle of equivalence was established experimentally with very high accuracy. For the first time (1890-1906) a precision check of the equality of inert and gravitational magnetic field was carried out by L. Eotvos, which found that M. matched error ~ 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 VB Braginsky and V.I. Panov reduced the error to 10-12.
The equivalence principle allows the most natural definition of the M. of a body weighing.
Initially, M. was considered (for example, by Newton) as a measure of the amount of matter. This definition has a clear meaning only for comparing homogeneous bodies built from the same material. It emphasizes the additivity of the M. - M. of a body is equal to the sum of M. of its parts. The M. of a homogeneous body is proportional to its volume, therefore, the concept density- M. unit volume of the body.
In classical physics, it was believed that the magnetic field of a body does not change in any processes. This corresponded to the conservation law of M. (matter) discovered by M.V. Lomonosov and A. L. Lavoisier... In particular, this law asserted that in any chemical reaction the sum of M. of the initial components is equal to the sum of M. of the final components.
The concept of M. has acquired a deeper meaning in the mechanics of specials. A. Einstein's theory of relativity (see. Relativity theory), considering the motion 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 momentum and particle velocity 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
:
Bearing in mind, in particular, this formula, they say that the magnetic field of a particle (body) grows with an increase in its speed. Such a relativistic increase in the magnitude of a particle as its velocity increases must be taken into account when designing particle accelerators high energies. M of rest m 0 (M in the frame of reference 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 a given type 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 the coefficient of proportionality between the momentum and the particle velocity, since the acceleration ceases to be parallel to the force that caused it, and the M. turns out to depend on the direction of the particle velocity.
According to the theory of relativity, the M. of a particle m is related to its energy E by the ratio:
M. rest determines the internal energy of the particle - the so-called rest energy E 0 = m 0 c 2
... Thus, energy is always associated with M. (and vice versa). Therefore, separately (as in classical physics) the law of conservation of M. and the law of conservation of energy do not exist separately - they are merged into a single law of conservation of total (that is, including the rest energy of particles) energy. An approximate separation into the law of conservation of energy and the law of conservation of M. is possible only in classical physics, when the particle velocities are small (v<< с
) и не происходят процессы превращения частиц.
In relativistic mechanics, M. is not an additive characteristic of a body. When two particles combine to form one composite stable state, an excess of energy is released (equal to bond energies) DE, which corresponds to M. Dm = DE / c 2
... Therefore, the M. of a composite particle is less than the sum of M. of its constituent particles by the value DE / s 2
(so-called mass defect). This effect is especially pronounced in nuclear reactions... For example, M. deuteron (d) is less than the sum of M. proton (p) and neutron (n); the M. defect Dm is associated with the energy E g of a gamma quantum (g), produced during the formation of a deuteron: p + n ® d + g, E g = Dm c 2
... A defect in M., arising during the formation of a composite particle, reflects the organic connection between M. and energy.
The unit of M. in the CGS system of units is gram and in International system of units SI - kilogram... M. atoms and molecules are usually measured in atomic mass units... It is customary to express the magnitude of elementary particles either in the units of the magnitude of the electron m e, or in energy units, indicating the rest energy of the corresponding particle. So, the M. of the electron is 0.511 MeV, the M. of the proton is 1836.1 m e, or 938.2 MeV, etc.
The nature of M. is one of the most important unsolved problems of modern physics. It is generally accepted that the magnetic field of an elementary particle is determined by the fields that are associated with it (electromagnetic, nuclear, and others). However, the quantitative theory of M. has not yet been created. There is also no theory that explains why the M. of elementary particles form a discrete spectrum of values, and even more so that allows you to determine this spectrum.
In astrophysics, the magnetic field of a body that creates a gravitational field is determined by the so-called gravitational radius body R gr = 2GM / s 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 will be invisible; therefore they were named “ black holes". Such celestial bodies should play an important role in the Universe.
Lit .: Jemmer 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 main. har to matter, which determines its inertial and gravitational. St. va. The concept of "M." was introduced into mechanics by I. Newton in the definition of the impulse (number of motion) of a body, impulse p is proportional ... ... Physical encyclopedia
- (lat.massa). 1) the amount of substance in the object, regardless of the form; body, matter. 2) in a hostel: a significant amount of something. Dictionary of foreign words included in the Russian language. Chudinov AN, 1910. MASS 1) in physics, the amount ... ... Dictionary of foreign words of the Russian language
- - 1) in the natural scientific sense, the amount of a substance contained in the body; the resistance of a body to a change in its movement (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 sense), 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 ... ... Great 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 inertia of a body). Equal to gravitats. mass (see WEIGHT). Physical encyclopedic dictionary. M .: Soviet encyclopedia. Editor-in-chief A ... Physical encyclopedia
- (heavy weight), physical a value that characterizes the body as a source of gravitation; equal to the inert mass. (see MASS). Physical encyclopedic dictionary. M .: Soviet encyclopedia. Chief editor A.M. Prokhorov. 1983 ... Physical encyclopedia
Phys. a value equal to the ratio of mass to number in va. The unit is M. m. (In SI) kg / mol. M = m / n, where M M. m. In kg / mol, m is the mass in va in kg, the number in va in moles. The numerical value of M. m., Express. in kg / mol, is equal to molecular weight divided by ... Big encyclopedic polytechnic dictionary - value, character of physical. objects or phenomena of the material world, common to many objects or phenomena in qualities. respect, but individual in quantities. relation for each of them. For example, mass, length, area, volume, electric force. current F ... Big Encyclopedic Polytechnic Dictionary