From the point of view of classical mechanics, the mass of a body does not depend on its movement. If the mass of a body at rest is equal to m 0, then for a moving body this mass will remain exactly the same. The theory of relativity shows that this is not actually the case. Body mass T, moving at speed v, expressed in terms of rest mass as follows:
m = m 0 / √(1 - v 2 /c 2) (5)
Let us immediately note that the speed appearing in formula (5) can be measured at any inertial system. In different inertial systems the body has different speed, in different inertial systems it will also have different masses.
Weight is the same relative value like speed, time, distance. We cannot talk about the magnitude of the mass until the frame of reference in which we study the body is fixed.
From what has been said, it is clear that when describing a body, one cannot simply say that its mass is such and such. For example, the sentence “the mass of the ball is 10 g” is completely indefinite from the point of view of the theory of relativity. The numerical value of the mass of the ball does not tell us anything until the inertial system in relation to which this mass is measured is indicated. Typically, the mass of a body is specified in an inertial system associated with the body itself, i.e., the rest mass is specified.
In table Figure 6 shows the dependence of body mass on its speed. It is assumed that the mass of the body at rest is 1 a. Speeds less than 6000 km/sec are not given in the table, since at such speeds the difference between the mass and the rest mass is negligible. At high speeds this difference becomes noticeable. The greater the speed of a body, the greater its mass. So, for example, when driving at a speed of 299,700 km/sec body weight increases almost 41 times. At high speeds, even a slight increase in speed significantly increases body weight. This is especially noticeable in Fig. 41, which graphically shows the dependence of mass on speed.
Rice. 41. Dependence of mass on speed (rest mass of a body is 1 g)
In classical mechanics, only slow motions are studied, for which the mass of a body differs completely insignificantly from the rest mass. When studying slow movements, the body mass can be considered equal to the rest mass. The mistake we make in this case is almost invisible.
If the speed of a body approaches the speed of light, then the mass grows unlimitedly or, as they say, the mass of the body becomes infinite. Only in one single case can a body acquire a speed equal to the speed of light.
From formula (5) it is clear that if the body moves at the speed of light, i.e. if v
= With and √(1 - v 2 /c 2), then the value must also be equal to zero m 0 .
If this were not the case, then formula (5) would lose all meaning, since dividing a finite number by zero is an unacceptable operation. A finite number divided by zero equals infinity - a result that has no definite physical meaning. However, we can make sense of the expression “zero divided by zero.” It follows that only objects whose rest mass is zero can move exactly at the speed of light. Such objects cannot be called bodies in the usual sense.
The equality of the rest mass to zero means that a body with such a mass cannot be at rest at all, but must always move with speed c. An object with zero rest mass is light, more precisely, photons (light quanta). Photons can never be at rest in any inertial frame; they always move at speed With. Bodies with a rest mass different from zero can be at rest or move at different speeds, but at lower speeds of light. They can never reach the speed of light.
Weight- a physical quantity that is inseparably inherent in matter and determines its inertial, energetic and gravitational properties. In classical physics it is strictly subject to the conservation law, on the basis of which classical mechanics is built. In quantum mechanics - special shape energy and, in this form, also the subject of the law of conservation (mass-energy).
Mass is denoted by the Latin letter m
The SI unit of mass is the kilogram. In the Gaussian system, mass is measured in grams. In atomic physics it is customary to equate mass to atomic mass unit, in physics solid- to the mass of the electron, in high energy physics the mass is measured in electronvolts. In addition to these units, there is great amount historical units of mass preserved in certain areas of use: pound, ounce, carat, ton etc. In astronomy, a unit for comparing masses celestial bodies serves as the mass of the Sun.
The mass of a body is a physical quantity that characterizes its inertial and gravitational properties.
In classical physics, mass is a measure of the amount of matter contained in a body. The law of conservation of mass is valid here: the mass of an isolated system of bodies does not change with time and is equal to the sum of the masses of the bodies that make it up.
In Newtonian mechanics, the mass of a body 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.
Mass is 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 (\(m_i\))
\[ m=\sum\limits_(i=1)^(n)(m_i) \]
In classical mechanics they consider:
- body weight is not dependent on the movement of the body, the influence of other bodies, or the location of the body;
- the law of conservation of mass is satisfied: the mass of a closed mechanical system of bodies is constant over time.
As a measure of the inertia of a body, mass is included in Newton's second law, written in a simplified form (for the case constant mass) form:
\[ \LARGE m = \dfrac(F)(a) \]
where \(a\) is acceleration, and \(F\) is the force that acts on the body
Types of mass
Strictly speaking, there are two different quantities that have common name"weight":
- Inert mass characterizes the ability of a body to resist a change in its state of motion under the influence of force. Assuming the force is the same, an object with less mass changes its state of motion more easily than an object with more mass. Inertial mass appears in a simplified form of Newton's second law, as well as in the formula for determining the momentum of a body in classical mechanics.
- Gravitational mass characterizes the intensity of interaction of a body with a gravitational field. It appears in Newton's law of universal gravitation.
Although inertial mass and gravitational mass are conceptually different concepts, all experiments known to date indicate that these two masses are proportional to each other. This makes it possible to construct a system of units so that the unit of measurement of all three masses is the same, and they are all equal to each other. Almost all unit systems are built on this principle.
IN general theory In relativity, inertial and gravitational masses are considered completely equivalent.
Inertia is the property of different material objects to acquire different accelerations under the same external influences from other bodies. Inherent different bodies to varying degrees. The property of inertia shows that time (distance) is required to change the speed of a body. The more difficult it is to change the speed of a body, the more inert it is.
Mass is a scalar quantity that is a measure of the inertia of a body during translational motion. (During rotational motion - moment of inertia). The more inert a body is, the greater its mass. The mass determined in this way is called inertial (in contrast to the gravitational mass determined from the law of universal gravitation).
Mass of elementary particles
Mass, or rather rest mass, is important characteristic elementary particles. The question of what causes the particle masses observed experimentally is important issue particle physics. For example, the mass of a neutron is slightly greater than the mass of a proton, which is due to the difference in the interaction of the quarks that make up these particles. The approximate equality of the masses of some particles makes it possible to combine them into groups, treating them as different states of one common particle with different values of the isotopic spin.
Javascript is disabled in your browser.To perform calculations, you must enable ActiveX controls!
In life, we very often say: “weighs 5 kilograms,” “weighs 200 grams,” and so on. And at the same time we don’t know that we are making a mistake in saying this. The concept of body weight is studied by everyone in the physics course in the seventh grade, but the erroneous use of some definitions has become so mixed up among us that we forget what we have learned and believe that body weight and mass are one and the same thing.
However, it is not. Moreover, body weight is a constant value, but body weight can change, decreasing down to zero. So what is the mistake and how to speak correctly? Let's try to figure it out.
Body weight and body weight: calculation formula
Mass is a measure of the inertia of a body, it is how the body reacts to an impact applied to it, or itself affects other bodies. And the weight of a body is the force with which the body acts on a horizontal support or vertical suspension under the influence of the Earth’s gravity.
Mass is measured in kilograms, and body weight, like any other force, is measured in newtons. The weight of a body has a direction, like any force, and is a vector quantity. But mass has no direction and is a scalar quantity.
The arrow that indicates body weight in pictures and graphs is always directed downward, just like the force of gravity.
Body weight formula in physics is written as follows:
where m is body mass
g - acceleration free fall= 9.81 m/s^2
But, despite the coincidence with the formula and direction of gravity, there is a serious difference between gravity and body weight. The force of gravity is applied to the body, that is, roughly speaking, it presses on the body, and the weight of the body is applied to the support or suspension, that is, here the body presses on the suspension or support.
But the nature of the existence of gravity and the weight of a body is the same as the attraction of the Earth. Strictly speaking, the weight of a body is a consequence of the force of gravity applied to the body. And, just like gravity, body weight decreases with increasing altitude.
Body weight in zero gravity
In a state of weightlessness, the weight of the body is zero. The body will not put pressure on the support or stretch the suspension and will not weigh anything. However, it will still have mass, since in order to give the body any speed, it will be necessary to apply a certain force, the greater the more mass bodies.
Under the conditions of another planet, the mass will also remain unchanged, and the weight of the body will increase or decrease, depending on the strength of the planet’s gravity. We measure body mass with scales, in kilograms, and to measure body weight, which is measured in Newtons, you can use a dynamometer, a special device for measuring force.
The problem of “normal” body weight seems quite relevant for many people. True, serious difficulties arise in defining the concept itself.
Most often, people evaluate their weight either according to existing “norms” designed for the “average”, average person (Table 1), or compare themselves with someone around them. However, both approaches to determining normal body weight are completely unacceptable.
The fact is that the “average” person does not exist in nature at all, and each of us is distinguished by our own characteristics, in particular genotypic ones (including body type, character metabolism etc.), state and level of health, etc. For example, with the same body length, normal weight for an asthenic person can be diagnosed for a hypersthenic person as “underweight”, and normal weight for a hypersthenic person will be a manifestation of obesity of varying degrees for an asthenic person. Hence, Each person should have their own “normal weight”. Its main criterion should be good health and well-being, sufficient tolerance physical activity, and high level performance and social adaptation.
Table 1. Standard formulas for assessing “normal” body weight|
Criterion |
Evaluation method |
Norm |
|
Broca's index |
Normal body weight for people with a height of 155 to 165 cm is equal to body length, from which one hundred units is subtracted; with a height of 166-175, 105 is deducted, with a height of 176 and above - 110 |
The remaining number of units should correspond to normal body weight in kilograms. For example: Height - 170 cm. Normal weight = 170 - 105 = 65 kg |
|
Bongard exponent |
Normal body weight (in kg) is equal to height (in cm) multiplied by chest circumference at nipple level (in cm) and divided by 240 |
For example: Chest circumference = 102 cm, height = 170 cm. Normal weight = 170 x 102 / 240 = 72.3 kg |
|
Quetelet index |
Body weight in grams divided by height in centimeters |
The norm for men is 350-400 g/cm, for women 325-375 g/cm |
|
Body Mass Index (BMI) |
Body weight in kilograms divided by the square of height in meters |
BMI = 18.5-23 - normal; 24-28 - 1st degree obesity; 29-35 - 2nd degree obesity; above 36 – 3rd degree obesity |
|
Body Index |
B = (P 2 x K)\1000, where B is weight, P is height in cm, K is body index |
The norm is 2.1 for women and 2.3 for men |
So what is “normal body weight”?
The main components of our body are bones, active mass and passive mass - mainly fat. “Active body mass” means the total mass of bones, muscles, internal organs, skin (without subcutaneous fat cells
chatki). It should be noted that bones are extremely light parts of our body, and our body weight is mainly determined by fat and muscle.
Muscle tissue, which makes up the vast majority of “active body mass,” burns calories even when a person is at rest. But fat does not need energy - it does not perform any physical functions. This does not mean that it has no physiological significance: As already noted (see section 6.1.), it performs numerous important functions. Body fat content to ensure these functions and in wildlife, and among our ancestors, until relatively recently, it was regulated naturally- the relationship between “income” and “expense”. If a person moved little, then a certain part of the energy of consumed food turned into fat, it became more difficult for the person to move, and therefore obtaining food was difficult. Consequently, he had to limit his food intake until his body weight returned to normal, his performance was restored, and he could again obtain food for himself. In a modern person, who loves to eat tasty and plentifully (and there is no need to run for food!), but moves little, fat reserves often turn out to be extremely excessive. Fat accumulation is associated with numerous adverse health consequences, including:
- metabolic disorders, the consequences of which are: atherosclerosis, diabetes mellitus, diseases of the joints, liver, varicose veins;
- cardiac dysfunction, due to the extremely significant load on it;
- difficulties in the functioning of internal organs due to the deposition of fat directly on them;
- fat in the body is a “sink for waste, etc.
The exception is a state of extreme exhaustion, when the volume of active mass in a person begins to decrease.
To what has been said should be added external aesthetic unattractiveness obese person.
Why does obesity occur?
First, let's understand the mechanism of formation of excess fat in the body. It turns out that fat cells are extremely conservative and, once they arise, disappear with great difficulty. It is fundamentally important that the most important age periods when fat cells are formed are intrauterine (i.e., during the development of the fetus itself) and the first three years after the birth of the child. Unfortunately, in everyday life, it is during these age periods that everything is done to ensure that as many fat cells as possible are formed in the body of the fetus and child - they try to feed both the pregnant woman and the baby as densely as possible. During subsequent periods age development due to increased growth, the excess of formed fat cells is not noticeable, but when growth stops (for girls this happens around 20 - 22 years old, for young people at 22 - 25), either a person noticeably reduces his motor activity, or certain hormonal factors intervene ( as happens at the age of puberty in girls) - these cells begin to increase in size many times over. This is obesity. It is called primary, since it is associated with a violation of the income/expense ratio with a predominance of the first part of this ratio: a person eats a lot, but spends little energy.
With age, when the course of metabolic processes slows down, the craving for food does not decrease, and physical activity progressively decreases, the ratio leans more and more towards the predominance of the arrival. In this case, fatty degeneration of muscle tissue occurs when muscle fibers are replaced by adipose tissue. This does not mean that age-related increase in body weight is natural - according to Academician. N.M. Amosov, and at 60 - 70 years old for a person leading a healthy active lifestyle, it should be the same as at 25 - 30 years old.
The described consequences of overeating and inactivity do not affect everyone, since different people the type of energy is different, which is due (in healthy people) mainly to genetic factors and the mother’s lifestyle during pregnancy. Thus, in thin people, energy metabolism per unit of time is more active, therefore, for example, in a healthy person of this constitution after a heavy meal it almost doubles, while in an obese person it is barely noticeable. To the effect of cold fat people do not respond to the same increase in energy costs as thin people. Consequently, other things being equal, an obese person absorbs more energy from the food he consumes than he needs to maintain life and perform everyday activities.
Depending on the severity of excess fat mass, obesity is classified as follows. If your body weight exceeds 9%, you are said to be overweight. Degree I obesity is considered to be excess weight in the range of 10-29%, II degree 30-49%, III 50-99% and, finally, IV 100 percent or more excess body weight.
Mass is a measure of inertia. The greater the mass of a body, the more inert it is, that is, it has greater inertia. The law of inertia states that if a body is not acted upon by other bodies, then it remains at rest or performs linear uniform motion.
When bodies interact, for example, collide, then rest or rectilinear uniform motion is disturbed. The body may begin to accelerate or, on the contrary, slow down. The speed that a body gains (or loses) after interacting with another body depends, among other things, on the ratio of the masses of the interacting bodies.
So if a rolling ball collides with a brick on its way, it will not just stop, but will most likely change its direction of movement and bounce. The brick will most likely remain in place, maybe fall. But if in the path of the ball there is cardboard box, equal in size to a brick, then the ball will no longer bounce off it at the same speed as from the brick. The ball can generally drag it in front of itself, continuing its movement, but slowing it down.
The ball, brick and box have different masses. The brick has more mass, and therefore it is more inert, so the ball can hardly change its speed. Rather, the brick reverses the speed of the ball. The box is less inert, so it is easier to move, and it itself cannot change the speed of the sword the way a brick could.
A classic example of comparing the masses of two bodies by estimating their inertia is as follows. Two trolleys at rest are held together by bending and linking elastic plates soldered to their ends. Next, the binding thread is burned out. The plates straighten, pushing away from each other. Thus, the carts also repel each other and move in opposite directions.
In this case, the following patterns exist. If the carts have equal masses, then they will acquire equal speeds and, before complete braking, will move away from the starting point to equal distances. If the carts have different masses, then the more massive (and therefore more inert) will travel a shorter distance, and the less massive (less inert) will travel a greater distance.
Moreover, there is a connection between the masses and velocities of interacting bodies that are initially at rest. The product of the mass and the acquired speed of one body is equal to the product of the mass and the acquired speed of the other body after interaction. Mathematically this can be expressed as follows:
m 1 v 1 = m 2 v 2
This formula says that the greater the mass of a body, the lower its speed, and the smaller the mass, the greater the speed of the body. The mass and speed of one body are inversely proportional dependence from each other (the larger one value, the smaller the other).
Usually the formula is written like this (it can be obtained by transforming the first formula):
m 1 / m 2 = v 2 / v 1
That is the ratio of the masses of bodies is inversely proportional to the ratio of their velocities.
Using this pattern, you can compare the masses of bodies by measuring the speeds they acquire after interaction. If, for example, after interaction, bodies at rest acquired speeds of 2 m/s and 4 m/s, and the mass of the second body is known (let it be 0.4 kg), then you can find out the mass of the first body: m1 = (v 2 /v 1) * m 2 = 4 / 2 * 0.4 = 0.8 (kg).