The concept of strength. Vector and scalar quantities

If a body accelerates, then something acts on it. How to find this “something”? For example, what kind of forces act on a body near the surface of the earth? This is the force of gravity directed vertically downward, proportional to the mass of the body and for heights much smaller than the radius of the earth $(\large R)$, almost independent of the height; it is equal

$(\large F = \dfrac (G \cdot m \cdot M)(R^2) = m \cdot g )$

$(\large g = \dfrac (G \cdot M)(R^2) )$

so-called acceleration due to gravity. In the horizontal direction the body will move at a constant speed, but the movement in the vertical direction is according to Newton's second law:

$(\large m \cdot g = m \cdot \left (\dfrac (d^2 \cdot x)(d \cdot t^2) \right) )$

after contracting $(\large m)$, we find that the acceleration in the direction $(\large x)$ is constant and equal to $(\large g)$. This is the well-known motion of a freely falling body, which is described by the equations

$(\large v_x = v_0 + g \cdot t)$

$(\large x = x_0 + x_0 \cdot t + \dfrac (1)(2) \cdot g \cdot t^2)$

How is strength measured?

In all textbooks and smart books, it is customary to express force in Newtons, but except in the models that physicists operate, Newtons are not used anywhere. This is extremely inconvenient.

Newton newton (N) is a derived unit of force in the International System of Units (SI).
Based on Newton's second law, the unit newton is defined as the force that changes the speed of a body weighing one kilogram by 1 meter per second in one second in the direction of the force.

Thus, 1 N = 1 kg m/s².

Kilogram-force (kgf or kg) is a gravitational metric unit of force equal to the force that acts on a body weighing one kilogram in the gravitational field of the earth. Therefore, by definition, a kilogram-force is equal to 9.80665 N. A kilogram-force is convenient in that its value is equal to the weight of a body weighing 1 kg.
1 kgf = 9.80665 newtons (approximately ≈ 10 N)
1 N ≈ 0.10197162 kgf ≈ 0.1 kgf

1 N = 1 kg x 1 m/s2.

Law of gravitation

Every object in the Universe is attracted to every other object with a force proportional to their masses and inversely proportional to the square of the distance between them.

$(\large F = G \cdot \dfrac (m \cdot M)(R^2))$

We can add that any body reacts to a force applied to it with acceleration in the direction of this force, in magnitude inversely proportional to the mass of the body.

$(\large G)$ — gravitational constant

$(\large M)$ — mass of the earth

$(\large R)$ — radius of the earth

$(\large G = 6.67 \cdot (10^(-11)) \left (\dfrac (m^3)(kg \cdot (sec)^2) \right) )$

$(\large M = 5.97 \cdot (10^(24)) \left (kg \right) )$

$(\large R = 6.37 \cdot (10^(6)) \left (m \right) )$

Within the framework of classical mechanics, gravitational interaction is described by Newton’s law of universal gravitation, according to which the force of gravitational attraction between two bodies of mass $(\large m_1)$ and $(\large m_2)$ separated by a distance $(\large R)$ is

$(\large F = -G \cdot \dfrac (m_1 \cdot m_2)(R^2))$

Here $(\large G)$ is the gravitational constant equal to $(\large 6.673 \cdot (10^(-11)) m^3 / \left (kg \cdot (sec)^2 \right) )$. The minus sign means that the force acting on the test body is always directed along the radius vector from the test body to the source of the gravitational field, i.e. gravitational interaction always leads to the attraction of bodies.
The gravity field is potential. This means that you can introduce the potential energy of gravitational attraction of a pair of bodies, and this energy will not change after moving the bodies along a closed loop. The potentiality of the gravitational field entails the law of conservation of the sum of kinetic and potential energy, which, when studying the motion of bodies in a gravitational field, often significantly simplifies the solution.
Within the framework of Newtonian mechanics, gravitational interaction is long-range. This means that no matter how a massive body moves, at any point in space the gravitational potential and force depend only on the position of the body at a given moment in time.

Heavier - Lighter

The weight of a body $(\large P)$ is expressed by the product of its mass $(\large m)$ and the acceleration due to gravity $(\large g)$.

$(\large P = m \cdot g)$

When on earth the body becomes lighter (presses less on the scales), this is due to a decrease masses. On the moon, everything is different; the decrease in weight is caused by a change in another factor - $(\large g)$, since the acceleration of gravity on the surface of the moon is six times less than on the earth.

mass of the earth = $(\large 5.9736 \cdot (10^(24))\ kg )$

moon mass = $(\large 7.3477 \cdot (10^(22))\ kg )$

acceleration of gravity on Earth = $(\large 9.81\ m / c^2 )$

gravitational acceleration on the Moon = $(\large 1.62 \ m / c^2 )$

As a result, the product $(\large m \cdot g )$, and therefore the weight, decreases by 6 times.

But it is impossible to describe both of these phenomena with the same expression “make it easier.” On the moon, bodies do not become lighter, but only fall less rapidly; they are “less epileptic”))).

Vector and scalar quantities

A vector quantity (for example, a force applied to a body), in addition to its value (modulus), is also characterized by direction. A scalar quantity (for example, length) is characterized only by its value. All classical laws of mechanics are formulated for vector quantities.

Figure 1.

In Fig. Figure 1 shows various options for the location of the vector $( \large \overrightarrow(F))$ and its projections $( \large F_x)$ and $( \large F_y)$ on the axis $( \large X)$ and $( \large Y )$ respectively:

• A. the quantities $( \large F_x)$ and $( \large F_y)$ are non-zero and positive
• B. the quantities $( \large F_x)$ and $( \large F_y)$ are non-zero, while $(\large F_y)$ is a positive quantity, and $(\large F_x)$ is negative, because the vector $(\large \overrightarrow(F))$ is directed in the direction opposite to the direction of the $(\large X)$ axis
• C.$(\large F_y)$ is a positive non-zero quantity, $(\large F_x)$ is equal to zero, because the vector $(\large \overrightarrow(F))$ is directed perpendicular to the axis $(\large X)$

moment of force

A moment of power is called the vector product of the radius vector drawn from the axis of rotation to the point of application of the force and the vector of this force. Those. According to the classical definition, the moment of force is a vector quantity. Within the framework of our problem, this definition can be simplified to the following: the moment of force $(\large \overrightarrow(F))$ applied to a point with coordinate $(\large x_F)$, relative to the axis located at point $(\large x_0 )$ is a scalar quantity equal to the product of the force modulus $(\large \overrightarrow(F))$ and the force arm - $(\large \left | x_F - x_0 \right |)$. And the sign of this scalar quantity depends on the direction of the force: if it rotates the object clockwise, then the sign is plus, if counterclockwise, then the sign is minus.

It is important to understand that we can choose the axis arbitrarily - if the body does not rotate, then the sum of the moments of forces about any axis is zero. The second important note is that if a force is applied to a point through which an axis passes, then the moment of this force about this axis is equal to zero (since the arm of the force will be equal to zero).

Let us illustrate the above with an example in Fig. 2. Let us assume that the system shown in Fig. 2 is in equilibrium. Consider the support on which the loads stand. It is acted upon by 3 forces: $(\large \overrightarrow(N_1),\ \overrightarrow(N_2),\ \overrightarrow(N),)$ points of application of these forces A, IN And WITH respectively. The figure also contains forces $(\large \overrightarrow(N_(1)^(gr)),\ \overrightarrow(N_2^(gr)))$. These forces are applied to the loads, and according to Newton's 3rd law

$(\large \overrightarrow(N_(1)) = - \overrightarrow(N_(1)^(gr)))$

$(\large \overrightarrow(N_(2)) = - \overrightarrow(N_(2)^(gr)))$

Now consider the condition for the equality of the moments of forces acting on the support relative to the axis passing through the point A(and, as we agreed earlier, perpendicular to the drawing plane):

$(\large N \cdot l_1 - N_2 \cdot \left (l_1 +l_2 \right) = 0)$

Please note that the moment of force $(\large \overrightarrow(N_1))$ was not included in the equation, since the arm of this force relative to the axis in question is equal to $(\large 0)$. If for some reason we want to select an axis passing through the point WITH, then the condition for equality of moments of forces will look like this:

$(\large N_1 \cdot l_1 - N_2 \cdot l_2 = 0)$

It can be shown that, from a mathematical point of view, the last two equations are equivalent.

Center of gravity

Center of gravity in a mechanical system is the point relative to which the total moment of gravity acting on the system is zero.

Center of mass

The point of the center of mass is remarkable in that if a great many forces act on the particles forming a body (no matter whether it is solid or liquid, a cluster of stars or something else) (meaning only external forces, since all internal forces compensate each other), then the resulting the force leads to such an acceleration of this point as if the entire mass of the body $(\large m)$ were in it.

The position of the center of mass is determined by the equation:

$(\large R_(c.m.) = \frac(\sum m_i\, r_i)(\sum m_i))$

This is a vector equation, i.e. in fact, three equations - one for each of the three directions. But consider only the $(\large x)$ direction. What does the following equality mean?

$(\large X_(c.m.) = \frac(\sum m_i\, x_i)(\sum m_i))$

Suppose the body is divided into small pieces with the same mass $(\large m)$, and the total mass of the body will be equal to the number of such pieces $(\large N)$ multiplied by the mass of one piece, for example 1 gram. Then this equation means that you need to take the $(\large x)$ coordinates of all the pieces, add them up and divide the result by the number of pieces. In other words, if the masses of the pieces are equal, then $(\large X_(c.m.))$ will simply be the arithmetic mean of the $(\large x)$ coordinates of all the pieces.

Mass and density

Mass is a fundamental physical quantity. Mass characterizes several properties of a body at once and in itself has a number of important properties.

• Mass serves as a measure of the substance contained in a body.
• Mass is a measure of the inertia of a body. Inertia is the property of a body to maintain its speed unchanged (in the inertial frame of reference) when external influences are absent or compensate each other. In the presence of external influences, the inertia of a body is manifested in the fact that its speed does not change instantly, but gradually, and the more slowly, the greater the inertia (i.e. mass) of the body. For example, if a billiard ball and a bus are moving at the same speed and are braked by the same force, then it takes much less time to stop the ball than to stop the bus.
• The masses of bodies are the reason for their gravitational attraction to each other (see the section “Gravity”).
• The mass of a body is equal to the sum of the masses of its parts. This is the so-called additivity of mass. Additivity allows you to use a standard of 1 kg to measure mass.
• The mass of an isolated system of bodies does not change with time (law of conservation of mass).
• The mass of a body does not depend on the speed of its movement. Mass does not change when moving from one frame of reference to another.
• Density of a homogeneous body is the ratio of the mass of the body to its volume:

$(\large p = \dfrac (m)(V) )$

Density does not depend on the geometric properties of the body (shape, volume) and is a characteristic of the substance of the body. The densities of various substances are presented in reference tables. It is advisable to remember the density of water: 1000 kg/m3.

Newton's second and third laws

The interaction of bodies can be described using the concept of force. Force is a vector quantity, which is a measure of the influence of one body on another.
Being a vector, force is characterized by its modulus (absolute value) and direction in space. In addition, the point of application of the force is important: the same force in magnitude and direction, applied at different points of the body, can have different effects. So, if you grab the rim of a bicycle wheel and pull tangentially to the rim, the wheel will begin to rotate. If you pull along the radius, there will be no rotation.

Newton's second law

The product of the body mass and the acceleration vector is the resultant of all forces applied to the body:

$(\large m \cdot \overrightarrow(a) = \overrightarrow(F) )$

Newton's second law relates acceleration and force vectors. This means that the following statements are true.

1. $(\large m \cdot a = F)$, where $(\large a)$ is the acceleration modulus, $(\large F)$ is the resulting force modulus.
2. The acceleration vector has the same direction as the resultant force vector, since the mass of the body is positive.

Newton's third law

Two bodies act on each other with forces equal in magnitude and opposite in direction. These forces have the same physical nature and are directed along a straight line connecting their points of application.

Superposition principle

Experience shows that if several other bodies act on a given body, then the corresponding forces add up as vectors. More precisely, the principle of superposition is valid.
The principle of superposition of forces. Let the forces act on the body$(\large \overrightarrow(F_1), \overrightarrow(F_2),\ \ldots \overrightarrow(F_n))$ If you replace them with one force$(\large \overrightarrow(F) = \overrightarrow(F_1) + \overrightarrow(F_2) \ldots + \overrightarrow(F_n))$ , then the result of the impact will not change.
The force $(\large \overrightarrow(F))$ is called resultant forces $(\large \overrightarrow(F_1), \overrightarrow(F_2),\ \ldots \overrightarrow(F_n))$ or resulting by force.

Forwarder or carrier? Three secrets and international cargo transportation

Forwarder or carrier: who to choose? If the carrier is good and the forwarder is bad, then the first. If the carrier is bad and the forwarder is good, then the latter. This choice is simple. But how can you decide when both candidates are good? How to choose from two seemingly equivalent options? The fact is that these options are not equivalent.

Horror stories of international transport

BETWEEN A HAMMER AND A HILL.

It is not easy to live between the customer of transportation and the very cunning and economical owner of the cargo. One day we received an order. Freight for three kopecks, additional conditions for two sheets, the collection is called.... Loading on Wednesday. The car is already in place on Tuesday, and by lunchtime the next day the warehouse begins to slowly throw into the trailer everything that your forwarder has collected for its recipient customers.

AN ENCHANTED PLACE - PTO KOZLOVICHY.

According to legends and experience, everyone who transported goods from Europe by road knows what a terrible place the Kozlovichi VET, Brest Customs, is. What chaos the Belarusian customs officers create, they find fault in every possible way and charge exorbitant prices. And it's true. But not all...

ON THE NEW YEAR'S TIME WE BROUGHT POWDERED MILK.

Loading with groupage cargo at a consolidation warehouse in Germany. One of the cargoes is milk powder from Italy, the delivery of which was ordered by the Forwarder.... A classic example of the work of a forwarder-“transmitter” (he doesn’t delve into anything, he just transmits along the chain).

Documents for international transport

International road transport of goods is very organized and bureaucratic; as a result, a bunch of unified documents are used to carry out international road transport of goods. It doesn’t matter if it’s a customs carrier or an ordinary one - he won’t travel without documents. Although this is not very exciting, we tried to simply explain the purpose of these documents and the meaning that they have. They gave an example of filling out TIR, CMR, T1, EX1, Invoice, Packing List...

Axle load calculation for road freight transport

The goal is to study the possibility of redistributing loads on the axles of the tractor and semi-trailer when the location of the cargo in the semi-trailer changes. And applying this knowledge in practice.

In the system we are considering there are 3 objects: a tractor $(T)$, a semi-trailer $(\large ((p.p.)))$ and a load $(\large (gr))$. All variables related to each of these objects will be marked with the superscript $T$, $(\large (p.p.))$ and $(\large (gr))$ respectively. For example, the tare weight of a tractor will be denoted as $m^(T)$.

Why don't you eat fly agarics? The customs officer exhaled a sigh of sadness.

What is happening in the international road transport market? The Federal Customs Service of the Russian Federation has already banned the issuance of TIR Carnets without additional guarantees in several federal districts. And she notified that from December 1 of this year she will completely terminate the agreement with the IRU as not meeting the requirements of the Customs Union and is putting forward financial claims that are not childish.
IRU in response: “The explanations of the Federal Customs Service of Russia regarding the alleged debt of ASMAP in the amount of 20 billion rubles are a complete fiction, since all the old TIR claims have been fully settled..... What do we, common carriers, think?

Stowage Factor Weight and volume of cargo when calculating the cost of transportation

The calculation of the cost of transportation depends on the weight and volume of the cargo. For sea transport, volume is most often decisive, for air transport - weight. For road transport of goods, a complex indicator is important. Which parameter for calculations will be chosen in a particular case depends on specific gravity of the cargo (Stowage Factor) .

It is necessary to know the point of application and direction of each force. It is important to be able to determine exactly what forces act on the body and in what direction. Force is denoted as , measured in Newtons. In order to distinguish between forces, they are designated as follows

Below are the main forces operating in nature. It is impossible to invent forces that do not exist when solving problems!

There are many forces in nature. Here we consider the forces that are considered in the school physics course when studying dynamics. Other forces are also mentioned, which will be discussed in other sections.

Gravity

Every body on the planet is affected by Earth's gravity. The force with which the Earth attracts each body is determined by the formula

The point of application is at the center of gravity of the body. Gravity always directed vertically downwards.

Friction force

Let's get acquainted with the force of friction. This force occurs when bodies move and two surfaces come into contact. The force arises from the fact that surfaces, when viewed under a microscope, are not as smooth as they appear. The friction force is determined by the formula:

The force is applied at the point of contact of two surfaces. Directed in the direction opposite to movement.

Ground reaction force

Let's imagine a very heavy object lying on a table. The table bends under the weight of the object. But according to Newton's third law, the table acts on the object with exactly the same force as the object on the table. The force is directed opposite to the force with which the object presses on the table. That is, up. This force is called the ground reaction. The name of the force "speaks" support reacts. This force occurs whenever there is an impact on the support. The nature of its occurrence at the molecular level. The object seemed to deform the usual position and connections of the molecules (inside the table), they, in turn, strive to return to their original state, “resist.”

Absolutely any body, even a very light one (for example, a pencil lying on a table), deforms the support at the micro level. Therefore, a ground reaction occurs.

There is no special formula for finding this force. It is denoted by the letter , but this force is simply a separate type of elasticity force, so it can also be denoted as

The force is applied at the point of contact of the object with the support. Directed perpendicular to the support.

Since we represent the body as a material point, force can be represented from the center

Elastic force

This force arises as a result of deformation (change in the initial state of the substance). For example, when we stretch a spring, we increase the distance between the molecules of the spring material. When we compress a spring, we decrease it. When we twist or shift. In all these examples, a force arises that prevents deformation - the elastic force.

Hooke's law

The elastic force is directed opposite to the deformation.

Since we represent the body as a material point, force can be represented from the center

When connecting springs in series, for example, the stiffness is calculated using the formula

When connected in parallel, the stiffness

Sample stiffness. Young's modulus.

Young's modulus characterizes the elastic properties of a substance. This is a constant value that depends only on the material and its physical state. Characterizes the ability of a material to resist tensile or compressive deformation. The value of Young's modulus is tabular.

Read more about properties of solids.

Body weight

Body weight is the force with which an object acts on a support. You say, this is the force of gravity! The confusion occurs in the following: indeed, often the weight of a body is equal to the force of gravity, but these forces are completely different. Gravity is a force that arises as a result of interaction with the Earth. Weight is the result of interaction with support. The force of gravity is applied at the center of gravity of the object, while weight is the force that is applied to the support (not to the object)!

There is no formula for determining weight. This force is designated by the letter.

The support reaction force or elastic force arises in response to the impact of an object on the suspension or support, therefore the weight of the body is always numerically the same as the elastic force, but has the opposite direction.

The support reaction force and weight are forces of the same nature; according to Newton’s 3rd law, they are equal and oppositely directed. Weight is a force that acts on the support, not on the body. The force of gravity acts on the body.

Body weight may not be equal to gravity. It may be more or less, or it may be that the weight is zero. This condition is called weightlessness. Weightlessness is a state when an object does not interact with a support, for example, the state of flight: there is gravity, but the weight is zero!

It is possible to determine the direction of acceleration if you determine where the resultant force is directed

Please note that weight is force, measured in Newtons. How to correctly answer the question: “How much do you weigh”? We answer 50 kg, not naming our weight, but our mass! In this example, our weight is equal to gravity, that is, approximately 500N!

Overload- ratio of weight to gravity

Archimedes' force

Force arises as a result of the interaction of a body with a liquid (gas), when it is immersed in a liquid (or gas). This force pushes the body out of the water (gas). Therefore, it is directed vertically upward (pushes). Determined by the formula:

In the air we neglect the power of Archimedes.

If the Archimedes force is equal to the force of gravity, the body floats. If the Archimedes force is greater, then it rises to the surface of the liquid, if less, it sinks.

Electrical forces

There are forces of electrical origin. Occurs in the presence of an electrical charge. These forces, such as the Coulomb force, Ampere force, Lorentz force, are discussed in detail in the section Electricity.

Schematic designation of forces acting on a body

Often a body is modeled as a material point. Therefore, in diagrams, various points of application are transferred to one point - to the center, and the body is depicted schematically as a circle or rectangle.

In order to correctly designate forces, it is necessary to list all the bodies with which the body under study interacts. Determine what happens as a result of interaction with each: friction, deformation, attraction, or maybe repulsion. Determine the type of force and correctly indicate the direction. Attention! The amount of forces will coincide with the number of bodies with which the interaction occurs.

The main thing to remember

1) Forces and their nature;
2) Direction of forces;
3) Be able to identify the acting forces

There are external (dry) and internal (viscous) friction. External friction occurs between contacting solid surfaces, internal friction occurs between layers of liquid or gas during their relative motion. There are three types of external friction: static friction, sliding friction and rolling friction.

Rolling friction is determined by the formula

The resistance force occurs when a body moves in a liquid or gas. The magnitude of the resistance force depends on the size and shape of the body, the speed of its movement and the properties of the liquid or gas. At low speeds of movement, the drag force is proportional to the speed of the body

At high speeds it is proportional to the square of the speed

Let's consider the mutual attraction of an object and the Earth. Between them, according to the law of gravity, a force arises

Now let's compare the law of gravity and the force of gravity

The magnitude of the acceleration due to gravity depends on the mass of the Earth and its radius! Thus, it is possible to calculate with what acceleration objects on the Moon or on any other planet will fall, using the mass and radius of that planet.

The distance from the center of the Earth to the poles is less than to the equator. Therefore, the acceleration of gravity at the equator is slightly less than at the poles. At the same time, it should be noted that the main reason for the dependence of the acceleration of gravity on the latitude of the area is the fact of the Earth’s rotation around its axis.

As we move away from the Earth's surface, the force of gravity and the acceleration of gravity change in inverse proportion to the square of the distance to the center of the Earth.

DEFINITION

Strength is a vector quantity that is a measure of the action of other bodies or fields on a given body, as a result of which a change in the state of this body occurs. In this case, a change in state means a change or deformation.

The concept of force refers to two bodies. You can always indicate the body on which the force acts and the body from which it acts.

Strength is characterized by:

• module;
• direction;
• application point.

The magnitude and direction of the force are independent of the choice.

The unit of force in the C system is 1 Newton.

In nature, there are no material bodies that are outside the influence of other bodies, and, therefore, all bodies are under the influence of external or internal forces.

Several forces can act on a body at the same time. In this case, the principle of independence of action is valid: the action of each force does not depend on the presence or absence of other forces; the combined action of several forces is equal to the sum of the independent actions of the individual forces.

Resultant force

To describe the motion of a body in this case, the concept of resultant force is used.

DEFINITION

Resultant force is a force whose action replaces the action of all forces applied to the body. Or, in other words, the resultant of all forces applied to the body is equal to the vector sum of these forces (Fig. 1).

Fig.1. Determination of resultant forces

Since the movement of a body is always considered in some coordinate system, it is convenient to consider not the force itself, but its projections onto the coordinate axes (Fig. 2, a). Depending on the direction of the force, its projections can be either positive (Fig. 2, b) or negative (Fig. 2, c).

Fig.2. Projections of force onto coordinate axes: a) on a plane; b) on a straight line (the projection is positive);
c) on a straight line (projection is negative)

Fig.3. Examples illustrating the vector addition of forces

We often see examples illustrating the vector addition of forces: a lamp hangs on two cables (Fig. 3, a) - in this case, equilibrium is achieved due to the fact that the resultant of the tension forces is compensated by the weight of the lamp; the block slides along an inclined plane (Fig. 3, b) - the movement occurs due to the resultant forces of friction, gravity and support reaction. Famous lines from the fable by I.A. Krylov “and the cart is still there!” - also an illustration of the equality of the resultant of three forces to zero (Fig. 3, c).

Examples of problem solving

EXAMPLE 1

In addition to gravity, bodies moving relative to the Earth's surface are also affected by the Coriolis force.

Story

Jordan Nemorarius, in his essay “On Gravities,” when considering loads on an inclined plane, decomposed their gravity forces into components normal and parallel to the inclined plane, and was close to the definition of static moment.

Spherically symmetrical body

Earth

Corner α (\displaystyle \alpha ) between gravity P → (\displaystyle (\vec (P))) and the force of gravitational attraction to the Earth F → (\displaystyle (\vec (F))) equal to:

It varies from zero (at the equator, where φ = 0 ∘ (\displaystyle \varphi =0^(\circ )) and at the poles, where φ = 90 ∘ (\displaystyle \varphi =90^(\circ ))) to 0.001 8 (\displaystyle 0(,)0018) glad or 6′ (\displaystyle 6")(at latitude 45 ∘ (\displaystyle 45^(\circ ))).

Movement of bodies under the influence of gravity

In the case when the module of movement of the body is much less than the distance to the center of the Earth, then the force of gravity can be considered constant, and the movement of the body is uniformly accelerated. If the initial velocity of the body is different from zero and its vector is not directed vertically, then under the influence of gravity the body moves along a parabolic trajectory.

When throwing a body from a certain height parallel to the surface of the Earth, the flight range increases with increasing initial speed. At large values ​​of the initial velocity, to calculate the trajectory of the body, it is necessary to take into account the spherical shape of the Earth and the change in the direction of gravity at different points of the trajectory.

At a certain speed value, called the first cosmic velocity, a body thrown tangentially to the surface of the Earth, under the influence of gravity in the absence of resistance from the atmosphere, can move around the Earth in a circle without falling on the Earth. At a speed exceeding the second escape velocity, the body moves away from the Earth's surface to infinity along a hyperbolic trajectory. At speeds intermediate between the first and second cosmic speeds, the body moves around the Earth along an elliptical trajectory.

Potential energy of a body raised above the Earth

The potential energy of a body raised above the Earth is the work of gravity taken with the opposite sign, performed when moving the body from the surface of the Earth to this position. It is equal E p = γ M m (1 R z − 1 R) (\displaystyle E_(p)=\gamma Mm((\frac (1)(R_(z)))-(\frac (1)(R)) )), Where γ (\displaystyle \gamma )- gravitational constant, M (\displaystyle M)- mass of the earth, m (\displaystyle m)- body weight, R z (\displaystyle R_(z))- radius of the Earth, R (\displaystyle R)- distance to the center of the Earth of the body.

When the body moves away from a distance not small compared to the radius of the Earth, the gravitational field can be considered uniform, that is, the acceleration of gravity is constant. In this case, when lifting a body with a mass m (\displaystyle m) to the height h (\displaystyle h) gravity does work from the Earth's surface A = − m g h (\displaystyle A=-mgh). Therefore, the potential energy of the body is: E p = m g h (\displaystyle E_(p)=mgh). The potential energy of a body can have both positive and negative values. Body at depth h (\displaystyle h) from the Earth's surface has a negative potential energy E p = − m g h (\displaystyle E_(p)=-mgh) .

When water evaporates from the Earth's surface, solar radiation is transformed into potential energy of water vapor in the atmosphere. Then, when atmospheric precipitation falls on land, it turns into kinetic energy during runoff and performs erosive work in the process of transporting denudation material across the entire land and makes possible the life of the organic world on Earth.

The potential energy of rock masses transported by tectonic processes is mainly spent on moving rock destruction products from elevated areas of the surface to lower ones.

Meaning in nature

Gravity plays an important role in the evolution of stars. For stars at the main sequence stage of their evolution, gravity is one of the important factors providing the conditions necessary for thermonuclear fusion. At the final stages of the evolution of stars, in the process of their collapse, thanks to the force of gravity, not compensated by the forces of internal pressure, the stars turn into neutron stars or black holes.

Gravity is very important for the formation of the structure of the internal structure of the Earth and other planets and the tectonic evolution of its surface. The greater the force of gravity, the greater the mass of meteorite material falls per unit of its surface. During the existence of the Earth, its mass has increased significantly due to gravity: every year 30-40 million tons of meteorite matter, mainly in the form of dust, settle on the Earth, which significantly exceeds the dispersion of the light components of the Earth’s upper atmosphere in space.

Without the potential energy of gravity, which continuously transforms into kinetic energy, the circulation of matter and energy on Earth would be impossible.

Gravity plays a very important role for life on Earth. It is only thanks to it that the Earth has an atmosphere. Due to the force of gravity acting on the air, atmospheric pressure exists.

All living organisms with a nervous system have receptors that determine the magnitude and direction of gravity and serve for orientation in space. In vertebrate organisms, including humans, the magnitude and direction of gravity is determined by the vestibular apparatus.

The presence of gravity led to the emergence in all multicellular terrestrial organisms of strong skeletons necessary to overcome it. In aquatic living organisms, gravity is balanced by hydrostatic force.

The role of gravity in the life processes of organisms is studied by gravitational biology.

Application in technology

Accurate measurements of gravity and its gradient (gravimetry) are used in studying the internal structure of the Earth and in gravity exploration of various minerals.

Stability of a body in a gravity field

For a body in a gravity field resting on one point (for example, when hanging a body by one point or placing a ball on a plane), for stable equilibrium it is necessary that the center of gravity of the body occupies the lowest position compared to all possible neighboring positions.

For a body in a field of gravity resting on several points (for example, a table) or on an entire platform (for example, a box on a horizontal plane), for stable equilibrium it is necessary that the vertical line drawn through the center of gravity pass inside the area of ​​the body's support. Support area body is a contour connecting the points of support or inside the platform on which the body rests.

Methods for measuring gravity

Gravity is measured using dynamic and static methods. Dynamic methods use observation of the movement of a body under the influence of gravity and measure the time of transition of the body from one predetermined position to another. They use: oscillations of a pendulum, free fall of a body, oscillations of a string with a load. Static methods use observation of changes in the equilibrium position of a body under the influence of gravity and some force that balances it and measure the linear or angular displacement of the body.

Measurements of gravity are either absolute or relative. Absolute measurements determine the total value of gravity at a given point. Relative measurements determine the difference between the force of gravity at a given point and some other, previously known value. Instruments designed for relative measurements of gravity are called gravimeters.

Dynamic methods for determining gravity can be both relative and absolute, static methods - only relative.

Gravity on other planets

See also

Notes

1. Sivukhin D.V. General physics course. - M.: Fizmatlit, 2005. - T. I. Mechanics. - P. 372. - 560 p. - ISBN 5-9221-0225-7.
2. Targ S. M. Gravity// Physical encyclopedia / Ch. ed. A. M. Prokhorov. - M.: Great Russian Encyclopedia, 1994. - T. 4. - P. 496. - 704 p. - 40,000 copies. - ISBN 5-85270-087-8.
3. , With. 49.
4. The maximum change in gravity due to the Moon's gravity is approximately 0 , 25 ⋅ 10 − 5 (\displaystyle 0(,)25\cdot 10^(-5)) m/s 2 , Sun 0 , 1 ⋅ 10 − 5 (\displaystyle 0(,)1\cdot 10^(-5)) m/s 2

The direction coincides with the applied force, and the modulus is proportional to the modulus of the force and inversely proportional to the mass of the material point.

The word “force” in Russian is polysemantic and is often used (by itself or in combinations, in science and everyday situations) in meanings different from the physical definition of the term.

General information

Strength Characteristics

In addition to the division according to the type of fundamental interactions, there are other classifications of forces, including: external-internal (that is, acting on material points (bodies) of a given mechanical system from material points (bodies) not belonging to this system and the forces of interaction between material points ( bodies) of a given system), potential or not (whether the field of the forces being studied is potential), elastic-dissipative, concentrated-distributed (applied at one or many points), constant or variable in time.

A system of forces is a set of forces acting on the body under consideration or on points of a mechanical system. Two systems of forces are called equivalent if their individual action on the same solid body or material point is the same, other things being equal.

A balanced system of forces (or a system of forces equivalent to zero) is a system of forces whose action on a rigid body or material point does not lead to a change in their kinematic state.

Dimension of force

Historical aspect of the concept of strength

In the ancient world

Humanity first began to perceive the concept of force through the direct experience of moving heavy objects. “Strength”, “power”, “work” were synonymous (as in modern language outside of natural science). The transfer of personal feelings to objects of nature led to anthropomorphism: all objects that can influence others (rivers, stones, trees) must be alive, living beings must contain the same force that a person felt in himself.

In antiquity

When Greek scientists began to think about the nature of movement, the concept of force arose as part of Heraclitus's teaching on statics as a balance of opposites. Empedocles and Anaxagoras tried to explain the cause of movement and came to concepts close to the concept of force. For Anaxagoras, “mind” moves matter external to it. In Empedocles, movement is caused by the struggle of two principles, “love” (philia) and “enmity” (phobia), which Plato considered as attraction and repulsion. Moreover, the interaction, according to Plato, was explained in terms of the four elements (fire, water, earth and air): close things are attracted, earth to earth, water to water, fire to fire. In ancient Greek science, each element also had its own place in nature, which it tried to occupy. Thus, the force of gravity, for example, was explained in two ways: the attraction of like things and the desire of elements to take their place. Unlike Plato, Aristotle consistently occupied the second position, which postponed the concept of the general force of gravity, which would explain the movement of earthly and celestial bodies, until the time of Newton.

To denote the concept of force, Plato used the term “dynamis” (“possibility” of movement). The term was used in an expanded sense, close to the modern concept of power: chemical reactions, heat and light were all also dynamises.

Aristotle considered two different forces: inherent in the body itself (“nature”, physis) and the force with which one body pulls or pushes another (the bodies must be in contact). It was this concept of force that formed the basis of Aristotelian mechanics, although dualism prevented the quantitative determination of the force of interaction between two bodies (since weight was a natural force not associated with interaction, and therefore could not be used as a standard). In the case of natural motion (the fall of a heavy body or the rise of a light body), Aristotle proposed a formula for speed in the form of the ratio of the densities of the moving body A and the medium through which the movement occurs, B: v=A/B (the obvious problem for the case of equal densities was noted already in VI century).

He studied forces in the process of constructing simple mechanisms in the 3rd century. BC e. Archimedes. Archimedes considered forces statically and purely geometrically, and therefore his contribution to the development of the concept of force is insignificant.

The Stoics contributed to the development of the concept of strength. According to their teaching, forces inextricably linked two bodies through long-range “sympathy” or (in Posidonius) through universal tension permeating all space. The Stoics came to these conclusions by observing the tides, where the interaction of the Moon, Sun and water in the ocean was difficult to explain from the position of Aristotelian short-range action (Aristotle himself believed that the Sun, setting in the ocean, causes winds that lead to tides).

In preclassical mechanics

Bacon called long-range forces species(usually this term specific to Bacon is not translated) and considered their distribution in the environment as a chain of close interactions. Such forces, according to Bacon, had a completely physical character; the closest equivalent in modern physics is a wave.

Ockham was the first to abandon the Aristotelian description of interaction as direct contact and declared the ability of the mover to influence the movable at a distance, citing magnets as one example.

The Aristotelian formula v=A/B was also subject to revision. Already in the 6th century, John Philoponus considered the difference A-B as the right side, which, in addition to the problematic situation with identical densities, also made it possible to describe motion in a vacuum. In the 14th century, Bradwardin proposed the formula v=log(A/B) .

Kepler's

In classical mechanics

Newton's

Modernity

The end of the 20th century was characterized by debate about whether the concept of force is necessary in science and whether forces exist in principle - or is it just a term introduced for convenience.

Bigelow et al. argued in 1988 that forces essentially determine cause-and-effect relationships and therefore cannot be discarded. M. Jammer objected to this that in the Standard Model and other physical theories, force is interpreted only as an exchange of angular momentum, the concept of force therefore comes down to a simpler “interaction” between particles. This interaction is described in terms of the exchange of additional particles (photons, gluons, bosons and possibly gravitons). Jammer gives the following simplified explanation: two skaters glide across the ice shoulder to shoulder, both holding a ball. A rapid and simultaneous exchange of balls will result in a repulsive interaction.

Newtonian mechanics

Newton set out to describe the motion of objects using the concepts of inertia and force. Having done this, he simultaneously established that all mechanical motion obeys general conservation laws. In Newton he published his famous work “Mathematical Principles of Natural Philosophy,” in which he outlined the three fundamental laws of classical mechanics (Newton’s laws).

Newton's first law

Newton's second law

Newton's second law is:

Where m (\displaystyle m)- mass of a material point, a → (\displaystyle (\vec (a)))− its acceleration, F → (\displaystyle (\vec (F)))- the resultant of the applied forces. It is considered to be "the second most famous formula in physics", although Newton himself never explicitly wrote his second law in this form. For the first time this form of the law can be found in the works of C. Maclaurin and L. Euler.