The force of gravity physics. Formula for gravity. Gravitational forces: definition

It's no secret that the law of universal gravitation was discovered by the great English scientist Isaac Newton, who, according to legend, was walking in the evening garden and thinking about the problems of physics. At that moment, an apple fell from the tree (according to one version, directly on the physicist’s head, according to another, it simply fell), which later became Newton’s famous apple, as it led the scientist to an insight, a eureka. The apple that fell on Newton’s head inspired him to discover the law of universal gravitation, because the Moon in the night sky remained motionless, but the apple fell, perhaps the scientist thought that some force was acting on the Moon (causing it to rotate in orbit), so on the apple, causing it to fall to the ground.

Now, according to some historians of science, this whole story about the apple is just a beautiful fiction. In fact, whether the apple fell or not is not so important; what is important is that the scientist actually discovered and formulated the law of universal gravitation, which is now one of the cornerstones of both physics and astronomy.

Of course, long before Newton, people observed both things falling to the ground and stars in the sky, but before him they believed that there were two types of gravity: terrestrial (acting exclusively within the Earth, causing bodies to fall) and celestial (acting on stars and moon). Newton was the first to combine these two types of gravity in his head, the first to understand that there is only one gravity and its action can be described by a universal physical law.

Definition of the law of universal gravitation

According to this law, all material bodies attract each other, and the force of attraction does not depend on physical or chemical properties tel. It depends, if everything is simplified as much as possible, only on the weight of the bodies and the distance between them. You also need to additionally take into account the fact that all bodies on Earth are affected by the gravitational force of our planet itself, which is called gravity (from Latin the word “gravitas” is translated as heaviness).

Let us now try to formulate and write down the law of universal gravitation as briefly as possible: the force of attraction between two bodies with masses m1 and m2 and separated by a distance R is directly proportional to both masses and inversely proportional to the square of the distance between them.

Formula for the law of universal gravitation

Below we present to your attention the formula of the law of universal gravitation.

G in this formula is the gravitational constant, equal to 6.67408(31) 10 −11, this is the magnitude of the impact of the gravitational force of our planet on any material object.

The law of universal gravitation and weightlessness of bodies

The law of universal gravitation discovered by Newton, as well as the accompanying mathematical apparatus, later formed the basis of celestial mechanics and astronomy, because with its help it is possible to explain the nature of the movement of celestial bodies, as well as the phenomenon of weightlessness. Being in outer space at a considerable distance from the force of attraction and gravity of such a large body as a planet, any material object (for example, a spaceship with astronauts on board) will find itself in a state of weightlessness, since the force of the Earth’s gravitational influence (G in the formula for the law of gravity) or some other planet will no longer influence him.

Law of universal gravitation, video

And in conclusion, an instructive video about the discovery of the law of universal gravitation.

According to Newton's laws, a body can move with acceleration only under the influence of force. Because Falling bodies move with acceleration directed downwards, then they are acted upon by the force of gravity towards the Earth. But not only the Earth has the property of acting on all bodies with a force of gravity. Isaac Newton suggested that there are gravitational forces between all bodies. These forces are called forces of universal gravity or gravitational forces.

Having extended the established patterns - the dependence of the force of attraction of bodies on the Earth on the distances between bodies and on the masses of interacting bodies, obtained as a result of observations - Newton discovered in 1682. law of universal gravitation:All bodies attract each other, the force of universal gravitation is directly proportional to the product of the masses of the bodies and inversely proportional to the square of the distance between them:

The vectors of universal gravitational forces are directed along the straight line connecting the bodies. The proportionality factor G is called gravitational constant (universal gravity constant) and is equal to

.

Gravity The gravitational force acting on all bodies from the Earth is called:

.

Let
is the mass of the Earth, and
– radius of the Earth. Let's consider the dependence of the acceleration of free fall on the height of rise above the Earth's surface:

Body weight. Weightlessness

Body weight - the force with which a body presses on a support or suspension due to the attraction of this body to the ground. The body weight is applied to the support (suspension). The amount of body weight depends on how the body moves with support (suspension).

Body weight, i.e. the force with which the body acts on the support and the elastic force with which the support acts on the body, in accordance with Newton’s third law, are equal in absolute value and opposite in direction.

If a body is at rest on a horizontal support or moves uniformly, only gravity and the elastic force from the support act on it, therefore the weight of the body is equal to gravity (but these forces are applied to different bodies):

.

With accelerated movement, the weight of the body will not be equal to the force of gravity. Let us consider the movement of a body of mass m under the influence of gravity and elasticity with acceleration. According to Newton's 2nd law:

If the acceleration of a body is directed downwards, then the weight of the body is less than the force of gravity; if the acceleration of a body is directed upward, then all bodies are greater than the force of gravity.

An increase in body weight caused by accelerated movement of a support or suspension is called overload.

If a body falls freely, then from the formula * it follows that the weight of the body is zero. The disappearance of weight when the support moves with the acceleration of free fall is called weightlessness.

The state of weightlessness is observed in an airplane or spacecraft when it moves with the acceleration of gravity, regardless of the speed of its movement. Outside the Earth's atmosphere, when the jet engines are turned off, only the force of universal gravity acts on the spacecraft. Under the influence of this force, the spacecraft and all the bodies in it move with the same acceleration; therefore, the phenomenon of weightlessness is observed in the ship.

Movement of a body under the influence of gravity. Movement of artificial satellites. First escape velocity

If the module of movement of the body is much less than the distance to the center of the Earth, then the force of universal gravity during movement can be considered constant, and the movement of the body is uniformly accelerated. The simplest case of body motion under the influence of gravity is free fall with zero initial speed. In this case, the body moves with free fall acceleration towards the center of the Earth. If there is an initial velocity that is not directed vertically, then the body moves along a curved path (parabola, if air resistance is not taken into account).

At a certain initial speed, a body thrown tangentially to the surface of the Earth, under the influence of gravity in the absence of an atmosphere, can move in a circle around the Earth without falling on it or moving away from it. This speed is called first escape velocity, and a body moving in this way is artificial earth satellite (AES).

Let us determine the first escape velocity for the Earth. If a body, under the influence of gravity, moves around the Earth uniformly in a circle, then the acceleration of gravity is its centripetal acceleration:

.

Hence the first escape velocity is equal to

.

The first escape velocity for any celestial body is determined in the same way. The acceleration of gravity at a distance R from the center of a celestial body can be found using Newton’s second law and the law of universal gravitation:

.

Consequently, the first escape velocity at a distance R from the center of a celestial body of mass M is equal to

.

To launch an artificial satellite into low-Earth orbit, it must first be taken out of the atmosphere. Therefore, spaceships launch vertically. At an altitude of 200 - 300 km from the Earth's surface, where the atmosphere is rarefied and has almost no effect on the movement of the satellite, the rocket makes a turn and imparts the satellite its first escape velocity in a direction perpendicular to the vertical.

To the question “What is force?” physics answers this way: “Force is a measure of the interaction of material bodies with each other or between bodies and other material objects - physical fields.” All forces in nature can be classified into four fundamental types of interactions: strong, weak, electromagnetic and gravitational. Our article talks about what gravitational forces are - a measure of the last and, perhaps, most widespread type of these interactions in nature.

Let's start with the gravity of the earth

Everyone alive knows that there is a force that attracts objects to the earth. It is commonly referred to as gravity, gravity, or gravity. Thanks to its presence, a person developed the concepts of “up” and “down”, which determine the direction of movement or the location of something relative to earth's surface. So in a particular case, on the surface of the earth or near it, gravitational forces manifest themselves, which attract objects with mass to each other, manifesting their effect at any distance, both small and very large, even by cosmic standards.

Gravity and Newton's third law

As is known, any force, if it is considered as a measure of the interaction of physical bodies, is always applied to one of them. So in the gravitational interaction of bodies with each other, each of them experiences such types of gravitational forces that are caused by the influence of each of them. If there are only two bodies (it is assumed that the action of all others can be neglected), then each of them, according to Newton’s third law, will attract the other body with the same force. So the Moon and the Earth attract each other, resulting in the ebb and flow of the Earth's seas.

Every planet in solar system experiences several forces of attraction from the Sun and other planets at once. Of course, it is the gravitational force of the Sun that determines the shape and size of its orbit, but astronomers also take into account the influence of other celestial bodies in their calculations of the trajectories of their movement.

Which will fall to the ground faster from a height?

The main feature of this force is that all objects fall to the ground at the same speed, regardless of their mass. Once upon a time, right up to the 16th century, it was believed that everything was the other way around - heavier bodies should fall faster than lighter ones. To dispel this misconception, Galileo Galilei had to perform his famous experiment of simultaneously dropping two cannonballs different weights from the leaning Leaning Tower of Pisa. Contrary to the expectations of witnesses to the experiment, both nuclei reached the surface at the same time. Today, every schoolchild knows that this happened due to the fact that gravity imparts to any body the same acceleration of free fall g = 9.81 m/s 2 regardless of the mass m of this body, and its value according to Newton’s second law is equal to F = mg.

Gravitational forces on the Moon and on other planets have different values ​​of this acceleration. However, the nature of the action of gravity on them is the same.

Gravity and body weight

If the first force is applied directly to the body itself, then the second to its support or suspension. In this situation, elastic forces always act on the bodies from the supports and suspensions. Gravitational forces applied to the same bodies act towards them.

Imagine a weight suspended above the ground by a spring. Two forces are applied to it: the elastic force of the stretched spring and the force of gravity. According to Newton's third law, the load acts on the spring with a force equal and opposite to the elastic force. This force will be its weight. A load weighing 1 kg has a weight equal to P = 1 kg ∙ 9.81 m/s 2 = 9.81 N (newton).

Gravitational forces: definition

The first quantitative theory of gravity, based on observations of planetary motion, was formulated by Isaac Newton in 1687 in his famous “Principles of Natural Philosophy.” He wrote that the gravitational forces that act on the Sun and planets depend on the amount of matter they contain. They spread over long distances and always decrease as the reciprocal of the square of the distance. How can we calculate these gravitational forces? The formula for the force F between two objects with masses m 1 and m 2 located at a distance r is:

• F=Gm 1 m 2 /r 2 ,
  where G is a constant of proportionality, a gravitational constant.

Physical mechanism of gravity

Newton was not completely satisfied with his theory, since it assumed interaction between attracting bodies at a distance. The great Englishman himself was sure that there must be some physical agent responsible for transferring the action of one body to another, which he quite clearly stated in one of his letters. But the time when the concept of a gravitational field that permeates all space was introduced came only four centuries later. Today, speaking about gravity, we can talk about the interaction of any (cosmic) body with the gravitational field of other bodies, the measure of which is the gravitational forces arising between each pair of bodies. The law of universal gravitation, formulated by Newton in the above form, remains true and is confirmed by many facts.

Gravity theory and astronomy

It was very successfully applied to solving problems of celestial mechanics in time XVIII and the beginning of the 19th century. For example, mathematicians D. Adams and W. Le Verrier, analyzing disturbances in the orbit of Uranus, suggested that it is subject to gravitational forces of interaction with an as yet unknown planet. They indicated its expected position, and soon Neptune was discovered there by astronomer I. Galle.

There was still one problem though. Le Verrier in 1845 calculated that the orbit of Mercury precesses by 35" per century, in contrast to the zero value of this precession obtained from Newton's theory. Subsequent measurements gave a more accurate value of 43". (The observed precession is actually 570"/century, but a careful calculation to subtract the influence from all other planets gives a value of 43".)

It was not until 1915 that Albert Einstein was able to explain this discrepancy within the framework of his theory of gravity. It turned out that the massive Sun, like any other massive body, bends space-time in its vicinity. These effects cause deviations in the orbits of planets, but on Mercury, as the smallest planet and closest to our star, they are most pronounced.

Inertial and gravitational masses

As noted above, Galileo was the first to observe that objects fall to the ground at the same speed, regardless of their mass. In Newton's formulas, the concept of mass comes from two different equations. His second law says that a force F applied to a body with mass m gives acceleration according to the equation F = ma.

However, the force of gravity F applied to a body satisfies the formula F = mg, where g depends on the other body interacting with the one in question (the earth usually when we talk about gravity). In both equations m is a coefficient of proportionality, but in the first case it is inertial mass, and in the second it is gravitational mass, and there is no obvious reason that they should be the same for any physical object.

However, all experiments show that this is indeed the case.

Einstein's theory of gravity

He took the fact of equality of inertial and gravitational masses as a starting point for his theory. He managed to construct the gravitational field equations, the famous Einstein equations, and with their help calculate the correct value for the precession of the orbit of Mercury. They also give a measured value for the deflection of light rays that pass near the Sun, and there is no doubt that they give the correct results for macroscopic gravity. Einstein's theory of gravity, or general theory of relativity (GR) as he called it, is one of the greatest triumphs of modern science.

Are gravitational forces acceleration?

If you cannot distinguish inertial mass from gravitational mass, then you cannot distinguish gravity from acceleration. The gravitational field experiment can instead be performed in an accelerating elevator in the absence of gravity. When an astronaut in a rocket accelerates away from the earth, he experiences a force of gravity that is several times greater than Earth's, with the vast majority of it coming from acceleration.

If no one can distinguish gravity from acceleration, then the former can always be reproduced by acceleration. A system in which acceleration replaces gravity is called inertial. Therefore, the Moon in near-Earth orbit can also be considered as an inertial system. However, this system will differ from point to point as the gravitational field changes. (In the example of the Moon, the gravitational field changes direction from one point to another.) The principle that one can always find an inertial system at any point in space and time at which physics obeys the laws in the absence of gravity is called the equivalence principle.

Gravity as a manifestation of the geometric properties of space-time

The fact that gravitational forces can be thought of as accelerations in inertial coordinate systems that differ from point to point means that gravity is a geometric concept.

We say that spacetime is curved. Consider a ball on a flat surface. It will rest or, if there is no friction, move uniformly in the absence of any forces acting on it. If the surface is curved, the ball will accelerate and move to the lowest point, taking the shortest path. Similarly, Einstein's theory states that four-dimensional space-time is curved, and a body moves in this curved space along a geodesic line that corresponds to the shortest path. Therefore, the gravitational field and the gravitational forces acting in it on physical bodies are geometric quantities that depend on the properties of space-time, which change most strongly near massive bodies.

Definition

Between any bodies that have masses, forces act that attract the above-mentioned bodies to each other. Such forces are called forces of mutual attraction.

Let's consider two material points (Fig. 1). They attract with forces directly proportional to the product of the masses of these material points and inversely proportional to the distance between them. So, the gravitational force () will be equal to:

where a material point of mass m 2 acts on a material point of mass m 1 with an attractive force - radius - a vector drawn from point 2 to point 1, the modulus of this vector is equal to the distance between material points (r); G=6.67 10 -11 m 3 kg -1 s -2 (in the SI system) – gravitational constant (gravity constant).

In accordance with Newton's third law, the force with which material point 2 is attracted to material point 1 () is equal to:

Gravity between bodies is carried out through a gravitational field (gravitational field). Gravitational forces are potential. This makes it possible to enter such energy characteristics gravitational field as a potential, which is equal to the ratio of the potential energy of a material point located at the field point under study to the mass of this point.

Formula for the force of attraction of bodies of arbitrary shape

In two bodies of arbitrary shape and size, we identify elementary masses that can be considered material points, and:

where are the matter densities of the material points of the first and second bodies, dV 1 , dV 2 are the elementary volumes of the selected material points. In this case, the force of attraction (), with which the element dm 2 acts on the element dm 1, is equal to:

Consequently, the force of attraction of the first body by the second can be found by the formula:

where integration must be carried out over the entire volume of the first (V 1) and second (V 2) bodies. If the bodies are homogeneous, then the expression can be slightly transformed and obtained:

Formula for the force of attraction of spherical solids

If the attractive forces are considered for two solid bodies of a spherical shape (or close to balls), the density of which depends only on the distances to their centers, formula (6) will take the form:

where m 1 ,m 2 are the masses of the balls, is the radius – the vector connecting the centers of the balls,

Expression (7) can be used if one of the bodies has a shape other than spherical, but its dimensions are much smaller than the dimensions of the second body - a ball. Thus, formula (7) can be used to calculate the forces of attraction of bodies to the Earth.

Units of gravity

The basic unit of measurement for the force of attraction (like any other force) in the SI system is: =H.

In GHS: =din.

Examples of problem solving

Example

Exercise. What is the force of attraction between two identical homogeneous spheres of mass equal to 1 kg? The distance between their centers is 1 m.

Solution. The basis for solving the problem is the formula:

To calculate the modulus of the attractive force, formula (1.1) is transformed to the form:

Let's carry out the calculations:

Answer.

Example

Exercise. With what force (in absolute value) does an infinitely long and thin and straight rod attract a material particle of mass m. The particle is located at a distance a from the rod. The linear mass density of the substance of the rod is equal to tau

Solution. Let's make a drawing

Let us select an elementary section of mass dm on the rod.

Gravity, also known as attraction or gravitation, is a universal property of matter that all objects and bodies in the Universe possess. The essence of gravity is that all material bodies attract all other bodies around them.

Earth gravity

If gravity is general concept and the quality that all objects in the Universe possess, then gravity is a special case of this comprehensive phenomenon. The earth attracts to itself all material objects located on it. Thanks to this, people and animals can safely move across the earth, rivers, seas and oceans can remain within their shores, and the air can not fly across the vast expanses of space, but form the atmosphere of our planet.

A fair question arises: if all objects have gravity, why does the Earth attract people and animals to itself, and not vice versa? Firstly, we also attract the Earth to us, it’s just that, compared to its force of attraction, our gravity is negligible. Secondly, the force of gravity depends directly on the mass of the body: the smaller the mass of the body, the lower its gravitational forces.

The second indicator on which the force of attraction depends is the distance between objects: the greater the distance, the less the effect of gravity. Thanks also to this, the planets move in their orbits and do not fall on each other.

It is noteworthy that the Earth, Moon, Sun and other planets owe their spherical shape precisely to the force of gravity. It acts in the direction of the center, pulling towards it the substance that makes up the “body” of the planet.

Earth's gravitational field

The Earth's gravitational field is a force energy field that is formed around our planet due to the action of two forces:

• gravity;
• centrifugal force, which owes its appearance to the rotation of the Earth around its axis (diurnal rotation).

Since both gravity and centrifugal force act constantly, the gravitational field is a constant phenomenon.

The field is slightly affected by the gravitational forces of the Sun, Moon and some other celestial bodies, as well as the atmospheric masses of the Earth.

The law of universal gravitation and Sir Isaac Newton

The English physicist, Sir Isaac Newton, according to a famous legend, one day while walking in the garden during the day, he saw the Moon in the sky. At the same time, an apple fell from the branch. Newton was then studying the law of motion and knew that an apple falls under the influence of a gravitational field, and the Moon rotates in orbit around the Earth.

And then the brilliant scientist, illuminated by insight, came up with the idea that perhaps the apple falls to the ground, obeying the same force thanks to which the Moon is in its orbit, and not rushing randomly throughout the galaxy. This is how the law of universal gravitation, also known as Newton’s Third Law, was discovered.

In the language of mathematical formulas, this law looks like this:

F=GMm/D 2 ,

Where F- the force of mutual gravity between two bodies;

M- mass of the first body;

m- mass of the second body;

D 2- the distance between two bodies;

G- gravitational constant equal to 6.67x10 -11.