In this paragraph we will remind you about gravity, centripetal acceleration and body weight
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.
The force with which a body is attracted to the Earth under the influence of the Earth's gravitational field is called gravity. According to the law of universal gravitation, on the surface of the Earth (or near this surface), a body of mass m is acted upon by the force of gravity
F t =GMm/R 2
where M is the mass of the Earth; R is the radius of the Earth.
If only the force of gravity acts on a body, and all other forces are mutually balanced, the body undergoes free fall. According to Newton's second law and formula F t =GMm/R 2 the gravitational acceleration module g is found by the formula
g=F t /m=GM/R 2 .
From formula (2.29) it follows that the acceleration of free fall does not depend on the mass m of the falling body, i.e. for all bodies in a given place on the Earth it is the same. From formula (2.29) it follows that Ft = mg. In vector form
F t = mg
In § 5 it was noted that since the Earth is not a sphere, but an ellipsoid of revolution, its polar radius is less than the equatorial one. From the formula F t =GMm/R 2 it is clear that for this reason the force of gravity and the acceleration of gravity caused by it at the pole is greater than at the equator.
The force of gravity acts on all bodies located in the gravitational field of the Earth, but not all bodies fall to the Earth. This is explained by the fact that the movement of many bodies is impeded by other bodies, for example supports, suspension threads, etc. Bodies that limit the movement of other bodies are called connections. Under the influence of gravity, the bonds are deformed and the reaction force of the deformed connection, according to Newton’s third law, balances the force of gravity.
The acceleration of gravity is affected by the rotation of the Earth. This influence is explained as follows. The reference systems associated with the Earth's surface (except for the two associated with the Earth's poles) are not, strictly speaking, inertial reference systems - the Earth rotates around its axis, and together with it such reference systems move in circles with centripetal acceleration. This non-inertiality of reference systems is manifested, in particular, in the fact that the value of the acceleration of gravity turns out to be different in different places on the Earth and depends on the geographic latitude of the place where the reference system associated with the Earth is located, relative to which the acceleration of gravity is determined.
Measurements carried out at different latitudes showed that the numerical values of the acceleration due to gravity differ little from each other. Therefore, with not very accurate calculations, we can neglect the non-inertiality of the reference systems associated with the Earth’s surface, as well as the difference in the shape of the Earth from spherical, and assume that the acceleration of gravity anywhere on the Earth is the same and equal to 9.8 m/s 2 .
From the law of universal gravitation it follows that the force of gravity and the acceleration of gravity caused by it decrease with increasing distance from the Earth. At a height h from the Earth's surface, the gravitational acceleration modulus is determined by the formula
g=GM/(R+h) 2.
It has been established that at an altitude of 300 km above the Earth's surface, the acceleration of gravity is 1 m/s2 less than at the Earth's surface.
Consequently, near the Earth (up to heights of several kilometers) the force of gravity practically does not change, and therefore the free fall of bodies near the Earth is a uniformly accelerated motion.
Body weight. Weightlessness and overload
The force in which, due to attraction to the Earth, a body acts on its support or suspension is called body weight. Unlike gravity, which is a gravitational force applied to a body, weight is an elastic force applied to a support or suspension (i.e., a link).
Observations show that the weight of a body P, determined on a spring scale, is equal to the force of gravity F t acting on the body only if the scales with the body relative to the Earth are at rest or moving uniformly and rectilinearly; In this case
Р=F t=mg.
If the body moves at an accelerated rate, then its weight depends on the value of this acceleration and on its direction relative to the direction of the acceleration of gravity.
When a body is suspended on a spring scale, two forces act on it: the force of gravity F t =mg and the elastic force F yp of the spring. If in this case the body moves vertically up or down relative to the direction of acceleration of free fall, then the vector sum of the forces F t and F up gives a resultant, causing acceleration of the body, i.e.
F t + F up =ma.
According to the above definition of the concept of “weight”, we can write that P = -F yp. From the formula: F t + F up =ma. taking into account that F T =mg, it follows that mg-ma=-F yp . Therefore, P=m(g-a).
The forces Ft and Fup are directed along one vertical straight line. Therefore, if the acceleration of body a is directed downward (i.e., it coincides in direction with the acceleration of free fall g), then in modulus
P=m(g-a)
If the acceleration of the body is directed upward (i.e., opposite to the direction of the acceleration of free fall), then
P = m = m(g+a).
Consequently, the weight of a body whose acceleration coincides in direction with the acceleration of free fall is less than the weight of a body at rest, and the weight of a body whose acceleration is opposite to the direction of the acceleration of free fall is greater than the weight of a body at rest. An increase in body weight caused by its accelerated movement is called overload.
In free fall a=g. From the formula: P=m(g-a)
it follows that in this case P = 0, i.e. there is no weight. Therefore, if bodies move only under the influence of gravity (i.e. freely falling), they are in a state weightlessness. A characteristic feature of this state is the absence of deformations and internal stresses in freely falling bodies, which are caused by gravity in bodies at rest. The reason for the weightlessness of bodies is that the force of gravity imparts equal accelerations to a freely falling body and its support (or suspension).
A particular, but extremely important for us, type of universal gravitational force is the force of attraction of bodies towards the Earth. This force is called gravity. According to the law of universal gravitation, it is expressed by the formula
where m is the mass of the body, M is the mass of the Earth, R is the radius of the Earth, h is the height of the body above the surface of the Earth. The force of gravity is directed vertically downward, towards the center of the Earth.
The force of gravity imparts an acceleration to the body, called the acceleration of gravity. According to Newton's second law
Taking into account expression (3.6.1) for the gravitational acceleration modulus we have
On the Earth's surface (h = 0) the gravitational acceleration modulus is equal to
and the force of gravity is
The modulus of free fall acceleration included in formulas (3.6.4) and (3.6.5) is approximately 9.8 m/s 2 .
Acceleration of gravity
From formula (3.6.3) it is clear that the acceleration of gravity does not depend on the mass of the body. It decreases as the body rises above the Earth's surface: the acceleration of gravity is inversely proportional to the square of the body's distance from the center of the Earth.
However, if the height h of the body above the Earth’s surface does not exceed 100 km, then in calculations that allow an error of ≈ 1.5%, this height can be neglected in comparison with the radius of the Earth (R = 6370 km). The acceleration of gravity at altitudes up to 100 km can be considered constant and equal to 9.8 m/s 2 .
And yet, at the surface of the Earth, the acceleration of gravity is not the same everywhere. It depends on latitude: more at the Earth's poles than at the equator. The fact is that the globe is somewhat flattened at the poles. The equatorial radius of the Earth is 21 km greater than the polar radius.
Another, more significant reason for the dependence of the acceleration of gravity on geographic latitude is the rotation of the Earth. Newton's second law, with the help of which formula (3.6.4) is obtained, is valid in an inertial frame of reference.
Such a system is, for example, the heliocentric system. The reference system associated with the Earth, strictly speaking, cannot be considered inertial. The Earth rotates around its axis and moves in a closed orbit around the Sun.
The rotation of the Earth and its oblateness at the poles leads to the fact that the acceleration of gravity relative to the geocentric reference system at different latitudes is different: at the poles g floor ≈ 9.83 m/s 2, at the equator g eq ≈ 9.78 m/s 2, at latitude 45° g = 9.81 m/s 2 . However, in our calculations we will assume the acceleration of gravity to be approximately equal to 9.8 m/s 2 .
Due to the rotation of the Earth around its axis, the acceleration of gravity in all places except the equator and poles is not directed exactly towards the center of the Earth.
In addition, the acceleration of gravity depends on the density of rocks located in the bowels of the Earth. In areas where rocks occur whose density is greater than the average density of the Earth (for example, iron ore), g is greater. And where there are oil deposits, g is less. Geologists use this when searching for minerals.
Earth mass
Without “terrestrial” experiments to determine the gravitational constant G, we would not be able to determine the mass of the Earth and other planets by any astronomical means.
Having determined experimentally the acceleration of gravity, we can use expression (3.6.4) to calculate the mass of the Earth:
Substituting into this formula R ≈ 6.4 10 6 m, g ≈ 9.8 m/s 2 and G = 6.67 10 -11 N m 2 /kg 2, we get
Center of gravity
The force of gravity (1) acts on all bodies. But to what point of the body is this force applied if the body cannot be considered a material point?
Let's take a body of arbitrary shape, for example a piece of plywood. Let's pierce several holes in it: at points A, B, D (Fig. 3.9, a).
Rice. 3.9
Let's hang this piece of plywood on a knitting needle passed through a hole at point A. The piece of plywood is acted upon by the force of gravity m and the force from the support (knitting needle) - the support reaction force. Under the influence of these two forces, the body is in equilibrium (at rest). Therefore, according to Newton's second law,
since the acceleration of the body is zero. From expression (3.6.7) it follows that
that is, the force of gravity m and the reaction force of the support are directed in opposite directions, and the lines of their action lie on the same straight line. This straight line is vertical and passes through point A (straight AK), since the reaction force of the spoke is applied to the piece of plywood at the suspension point, i.e., at point A. Consequently, the point of application of gravity (the beginning of the gravity vector) acting on the piece plywood, lies on straight AK.
Now let's hang the same piece of plywood at point B (Fig. 3.9.6). Using similar reasoning, we will come to the conclusion that the point of application of gravity lies on the straight line BL. But since the point of application of gravity lies on both the straight line BL and the straight line AK, then it must coincide with the point C of their intersection. By hanging a piece of plywood at point D (Fig. 3.9, c) and drawing a vertical line through it, we will make sure that it also passes through point C. Thus, for any position of the body in space, the point of application of the force of gravity acting on the body is one and same point. This point is called the center of gravity of the body.
The center of gravity of a body is the point of application of the force of gravity acting on the body at any position in space.
We must understand well that the force of gravity acts on all the particles that make up the body. But if the position of the center of gravity is known, then we can “forget” that all parts of the body are affected by gravity forces, and assume that there is only one force applied at the center of gravity.
Guided by considerations of symmetry, we can indicate the position of the center of gravity of homogeneous bodies of simple shape (Fig. 3.10):
- disk and ball - in the center;
- a plate in the shape of a parallelogram and a beam in the shape of a parallelepiped - at the point of intersection of their diagonals;
- cylinder - in the middle of its axis.
Rice. 3.10
The force of attraction of bodies towards the Earth - the force of gravity - is one of the manifestations of the force of universal gravity. This force is applied at a point called the center of gravity of the body.
Self-test questions
- Where is the acceleration of free fall greater: in Moscow or St. Petersburg?
- It is known that the Moon is attracted to the Earth with a force F = 2 10 20 N. Calculate the mass of the Moon.
- Can the center of gravity be outside the body?
- Where is the center of gravity of a uniform triangular plate?
- Cut out several plates of arbitrary shape from cardboard and experimentally find their center of gravity.
To change the position of a body or the speed of its movement, a force must be applied to the body. In mechanics, force is the measure of the action of one body on another. Force is usually characterized by magnitude, direction and point of application.
The magnitude of the force is measured with a dynamometer and is most often expressed in kilograms.
When lifting a load, the magnitude of the force depends on the distance to the fulcrum. To determine the action of a force on a body, you need to know the point of its application to this body. The point of application of force is of great importance in technology.
Graphical way of representing forces. Force can be represented graphically as a line with an arrow on an arbitrarily chosen scale. The arrow indicates the direction of the force. The beginning of the line is called the point of application of the force. The straight line on which the segment representing the force lies is called line of action of the force.
The rule of addition and expansion of forces. Forces, the joint action of which can be replaced by a resultant force that has the same effect on the body as a given system of forces, are called components. Adding forces means finding the resultant.
Resultanttwo or more forces directed along one straight line is equal in e-magnitude to their algebraic sum.
Resultant two forces that have a common point of application and act at an angle to each other, is equal in magnitude and direction to the diagonal of a parallelogram built on these forces on the sides.
This resultant is called the geometric (or vector) sum of the component forces. As the angle between the forces changes, the magnitude of the resultant also changes.
The action opposite to the addition of forces is called disintegration of power into components. To decompose a force into two components, it is necessary to know their lines of action, which intersect at some point, or the magnitude and direction of one of the component forces.
Center of gravity. For small body sizes, the gravitational forces acting on its particles can be taken as parallel. In this case, the center of parallel forces is called centergravity. Consequently, the center of gravity is the center of all parallel forces acting on individual particles of the body.
The center of gravity occupies a certain position in the body, no matter how the body is rotated relative to the direction of gravity. The resultant of all gravity forces acting on individual particles of a body, applied at the center of gravity, is the weight of the body.
Each body has its own center of gravity. For example, the center of gravity of a homogeneous rod is at its middle; the center of gravity of the circle coincides with its center; the center of gravity of the area of a triangle lies at the point of intersection of its medians, and the center of gravity of a ball lies at its geometric center.
Stability of balance. Body position is considered stand firmyou m, if the body returns to its original position after it has been removed from it by some force. A ball suspended at one point located on the same vertical with the center of gravity is in a stable position or in a position of stable equilibrium.
The position is called unstable, if a body removed from an equilibrium position cannot be returned by its weight to its initial position. If the fulcrum coincides with the center of gravity, then the body remains at rest in any position (for example, a ball lying on a horizontal plane). This position is called without timepersonal, or a state of indifferent equilibrium.
Moment of power. The moment of force characterizes rotational motion.
If on a beam resting on a fixed support at a point WITH, place a load Q at a distance from point C sun, then the beam will begin to move counterclockwise around the point WITH.
Moments of forces rotating a body counterclockwise are conventionally considered negative, and moments of forces rotating a body clockwise are considered positive. To maintain the balance of the system, it is necessary to reach the end of the beam, for example at the point A, apply force R, directed in the direction opposite to the direction of gravity Q. The greater the distance from the application point A to the fulcrum C, the smaller the force P must be to maintain balance. Distances AC And Sun called shoulders. Let's denote the shoulder AC letter V. Product of force P per arm V is called the moment of this force relative to the fulcrum. For the beam to be balanced, it is necessary that the algebraic sum of the moments of all acting forces relative to the support point be equal to zero. Let's denote the shoulder Sun letter a, then Qa-Рв = 0.
The conditions of equilibrium of forces are widely used in the calculation of new machines.
In a technical system, a moment of force of 1 is used as a unit of measurement of moment of force. kgf, having a leverage of 1 m.
Centrifugal and centripetal forces. When a ball tied to a thread rotates, centrifugal and centripetal forces simultaneously arise; When the rotation stops, they disappear. The force holding the ball on the circle is directed along the thread towards the center of rotation and is called centripetal. The force applied to the thread, counteracting the centripetal force, is called centrifugal. Centrifugal and centripetal forces are usually equal to each other, but in opposite directions.
In technology, centrifugal force plays an important role. If the center of gravity of rotating parts (bearings and rollers) is shifted relative to the axis, then the magnitude of the centrifugal force can exceed the weight of the body itself by tens and hundreds of times. As a result, the bearings and journals of the rollers will wear out, which will lead to equipment failure.
Centrifugal force can be useful for machines, for example, a centrifuge is designed to separate bulk solids during ore dressing. When the centrifuge rotates, particles with the highest specific gravity are located at the periphery, and particles with a lower specific gravity are closer to the axis of rotation. In a centrifugal pump, the movement of fluid and the required pressure are created due to the centrifugal force resulting from the rotation of the impeller in the pump housing.
The point of application of gravity is called the center of gravity.
The point of application of gravity (center of gravity) is easy to determine if the body is fixed at one point so that it can rotate freely. If the body is in a position of equilibrium, then the center of gravity should be on the vertical passing through the point of attachment of the body.
Since the bricks are homogeneous, the point of application of gravity for each brick will lie in the middle of its length.
In this case, the rod AB will move forward and the trajectories of the points of application of gravity forces mtg and m - ig will be horizontal straight lines.
Decomposition of force into two parallel components. of the earth on a solid body is as if the point of application of gravity lay at the center of gravity of the body. We will use this further, replacing the action of gravity applied to individual parts of a solid body by the action of one force applied at its center of gravity and equal to the force of gravity acting on the entire body.
For the reduction point we will take the center of mass S of the link, which is the point of application of the force of gravity - Fg of the link and the force of inertia Ri. The main vector of forces acting on the link, F F0 Рг Fg Рi - The value and direction of the force F can be obtained analytically using the operator function SMVKT (see Chap.
These formulas are approximate, since the coordinates xk, yk, zk of the point of application of gravity Pk k - vi of a material particle are determined with an accuracy up to the size of this particle.
Thus, the effect of the Earth's gravity on a solid body is as if the point of application of gravity lay at the center of gravity of the body.
These formulas are approximate, since the coordinates Xb, z /, Zk of the point of application of gravity Pk k - b of a material particle are determined with an accuracy up to the size of this particle.
Determination of the roll center for various types of suspensions. The gravity force GK and the centrifugal force Pku are applied to the center of gravity of the sprung masses. The point of application of the gravity force GH and the centrifugal force Rau of the unsprung masses is located at a height approximately equal to the radius of the wheel.
Among the given forces in the tasks there may be: concentrated loads, depicted in the drawings for the tasks in the form of force vectors; weights of structural elements; distributed loads with a given intensity. If in problems a given pair of forces acts on a body or a system of bodies, then they are usually specified by the magnitude of the moment and the direction of rotation. Points of application of concentrated loads are always indicated in the conditions for the problem. The points of application of gravity forces are, as a rule, not indicated. It is believed that everyone solving the problem will apply this force at the center of gravity of the body in question. It is necessary to dwell on distributed loads in more detail.
From the equilibrium conditions it follows that the completely submerged face of the parallelepiped must be horizontal. When the parallelepiped deviates from its equilibrium position, the center of gravity of the displaced volume moves in the same direction where the parallelepiped tilted. Due to the fact that the point of application of gravity O and the point of application of lifting force C do not lie on the same vertical, moments of gravity and lifting force arise. If the parallelepiped face EF completely immersed in the liquid is larger than the partially immersed DE and GF (Fig. 283), then the resulting moment will return the body to the equilibrium position - the equilibrium will be stable. Otherwise (Fig. 284), when the face EF completely immersed in the liquid is smaller than the partially immersed faces DE and GF, the resulting moment will tilt the body even more - the equilibrium will be unstable.
It is necessary to know the point of application and direction of each force. It is important to be able to determine which 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 occurs because 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 the body is represented 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 the body is represented 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 it is less, it sinks.
Electric 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.
