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1 BASIC FORMULAS IN PHYSICS FOR STUDENTS OF TECHNICAL UNIVERSITIES.. Physical foundations of mechanics. Instantaneous speed dr r- radius vector of the material point, t- time, Module of instantaneous speed s- distance along the trajectory of motion, Path length Acceleration: instantaneous tangential normal total τ- unit vector tangent to the trajectory; R is the radius of curvature of the trajectory, n is the unit vector of the main normal. ANGULAR SPEED ds = S t t t d a d a a n n R a a a, n a a a n d φ - angular displacement. Angular acceleration d.. Relationship between linear and.. angular quantities s= φr, υ= ωr, and τ = εr, and n = ω R.3. Impulse.4. material point p mass of the material point. Basic equation of the dynamics of a material point (Newton's second law)

2 a dp Fi, Fi Law of conservation of momentum for an isolated mechanical system Radius vector of the center of mass Dry friction force μ - friction coefficient, N - normal pressure force. Elastic force k- coefficient of elasticity (stiffness), Δl- deformation..4.. Force of gravitational r F i i onst r i N F in =k Δl, i i.4.. interactions.4.3. F G r and are the particle masses, G is the gravitational constant, r is the distance between particles. Work of force A FdS da Power N F Potential energy: k(l) of an elastically deformed body P = gravitational interaction of two particles P = G r body in a uniform gravitational field g - gravitational field strength (gravitational acceleration), h - distance from the zero level. P=gh

3.4.4. Gravitational tension.4.5. Earth's field g= G (R h) 3 mass of the Earth, R 3 - radius of the Earth, h - distance from the surface of the Earth. Potential of the Earth's gravitational field 3 Kinetic energy of a material point φ= G T= (R 3 3 h) p Law of conservation of mechanical energy for a mechanical system E=T+P=onst Moment of inertia of a material point J=r r- distance to the axis of rotation. Moments of inertia of bodies with mass relative to an axis passing through the center of mass: a thin-walled cylinder (ring) of radius R, if the axis of rotation coincides with the axis of the cylinder J o = R solid cylinder (disk) of radius R, if the axis of rotation coincides with the axis of the cylinder J o = R a ball of radius R J о = 5 R a thin rod of length l, if the axis of rotation is perpendicular to the rod J о = l Moment of inertia of a body with mass relative to an arbitrary axis (Steiner’s theorem) J=J +d

4 J is the moment of inertia about a parallel axis passing through the center of mass, d is the distance between the axes. The moment of force acting on a material point relative to the origin of coordinates r is the radius vector of the point of application of the force. Momentum of the system.4.8. relative to the Z axis r F N.4.9. L z J iz iz i.4.. Basic equation of dynamics.4.. of rotational motion Law of conservation of angular momentum for an isolated system Work during rotational motion dl, J.4.. Σ J i ω i =onst A d Kinetic energy of a rotating body J T= L J Relativistic length contraction l l lо the length of a body at rest c is the speed of light in vacuum. Relativistic time dilation t t t o proper time. Relativistic mass o rest mass Rest energy of the particle E o = o c

5.4.3. Total relativistic energy.4.4. particles.4.5. E=.4.6. Relativistic impulse P=.4.7. Kinetic energy.4.8. relativistic particle.4.9. T = E- E o = Relativistic relationship between total energy and momentum E = p c + E o The law of addition of velocities in relativistic mechanics and and and - velocities in two inertial reference systems moving relative to each other with a speed υ coinciding in direction with and (sign -) or oppositely directed (sign +) u u u Physics of mechanical vibrations and waves. The displacement of the oscillating material point s Aos(t) A is the amplitude of the oscillation, is the natural cyclic frequency, φ o is the initial phase. Cyclic frequency T

6 T period of oscillation - frequency Speed ​​of an oscillating material point Acceleration of an oscillating material point Kinetic energy of a material point performing harmonic oscillations v ds d s a v T Potential energy of a material point performing harmonic oscillations Ï kx stiffness coefficient (elasticity coefficient) Total energy of a material point performing harmonic oscillations oscillations A sin(t) dv E T Ï A os(t) A A A sin (t) os (t) d s Differential equation s of free harmonic undamped oscillations of quantity s d s ds Differential equation s of free damped oscillations of quantity s, - damping coefficient A(t) T Logarithmic decrement ln T A(T t) of damping, relaxation time d s ds Differential equation s F ost Period of oscillation of pendulums: spring T, k

7 physical T J, gl - mass of the pendulum, k - spring stiffness, J - moment of inertia of the pendulum, g - gravitational acceleration, l - distance from the suspension point to the center of mass. Equation of a plane wave propagating in the direction of the Ox axis, v speed of wave propagation Wave length T - wave period, v - speed of wave propagation, oscillation frequency Wave number Speed ​​of sound propagation in gases γ - ratio of the heat capacities of gas, at constant pressure and volume, R- molar gas constant, T- thermodynamic temperature, M- molar mass of gas x (x, t) Aos[ (t) ] v v T v vt v RT Molecular physics and thermodynamics..4.. Amount of substance N N A, N- number of molecules, N A - Avogadro's constant - mass of substance M molar mass. Clapeyron-Mendeleev equation p = ν RT,

8 p is the gas pressure, is its volume, R is the molar gas constant, T is the thermodynamic temperature. Equation of molecular kinetic theory of gases Р= 3 n<εпост >= 3 no<υ кв >n is the concentration of molecules,<ε пост >- average kinetic energy of translational motion of a molecule. o - molecular mass<υ кв >- root mean square speed. Average molecular energy<ε>= i kt i - number of degrees of freedom k - Boltzmann constant. Internal energy of an ideal gas U= i νrt Molecular velocities: root mean square<υ кв >= 3kT = 3RT ; arithmetic mean<υ>= 8 8RT = kt ; most likely<υ в >= Average free length kt = RT ; path of a molecule d-effective diameter of a molecule Average number of collisions (d n) of a molecule per unit time z d n v

9 Distribution of molecules in a potential force field P is the potential energy of a molecule. Barometric formula p - gas pressure at height h, p - gas pressure at a level taken as zero, - molecular mass, Fick's Diffusion Law j - mass flow density, n n exp kt gh p p exp kt j d ds d =-D dx d - density gradient, dx D - diffusion coefficient, ρ - density, d - gas mass, ds - elementary area perpendicular to the Ox axis. Fourier's law of thermal conductivity j - heat flux density, Q j Q dq ds dt =-æ dx dt - temperature gradient, dx æ - thermal conductivity coefficient, Internal friction force η - dynamic viscosity coefficient, dv df ds dz d - velocity gradient, dz Coefficient diffusion D= 3<υ><λ>Coefficient of dynamic viscosity (internal friction) v 3 D Thermal conductivity coefficient æ = 3 сv ρ<υ><λ>=ηс v

10 s v specific isochoric heat capacity, Molar heat capacity of an ideal gas isochoric isobaric First law of thermodynamics i C v R i C p R dq=du+da, da=pd, du=ν C v dt Work of gas expansion during an isobaric process A=p( -)= ν R(T -T) isothermal p А= ν RT ln = ν RT ln p adiabatic A C T T) γ=с р/С v (RT A () p A= () Poisson's equations Carnot cycle efficiency. 4.. Q n and T n - the amount of heat received from the heater and its temperature; Q x and T x - the amount of heat transferred to the refrigerator and its temperature. The change in entropy during the transition of the system from state to state P γ =onst T γ- =onst. T γ r - γ =onst Qí Q Q S S í õ Tí T T dq T í õ

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Dynamics- a branch of physics that studies the causes of the motion of bodies.

Newton's first law states that there are inertial frames of reference relative to which bodies maintain a constant speed if they are not acted upon by other bodies.

states that the acceleration acquired by a body under the action of a force is directly proportional to the magnitude of the force and inversely proportional to the mass of the body.

states that interacting bodies act on each other with forces whose vectors are equal in magnitude and opposite in direction.

Law of Gravity states: the force of gravitational attraction between two material points is directly proportional to the product of their masses and inversely proportional to the square of the distance between them. The coefficient of proportionality is the gravitational constant.

Hooke's lawestablishes the proportionality of the modulus of the elastic force to the elongation modulus of the body if its deformation is elastic. The proportionality coefficient is the stiffness coefficient of the body.

Amonton-Coulomb law establishes the proportionality of the sliding friction force or the maximum static friction force to the force of the normal support reaction. The proportionality coefficient is the friction coefficient.

Impulse of poweris called the product of the velocity vector and the time interval of its action. Unit of force impulse modulus – 1 kg m/s .

Body impulse(quantity of motion) is the product of a body’s mass and its velocity vector. Unit of body impulse modulus – 1 kg m/s .

Law of conservation of momentum states: the sum of the momenta of bodies before their interaction is equal to the sum of the impulses of the same bodies after interaction, if the system is closed.

Change in body kinetic energy equal to the resultant work of all forces. The kinetic energy of a body moving in space without rotation is equal to half the product of its mass and the square of its speed. Unit of measurement – 1 J .

Change in body potential energy equal to the work of the potential force in question taken with the opposite sign. Potential energy under the action of gravity is equal to the product of the gravity modulus and the distance from the body to the selected zero energy level. The potential energy under the action of an elastic force is equal to half the product of the stiffness coefficient and the square of the elongation of the body compared to its undeformed state. The unit for measuring potential energy of any kind is 1 J .

Dynamics. Tables.

The session is approaching, and it’s time for us to move from theory to practice. Over the weekend we sat down and thought that many students would benefit from having a collection of basic physics formulas at their fingertips. Dry formulas with explanation: short, concise, nothing superfluous. A very useful thing when solving problems, you know. And during an exam, when exactly what was memorized the day before might “jump out of your head,” such a selection will serve an excellent purpose.

The most problems are usually asked in the three most popular sections of physics. This mechanics, thermodynamics And molecular physics, electricity. Let's take them!

Basic formulas in physics dynamics, kinematics, statics

Let's start with the simplest. The good old favorite straight and uniform movement.

Kinematics formulas:

Of course, let's not forget about motion in a circle, and then we'll move on to dynamics and Newton's laws.

After dynamics, it’s time to consider the conditions of equilibrium of bodies and liquids, i.e. statics and hydrostatics

Now we present the basic formulas on the topic “Work and Energy”. Where would we be without them?

Basic formulas of molecular physics and thermodynamics

Let's finish the mechanics section with formulas for oscillations and waves and move on to molecular physics and thermodynamics.

The efficiency factor, the Gay-Lussac law, the Clapeyron-Mendeleev equation - all these formulas dear to the heart are collected below.

By the way! There is now a discount for all our readers 10% on .

Basic formulas in physics: electricity

It's time to move on to electricity, even though it is less popular than thermodynamics. Let's start with electrostatics.

And, to the beat of the drum, we end with formulas for Ohm’s law, electromagnetic induction and electromagnetic oscillations.

That's all. Of course, a whole mountain of formulas could be cited, but this is of no use. When there are too many formulas, you can easily get confused and even melt your brain. We hope our cheat sheet of basic physics formulas will help you solve your favorite problems faster and more efficiently. And if you want to clarify something or haven’t found the right formula: ask the experts student service. Our authors keep hundreds of formulas in their heads and crack problems like nuts. Contact us, and soon any task will be up to you.

If kinematics only describes the movement of bodies, then dynamics studies the causes of this movement under the influence of forces acting on the body.

Dynamics– a branch of mechanics that studies the interactions of bodies, the causes of movement and the type of movement that occurs. Interaction- a process during which bodies exert mutual influence on each other. In physics, all interactions are necessarily paired. This means that bodies interact with each other in pairs. That is, every action necessarily generates a reaction.

Strength is a quantitative measure of the intensity of interaction between bodies. Force causes a change in the speed of the body as a whole or its parts (deformation). Force is a vector quantity. The straight line along which the force is directed is called the line of action of the force. Force is characterized by three parameters: the point of application, the magnitude (numerical value) and direction. In the International System of Units (SI), force is measured in Newtons (N). Calibrated springs are used to measure forces. Such calibrated springs are called dynamometers. Strength is measured by the stretch of a dynamometer.

A force that has the same effect on a body as all the forces acting on it taken together is called resultant force. It is equal to the vector sum of all forces acting on the body:

To find the vector sum of several forces, you need to make a drawing, where you correctly draw all the forces and their vector sum, and using this drawing, using knowledge from geometry (mainly the Pythagorean theorem and the cosine theorem), find the length of the resulting vector.

Types of forces:

1. Gravity. Applied to the center of mass of the body and directed vertically downwards (or what is the same: perpendicular to the horizon line), and is equal to:

Where: g- free fall acceleration, m- body weight. Don’t get confused: the force of gravity is perpendicular to the horizon, and not to the surface on which the body lies. Thus, if the body lies on an inclined surface, the force of gravity will still be directed straight down.

2. Friction force. It is applied to the surface of contact of the body with the support and is directed tangentially to it in the direction opposite to the one where other forces are pulling or trying to pull the body.

3. Viscous friction force (medium resistance force). Occurs when a body moves in a liquid or gas and is directed against the speed of movement.

4. Ground reaction force. Acts on the body from the side of the support and is directed perpendicular to the support from it. When a body rests on an angle, the reaction force of the support is directed perpendicular to the surface of the body.

5. Thread tension force. Directed along the thread away from the body.

6. Elastic force. Occurs when the body is deformed and is directed against the deformation.

Pay attention and note for yourself the obvious fact: if the body is at rest, then the resultant of the forces is equal to zero.

Force projections

In most dynamics problems, more than one force acts on a body. In order to find the resultant of all forces in this case, you can use the following algorithm:

1. Let's find the projections of all forces onto the OX axis and sum them up, taking into account their signs. So we get the projection of the resultant force on the OX axis.
2. Let's find the projections of all forces onto the OY axis and sum them up, taking into account their signs. This way we get the projection of the resultant force onto the OY axis.
3. The resultant of all forces will be found according to the formula (Pythagorean theorem):

In doing so, pay special attention to the following:

1. If the force is perpendicular to one of the axes, then the projection onto this axis will be equal to zero.
2. If, when projecting a force onto one of the axes, the sine of the angle “pops up,” then when projecting the same force onto another axis there will always be a cosine (of the same angle). When projecting, it is easy to remember which axis the sine or cosine will be on. If the angle is adjacent to the projection, then when the force is projected onto this axis there will be a cosine.
3. If the force is directed in the same direction as the axis, then its projection onto this axis will be positive, and if the force is directed in the direction opposite to the axis, then its projection onto this axis will be negative.

Newton's laws

The laws of dynamics, which describe the influence of various interactions on the motion of bodies, were in one of their simplest forms first clearly and clearly formulated by Isaac Newton in the book “Mathematical Principles of Natural Philosophy” (1687), therefore these laws are also called Newton’s Laws. Newton's formulation of the laws of motion is valid only in inertial reference systems (IRS). ISO is a reference system associated with a body moving by inertia (uniformly and rectilinearly).

There are other restrictions on the applicability of Newton's laws. For example, they give accurate results only as long as they are applied to bodies whose speeds are much less than the speed of light, and whose sizes significantly exceed the sizes of atoms and molecules (a generalization of classical mechanics to bodies moving at an arbitrary speed is relativistic mechanics, and to bodies , whose dimensions are comparable to atomic ones - quantum mechanics).

Newton's first law (or law of inertia)

Formulation: In ISO, if no forces act on the body or the action of the forces is compensated (that is, the resultant of the forces is zero), then the body maintains a state of rest or uniform linear motion.

The property of bodies to maintain their speed in the absence of the action of other bodies on it is called inertia. Therefore, Newton's first law is called the law of inertia. So, the reason for a change in the speed of movement of a body as a whole or its parts is always its interaction with other bodies. To quantitatively describe changes in the movement of a body under the influence of other bodies, it is necessary to introduce a new quantity - body mass.

Weight is a property of a body that characterizes its inertia (the ability to maintain a constant speed. In the International System of Units (SI), body mass is measured in kilograms (kg). Body mass is a scalar quantity. Mass is also a measure of the amount of substance:

Newton's second law - the fundamental law of dynamics

When starting to formulate the second law, we should remember that two new physical quantities are introduced in dynamics - body mass and force. The first of these quantities – mass – is a quantitative characteristic of the inert properties of a body. It shows how the body reacts to external influences. The second - force - is a quantitative measure of the action of one body on another.

Formulation: The acceleration acquired by a body in ISO is directly proportional to the resultant of all forces acting on the body, and inversely proportional to the mass of this body:

However, when solving problems in dynamics, it is advisable to write Newton’s second law in the form:

If several forces simultaneously act on a body, then the force in the formula expressing Newton’s second law must be understood as the resultant of all forces. If the resultant force is zero, then the body will remain in a state of rest or uniform linear motion, because the acceleration will be zero (Newton's first law).

Newton's third law

Formulation: In ISO, bodies act on each other with forces equal in magnitude and opposite in direction, lying on the same straight line and having the same physical nature:

These forces are applied to different bodies and therefore cannot balance each other. Please note that you can only add forces that simultaneously act on one of the bodies. When two bodies interact, forces arise that are equal in magnitude and opposite in direction, but they cannot be added, because they are attached to different bodies.

Algorithm for solving dynamics problems

Dynamics problems are solved using Newton's laws. The following procedure is recommended:

1. After analyzing the condition of the problem, establish which forces act on which bodies;

2. Show in the figure all forces in the form of vectors, that is, directed segments applied to the bodies on which they act;

3. Choose a reference system, in which case it is useful to direct one coordinate axis to the same direction as the acceleration of the body in question, and the other - perpendicular to the acceleration;

4. Write Newton's II law in vector form:

5. Go to the scalar form of the equation, that is, write down all its terms in the same order in projections onto each of the axes, without vector signs, but taking into account that the forces directed against the selected axes will have negative projections, and thus on the left side Newton's law they will be subtracted, not added. The result will be expressions like:

6. Create a system of equations, supplementing the equations obtained in the previous paragraph, if necessary, with kinematic or other simple equations;

8. If several bodies are involved in the movement, the analysis of forces and recording of equations is carried out for each of them separately. If a dynamics problem describes several situations, then a similar analysis is performed for each situation.

When solving problems, also consider the following: the direction of the body's velocity and the resultant forces do not necessarily coincide.

Elastic force

Deformation refers to any change in the shape or size of the body. Elastic deformations are those in which the body completely restores its shape after the cessation of the deforming force. For example, after the load was removed from the spring, its undeformed length did not change. When a body undergoes elastic deformation, a force arises that tends to restore the previous size and shape of the body. It is called elastic force. The simplest type of deformation is unilateral tensile or compression deformation.

For small deformations, the elastic force is proportional to the deformation of the body and is directed in the direction opposite to the direction of movement of the body particles during deformation:

Where: k– body rigidity, X– the amount of stretching (or compression, deformation of the body), it is equal to the difference between the final and initial length of the deformed body. And it is not equal to either its initial or final length separately. Stiffness does not depend either on the magnitude of the applied force or on the deformation of the body, but is determined only by the material from which the body is made, its shape and dimensions. In the SI system, stiffness is measured in N/m.

The statement about the proportionality of the force of elasticity and deformation is called Hooke's law. Spiral springs are often used in technology. When springs are stretched or compressed, elastic forces arise, which also obey Hooke's law. The coefficient k is called the spring stiffness. Within the limits of applicability of Hooke's law, springs are capable of greatly changing their length. Therefore, they are often used to measure forces. A spring whose tension is calibrated in units of force is called a dynamometer.

Thus, each specific body (not material) has its own rigidity and it does not change for a given body. Thus, if in a dynamics problem you stretched the same spring several times, you must understand that its stiffness was the same in all cases. On the other hand, if in the problem there were several springs of different sizes, but, for example, they were all steel, then nevertheless they will all have different stiffnesses. Since stiffness is not a material characteristic, it cannot be found in any tables. The stiffness of each specific body will either be given to you in the dynamics problem, or its value should be the subject of some additional research when solving this problem.

When compressed, the elastic force prevents compression, and when stretched, it prevents stretching. Let's also consider how we can express the stiffness of several springs connected in a certain way. When connecting springs in parallel The overall stiffness coefficient is calculated using the formula:

When connecting springs in series The overall stiffness coefficient can be found from the expression:

Body weight

The force of gravity with which bodies are attracted to the Earth must be distinguished from the weight of the body. The concept of weight is widely used in everyday life in the wrong sense, weight means mass, but this is not true.

Body weight is the force with which the body acts on a support or suspension. Weight is a force, which, like all forces, is measured in newtons (and not in kilograms), and is designated P. In this case, it is assumed that the body is motionless relative to the support or suspension. According to Newton's third law, weight is often equal to either the reaction force of the support (if the body is lying on a support), or the tension force of a thread or the elastic force of a spring (if the body is hanging on a thread or spring). Let's make a reservation right away - weight is not always equal to gravity.

Weightlessness is a state that occurs when body weight is zero. In this state, the body does not act on the support, but the support acts on the body.

An increase in body weight caused by accelerated movement of a support or suspension is called overload. Overload is calculated using the formula:

Where: P– weight of the body experiencing overload, P 0 – weight of the same body at rest. Overload is a dimensionless quantity. This is clearly seen from the formula. Therefore, do not believe science fiction writers who measure it in their books in g.

Remember that weight is never shown in pictures. It is simply calculated using formulas. And the pictures depict the tension force of the thread or the reaction force of the support, which, according to Newton’s third law, are numerically equal to the weight, but are directed in the other direction.

So, let us note once again three essential points that are often confused:

• Even though the weight and ground reaction force are equal in magnitude and opposite in direction, their sum is not zero. These forces cannot be added at all, because they are applied to different bodies.
• Body mass and body weight should not be confused. Mass is a characteristic of the body, measured in kilograms; weight is the force exerted on a support or suspension, measured in Newtons.
• If you need to find the weight of a body R, then first find the ground reaction force N, or thread tension T, and according to Newton's third law, the weight is equal to one of these forces and opposite in direction.

Friction force

Friction- one of the types of interaction between bodies. It occurs in the area of ​​​​contact of two bodies during their relative movement or an attempt to cause such movement. Friction, like all other types of interaction, obeys Newton’s third law: if a friction force acts on one of the bodies, then a force of the same magnitude, but directed in the opposite direction, also acts on the second body.

Dry friction that occurs when bodies are at relative rest is called static friction. Static friction force always equal in magnitude to the external causing force and directed in the opposite direction to it. The static friction force cannot exceed a certain maximum value, which is determined by the formula:

Where: μ is a dimensionless quantity called the coefficient of static friction, and N– ground reaction force.

If the external force is greater than the maximum value of the frictional force, relative slip occurs. The friction force in this case is called sliding friction force. It is always directed in the direction opposite to the direction of movement. The sliding friction force can be considered equal to the maximum static friction force.

Proportionality factor μ therefore also called sliding friction coefficient. Friction coefficient μ – dimensionless quantity. The friction coefficient is positive and less than unity. It depends on the materials of the contacting bodies and on the quality of processing of their surfaces. Thus, the friction coefficient is a certain specific number for each specific pair of interacting bodies. You won't be able to find it in any tables. For you, it must either be given in the problem, or you yourself must find it while solving it from some formulas.

If, as part of solving a problem, you get a friction coefficient greater than one or negative, you are solving this problem in dynamics incorrectly.

If the condition of the problem asks to find the minimum force under the influence of which movement begins, then they look for the maximum force under the influence of which movement does not yet begin. This makes it possible to equate the acceleration of bodies to zero, which means significantly simplifying the solution of the problem. In this case, the friction force is assumed to be equal to its maximum value. In this way, the moment is considered at which an increase in the desired force by a very small amount will immediately cause movement.

Features of solving problems in dynamics with several bodies

Bound bodies

An algorithm for solving problems in dynamics in which several bodies connected by threads are considered:

1. Make a drawing.
2. Write down Newton's second law for each body separately.
3. If the thread is inextensible (and this will be the case in most problems), then the accelerations of all bodies will be identical in magnitude.
4. If the thread is weightless, the block has no mass, and there is no friction in the axis of the block, then the tension force is the same at any point of the thread.

Movement of the body through the body

In problems of this type, it is important to take into account that the friction force on the surface of contacting bodies acts on both the upper body and the lower body, that is, friction forces occur in pairs. Moreover, they are directed in different directions and have equal magnitude, determined by the weight of the upper body. If the lower body also moves, then it must be taken into account that it is also affected by the friction force from the support.

Rotational movement

When a body moves in a circle, regardless of the plane in which the movement occurs, the body will move with centripetal acceleration, which will be directed towards the center of the circle along which the body is moving. However, the concept of a circle should not be taken literally. A body can only move through a circular arc (for example, move along a bridge). In all problems of this type, one of the axes is necessarily selected in the direction of centripetal acceleration, i.e. to the center of a circle (or arc of a circle). It is advisable to direct the second axis perpendicular to the first. Otherwise, the algorithm for solving these problems coincides with solving other problems in dynamics:

1. Having selected the axes, write down Newton's law in projections onto each axis, for each of the bodies participating in the problem, or for each of the situations described in the problem.

2. If necessary, supplement the system of equations with the necessary equations from other topics in physics. It is especially important to remember the formula for centripetal acceleration:

3. Solve the resulting system of equations using mathematical methods.

There are also a number of tasks involving rotation in a vertical plane on a rod or thread. At first glance, it may seem that such tasks will be the same. This is wrong. The fact is that the rod can experience both tensile and compressive deformations. The thread cannot be compressed; it immediately bends, and the body simply collapses on it.

Movement on a thread. Since the thread only stretches, when a body moves on the thread in a vertical plane, only tensile deformation will occur in the thread and, as a consequence, the elastic force arising in the thread will always be directed towards the center of the circle.

Movement of the body on the rod. The rod, unlike the thread, can be compressed. Therefore, at the top point of the trajectory, the speed of the body attached to the rod can be equal to zero, in contrast to the thread, where the speed must be no less than a certain value so that the thread does not fold. The elastic forces arising in the rod can be directed both towards the center of the circle and in the opposite direction.

Turning the car. If a body moves along a solid horizontal surface in a circle (for example, a car is going through a turn), then the force that keeps the body on the trajectory will be the friction force. In this case, the friction force is directed towards the turn, and not against it (the most common mistake), it helps the car turn. For example, when a car turns right, the friction force is directed in the direction of the turn (to the right).

The law of universal gravitation. Satellites

All bodies attract each other with forces directly proportional to their masses and inversely proportional to the square of the distance between them. Thus law of universal gravitation in formula form looks like this:

This recording of the law of universal gravitation is valid for material points, balls, spheres, for which r measured between centers. Proportionality factor G is the same for all bodies in nature. They call him gravitational constant. In the SI system it is equal to:

One of the manifestations of the force of universal gravity is the force of gravity. This is the common name for the force of attraction of bodies towards the Earth or another planet. If M– mass of the planet, R n is its radius, then acceleration of free fall at the surface of the planet:

If you move away from the surface of the Earth at some distance h, then the acceleration of free fall at this height will become equal (with the help of simple transformations you can also obtain the relationship between the acceleration of free fall on the surface of the planet and the acceleration of free fall at a certain height above the surface of the planet):

Let us now consider the question of artificial satellites of planets. Artificial satellites move outside the atmosphere (if the planet has one), and they are affected only by gravitational forces from the planet. Depending on the initial speed, the trajectory of a cosmic body can be different. We will consider here only the case of an artificial satellite moving in a circular orbit at almost zero altitude above the planet. The orbital radius of such satellites (the distance between the center of the planet and the point where the satellite is located) can be approximately taken equal to the radius of the planet R n. Then the centripetal acceleration of the satellite imparted to it by gravitational forces is approximately equal to the acceleration of gravity g. The speed of a satellite in orbit near the surface (at zero altitude above the planet's surface) is called first escape velocity. The first escape velocity is found by the formula:

The motion of a satellite can be thought of as a free fall, similar to the motion of projectiles or ballistic missiles. The only difference is that the speed of the satellite is so high that the radius of curvature of its trajectory is equal to the radius of the planet. For satellites moving along circular trajectories at a considerable distance from the planet, gravitational attraction weakens in inverse proportion to the square of the radius r trajectories. The speed of the satellite in this case is found using the formula:

Kepler's law for the periods of revolution of two bodies rotating around one attractive center:

If we are talking about planet Earth, then it is easy to calculate that with a radius r educational materials on this site. To do this, you need nothing at all, namely: devote three to four hours every day to preparing for the CT in physics and mathematics, studying theory and solving problems. The fact is that the CT is an exam where it is not enough just to know physics or mathematics, you also need to be able to quickly and without failures solve a large number of problems on different topics and of varying complexity. The latter can only be learned by solving thousands of problems.

Successful, diligent and responsible implementation of these three points will allow you to show an excellent result at the CT, the maximum of what you are capable of.

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