Kinematic formulas 10. Kinematics. Basic formulas. Difficulty in describing movement

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 do well to have a collection of basic physics formulas handy. Dry formulas with explanation: short, concise, nothing more. A very useful thing when solving problems, you know. Yes, and in the exam, when exactly what was cruelly memorized the day before can “jump out” of my head, such a selection will serve you well.

Most of the tasks are usually given in the three most popular sections of physics. it Mechanics, thermodynamics and Molecular physics, electricity. Let's take them!

Basic formulas in physics dynamics, kinematics, statics

Let's start with the simplest. Good old favorite rectilinear and uniform movement.

Kinematic formulas:

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

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

Now we give 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 section of mechanics with formulas for vibrations and waves and move on to molecular physics and thermodynamics.

Efficiency, Gay-Lussac's law, the Clapeyron-Mendeleev equation - all these sweet formulas are collected below.

By the way! There is a discount for all our readers 10% on the any kind of work.

Basic formulas in physics: electricity

It's time to move on to electricity, although thermodynamics loves it less. Let's start with electrostatics.

And, to the drum roll, we finish with the formulas for Ohm's law, electromagnetic induction and electromagnetic oscillations.

That's all. Of course, a whole mountain of formulas could be given, but this is useless. When there are too many formulas, you can easily get confused, and then completely melt the brain. We hope that our cheat sheet of basic formulas in physics will help you solve your favorite problems faster and more efficiently. And if you want to clarify something or have not found the formula you need: ask the experts student service. Our authors keep hundreds of formulas in their heads and click tasks like nuts. Contact us, and soon any task will be "too tough" for you.

First of all, it should be noted that we are talking about a geometric point, that is, a region of space that has no dimensions. It is for this abstract image (model) that all the definitions and formulas presented below are valid. However, for the sake of brevity, I will often refer to the motion body, object or particles. I do this only to make it easier for you to read. But always remember that we are talking about a geometric point.

Radius vector points is a vector whose beginning coincides with the origin of the coordinate system and whose end coincides with the given point. The radius vector is usually denoted by the letter r. Unfortunately, some authors refer to it as s. Strongly advise do not use designation s for the radius vector. The fact is that the vast majority of authors (both domestic and foreign) use the letter s to denote a path, which is a scalar and, as a rule, has nothing to do with the radius vector. If you denote the radius vector as s you can easily get confused. Once again, we, like all normal people, will use the following notation: r is the radius vector of the point, s is the path traveled by the point.

Displacement vector(often just say - moving) - this is vector, the beginning of which coincides with the point of the trajectory where the body was when we began to study this movement, and the end of this vector coincides with the point of the trajectory where we finished this study. We will denote this vector as Δ r. The use of the symbol Δ is obvious: Δ r is the difference between the radius vector r the end point of the studied segment of the trajectory and the radius vector r 0 point of the beginning of this segment (Fig. 1), that is, Δ r= r − r 0 .

Trajectory is the line along which the body moves.

Path- this is the sum of the lengths of all sections of the trajectory successively traversed by the body during movement. It is denoted either ∆S, if we are talking about a section of the trajectory, or S, if we are talking about the entire trajectory of the observed movement. Sometimes (rarely) the path is also denoted by another letter, for example, L (just don't denote it as r, we already talked about this). Remember! The path is positive scalar! The path in the process of movement can only increase.

Average travel speed v Wed

v cf = ∆ r/Δt.

Instantaneous movement speed v is the vector defined by the expression

v=d r/dt.

Average travel speed v cp is the scalar defined by the expression

Vav = ∆s/∆t.

Other notations are often used, for example, .

Instantaneous travel speed v is the scalar defined by the expression

The modulus of the instantaneous speed of movement and the instantaneous speed of the path are the same, since dr = ds.

Average acceleration a

a cf = ∆ v/Δt.

Instant Boost(or simply, acceleration) a is the vector defined by the expression

a=d v/dt.

Tangential (tangential) acceleration aτ (the subscript is the Greek lowercase letter tau) is vector, which is vector projection instantaneous acceleration on the tangential axis.

Normal (centripetal) acceleration a n is vector, which is vector projection instantaneous acceleration on the normal axis .

Tangential acceleration modulus

| aτ | = dv/dt,

That is, it is the derivative of the module of instantaneous velocity with respect to time.

Normal acceleration module

| a n | = v 2 /r,

Where r is the value of the radius of curvature of the trajectory at the point where the body is located.

Important! I would like to draw your attention to the following. Do not get confused with the notation regarding tangential and normal accelerations! The fact is that in the literature on this subject there is traditionally a complete leapfrog.

Remember!

a t is vector tangential acceleration,

a n is vector normal acceleration.

aτ and a n are vector full acceleration projections a on the tangent axis and the normal axis, respectively,

A τ is the projection (scalar!) of the tangential acceleration onto the tangential axis,

A n is the projection (scalar!) of the normal acceleration onto the normal axis,

| aτ | is module vector tangential acceleration,

| a n | - this is module vector normal acceleration.

Especially do not be surprised if, reading in the literature about curvilinear (in particular, rotational) motion, you find that the author understands a τ as a vector, and its projection, and its modulus. The same applies to a n . Everything, as they say, "in one bottle." And, unfortunately, this is all too often the case. Even textbooks for higher education are no exception, in many of them (believe me - in most!) There is complete confusion about this.

So, without knowing the basics of vector algebra or neglecting them, it is very easy to get completely confused when studying and analyzing physical processes. Therefore, knowledge of vector algebra is the most important condition for success in the study of mechanics. And not just mechanics. In the future, when studying other branches of physics, you will repeatedly be convinced of this.

Instantaneous angular velocity(or simply, angular velocity) ω is the vector defined by the expression

ω =d φ /dt,

Where d φ - an infinitesimal change in the angular coordinate (d φ - vector!).

Instantaneous angular acceleration(or simply, angular acceleration) ε is the vector defined by the expression

ε =d ω /dt.

Connection between v, ω and r:

v = ω × r.

Connection between v, ω and r:

Connection between | aτ |, ε and r:

| aτ | = ε r.

Now let's move on to kinematic equations specific types of movement. These equations must be learned by heart.

Kinematic equation of uniform and rectilinear motion looks like:

r = r 0 + v t,

Where r is the radius vector of the object at time t, r 0 - the same at the initial time t 0 (at the start of observations).

Kinematic equation of motion with constant acceleration looks like:

r = r 0 + v 0 t + a t 2 /2, where v 0 the speed of the object at the moment t 0 .

The equation for the speed of a body when moving with constant acceleration looks like:

v = v 0 + a t.

Kinematic equation of uniform circular motion in polar coordinates looks like:

φ = φ 0 + ω z t,

Where φ is the angular coordinate of the body at a given time, φ 0 is the angular coordinate of the body at the time of the start of observation (at the initial time), ω z is the projection of the angular velocity ω on the Z-axis (usually this axis is chosen perpendicular to the plane of rotation).

Kinematic equation of circular motion with constant acceleration in polar coordinates looks like:

φ = φ 0 + ω 0z t + ε z t 2 /2.

Kinematic equation of harmonic vibrations along the X axis looks like:

X \u003d A Cos (ω t + φ 0),

Where A is the amplitude of the oscillations, ω is the cyclic frequency, φ 0 is the initial phase of the oscillations.

The projection of the velocity of a point oscillating along the X axis onto this axis is equal to:

V x = − ω A Sin (ω t + φ 0).

The projection of the acceleration of a point oscillating along the X axis onto this axis is equal to:

A x \u003d - ω 2 A Cos (ω t + φ 0).

Connection between the cyclic frequency ω, the ordinary frequency ƒ and the oscillation period T:

ω \u003d 2 πƒ \u003d 2 π / T (π \u003d 3.14 - the number of pi).

Mathematical pendulum has an oscillation period T, determined by the expression:

In the numerator of the radical expression is the length of the pendulum thread, in the denominator is the acceleration of free fall

Connection between absolute v abs, relative v rel and figurative v lane speeds:

v abs = v rel + v per.

Here, perhaps, are all the definitions and formulas that may be needed when solving problems in kinematics. The information provided is for reference only and cannot replace an e-book where the theory of this section of mechanics is presented in an accessible, detailed and, I hope, fascinating way.

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 do well to have a collection of basic physics formulas handy. Dry formulas with explanation: short, concise, nothing more. A very useful thing when solving problems, you know. Yes, and in the exam, when exactly what was cruelly memorized the day before can “jump out” of my head, such a selection will serve you well.

Most of the tasks are usually given in the three most popular sections of physics. it Mechanics, thermodynamics and Molecular physics, electricity. Let's take them!

Basic formulas in physics dynamics, kinematics, statics

Let's start with the simplest. Good old favorite rectilinear and uniform movement.

Kinematic formulas:

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

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

Now we give 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 section of mechanics with formulas for vibrations and waves and move on to molecular physics and thermodynamics.

Efficiency, Gay-Lussac's law, the Clapeyron-Mendeleev equation - all these sweet formulas are collected below.

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

Basic formulas in physics: electricity

It's time to move on to electricity, although thermodynamics loves it less. Let's start with electrostatics.

And, to the drum roll, we finish with the formulas for Ohm's law, electromagnetic induction and electromagnetic oscillations.

That's all. Of course, a whole mountain of formulas could be given, but this is useless. When there are too many formulas, you can easily get confused, and then completely melt the brain. We hope that our cheat sheet of basic formulas in physics will help you solve your favorite problems faster and more efficiently. And if you want to clarify something or have not found the formula you need: ask the experts student service. Our authors keep hundreds of formulas in their heads and click tasks like nuts. Contact us, and soon any task will be "too tough" for you.