What is the difference between move and path. Moving. Give the definition of an absolutely rigid body
At first glance, movement and path are similar concepts. However, in physics, there are key differences between displacement and path, although both concepts are associated with a change in the position of a body in space and are often (usually in rectilinear motion) numerically equal to each other.
To understand the differences between displacement and path, let us first give them the definitions that physics gives them.
Body movement - this is directed line segment (vector)whose beginning coincides with the starting position of the body, and the end coincides with the ending position of the body.
Body path - this is distancethat the body has passed in a certain period of time.
Let's imagine that you have become at your entrance to a certain point. We walked around the house and returned to the starting point. So: your movement will be equal to zero, and the path will not. The path will be equal to the length of the curve (for example, 150 m), along which you walked around the house.
However, back to the coordinate system. Let a point body move rectilinearly from point A with coordinate x 0 \u003d 0 m to point B with coordinate x 1 \u003d 10 m. The body movement in this case will be 10 m.Since the movement was rectilinear, then 10 meters will be equal to the done body way.
If the body moved rectilinearly from the initial (A) point with the coordinate x 0 \u003d 5 m, to the end point (B) with the coordinate x 1 \u003d 0, then its displacement will be -5 m, and the path 5 m.
The displacement is found as a difference, where the initial coordinate is subtracted from the final coordinate. If the end coordinate is less than the start coordinate, that is, the body moved in the opposite direction with respect to the positive direction of the X axis, then the displacement will be negative.
Since displacement can have both positive and negative values, displacement is a vector quantity. In contrast, the path is always a positive or zero value (the path is a scalar), since the distance cannot be negative in principle.
Let's take another example. The body moved rectilinearly from point A (x 0 \u003d 2 m) to point B (x 1 \u003d 8 m), then also rectilinearly moved from B to point C with coordinate x 2 \u003d 5 m.What are the common paths (A → B → C) done by this body and its total displacement?
Initially, the body was at a point with a coordinate of 2 m, at the end of its movement it turned out to be at a point with a coordinate of 5 m. Thus, the movement of the body was 5 - 2 \u003d 3 (m). It is also possible to calculate the total displacement as the sum of two displacements (vectors). The movement from A to B was 8 - 2 \u003d 6 (m). Moving from point B to C was 5 - 8 \u003d -3 (m). Adding both displacements, we get 6 + (-3) \u003d 3 (m).
The total path is calculated by adding the two distances traveled by the body. The distance from point A to B is 6 m, and from B to C the body has traveled 3 m.Total we get 9 m.
Thus, in this problem, the path and movement of the body differ from each other.
The problem considered is not entirely correct, since it is necessary to indicate the moments of time at which the body is at certain points. If x 0 corresponds to the moment of time t 0 \u003d 0 (the moment of the beginning of observations), then let, for example, x 1 correspond to t 1 \u003d 3 s, and x 2 corresponds to t 2 \u003d 5 s. That is, the time interval between t 0 and t 1 is 3 s, and between t 0 and t 2 is 5 s. In this case, it turns out that the body's path in a time interval of 3 seconds was 6 meters, and in an interval of 5 seconds - 9 meters.
Time appears in the definition of the path. In contrast, time does not really matter for moving.
Mechanics.
weight (kg)
Electric charge (C)
Trajectory
Distance traveledor just a path ( l) -
Moving- it is vectorS
Give a definition and indicate the unit of measure for speed.
Speed- vector physical quantity characterizing the speed of movement of a point and the direction of this movement. [V] \u003d m · s
Define and specify the unit for acceleration.
Acceleration- vector physical quantity characterizing the speed of the change in the modulus and direction of the speed and equal to the increment of the speed vector per unit time:
Define and specify the unit of measure for the radius of curvature.
Radius of curvature is a scalar physical quantity inverse to the curvature C at a given point of the curve and equal to the radius of the circle tangent to the trajectory at this point. The center of such a circle is called the center of curvature for a given point of the curve. The radius of curvature is determined: R \u003d С -1 \u003d, 
[R] \u003d 1m / rad.
Define and specify the unit of measure for curvature
Trajectories.
Trajectory curvature - physical quantity equal to 
, where is the angle between tangents drawn at 2 points of the trajectory; is the length of the trajectory between these points. Than< , тем кривизна меньше. В окружности 2 пи радиант = .
Give a definition and indicate the unit of measurement for the angular velocity.
Angular velocity- vector physical quantity characterizing the rate of change of the angular position and equal to the angle of rotation per unit. time: 
... [w] \u003d 1 rad / s \u003d 1s -1
Define and indicate the unit of measure for the period.
Period(T) is a scalar physical quantity equal to the time of one complete revolution of a body around its axis or the time of a complete revolution of a point along a circle. where N is the number of revolutions in a time equal to t. [T] \u003d 1c.
Define and indicate the unit of measurement for frequency.
Call frequency- scalar physical quantity equal to the number of revolutions per unit of time:. \u003d 1 / s.
Give a definition and indicate the unit of measurement for body impulse (momentum).
Pulse - vector physical quantity equal to the product of mass by the velocity vector. ... [p] \u003d kg · m / s.
Define and indicate the unit of measurement for the impulse of force.
Impulse of force - vector physical quantity equal to the product of the force by the time of its action. [N] \u003d H · s. 
Define and indicate the unit of measure for the work.
Work force- a scalar physical quantity characterizing the action of a force and equal to the scalar product of the force vector by the displacement vector: where is the projection of the force on the direction of displacement, is the angle between the directions of force and displacement (velocity). [A] \u003d \u003d 1N · m.
Give a definition and indicate the unit of measure for power.
Power- scalar physical quantity characterizing the speed of performing work and equal to the work produced per unit of time:. [N] \u003d 1 W \u003d 1J / 1s.
Define potential forces.
Potentialor conservative forces - forces, the work of which when moving the body does not depend on the trajectory of the body and is determined only by the initial and final positions of the body.
Give a definition of dissipative (non-potential) forces.
Non-potential forces are forces, under the action of which on a mechanical system its total mechanical energy decreases, passing into other non-mechanical forms of energy.
Give definition of shoulder strength.
Shoulder of strengthcalled distance between an axis and a line along which the force acts(distance xmeasured along the O axis xperpendicular to the given axis and force).
Give the definition of the moment of force relative to the point.
Moment of force relative to some point O- vector physical quantity equal to the vector product of the radius vector drawn from a given point O to the point of application of the force and the force vector.M \u003d r * F \u003d. [M] SI \u003d 1N · m \u003d 1kg · m 2 / s 2
Give the definition of an absolutely rigid body.
Absolutely solid- a body whose deformations can be neglected.
Conservation of momentum.
Impulse conservation law:the momentum of a closed system of bodies is a constant value. 
Mechanics.
1. Specify the unit of measurement for concepts: force (1 N \u003d 1 kg · m / s 2)
weight (kg)
Electric charge (C)
Give a definition of the concepts: movement, path, trajectory.
Trajectory- an imaginary line along which the body moves
Distance traveledor just a path ( l) -the length of the path along which the body moved
Moving- it is vectorSfrom the start point to the end point
The position of a material point is determined in relation to some other, arbitrarily chosen body, called reference body... Contacts him frame of reference - a set of coordinate systems and clocks associated with the reference body.
In the Cartesian coordinate system, the position of point A at a given time in relation to this system is characterized by three coordinates x, y and z or a radius vector r – a vector drawn from the origin of the coordinate system to a given point. When a material point moves, its coordinates change over time. r=r(t) or x \u003d x (t), y \u003d y (t), z \u003d z (t) - material point kinematic equations.
The main task of mechanics- knowing the state of the system at some initial moment of time t 0, as well as the laws governing the motion, determine the state of the system at all subsequent moments of time t.
Trajectory movement of a material point - a line described by this point in space. Depending on the shape of the trajectory, there are straightforward and curvilinear point movement. If the trajectory of a point is a flat curve, i.e. lies entirely in one plane, then the movement of the point is called flat.
The length of the AB trajectory segment traversed by a material point from the moment the time starts is called long path Δs and is a scalar function of time: Δs \u003d Δs (t). Unit of measurement - meter(m) - the length of the path traveled by light in a vacuum for 1/299792458 s.
IV. Vector way to define motion
Radius vector r – a vector drawn from the origin of the coordinate system to a given point. Vector Δ r=r-r 0 drawn from the initial position of the moving point to its position at a given time is called displacement (increment of the radius vector of the point for the considered period of time).
Average speed vector< v> is called the increment relation Δ r the radius vector of the point to the time interval Δt: (1). The direction of the average speed coincides with the direction of Δ r. With an unlimited decrease in Δt, the average velocity tends to the limiting value, which is called instant speedv... Instantaneous speed is the speed of the body at a given moment in time and at a given point of the trajectory: (2). Instant speed v is a vector quantity equal to the first time derivative of the radius vector of the moving point.
To characterize the rate of change of speed vpoint in mechanics, a vector physical quantity is introduced, called acceleration.
Average acceleration of uneven motion in the interval from t to t + Δt is a vector quantity equal to the ratio of the change in speed Δ v to the time interval Δt:
Instant acceleration a material point at time t will be the limit of the average acceleration: (4). Acceleration and is a vector quantity equal to the first derivative of the velocity with respect to time.
V. Coordinate way of specifying movement
The position of the point M can be characterized by the radius - vector r or three coordinates x, y and z: М (x, y, z). Radius - vector can be represented as the sum of three vectors directed along the coordinate axes: (5).
From the definition of speed 
(6). Comparing (5) and (6) we have: (7). Taking into account (7), formula (6) can be written (8). The speed module can be found: (9).
Similarly for the acceleration vector:
(10),
(11),
The natural way to define motion (describing motion using trajectory parameters)
The movement is described by the formula s \u003d s (t). Each point of the trajectory is characterized by its own value s. Radius - vector is a function of s and the trajectory can be given by the equation r=r(s). Then r=r(t) can be represented as a complex function r... Let us differentiate (14). The quantity Δs is the distance between two points along the trajectory, | Δ r| - the distance between them in a straight line. As the points get closer, the difference decreases. where τ
Is the unit vector tangent to the trajectory. , then (13) has the form v=τ
v (15). Therefore, the speed is directed tangentially to the trajectory. 
Acceleration can be directed at any angle to the tangent to the motion path. From the definition of acceleration 
(sixteen). If a τ
is the tangent to the trajectory, then is the vector perpendicular to this tangent, i.e. directed along the normal. The unit vector, in the normal direction is denoted n... The vector value is 1 / R, where R is the radius of curvature of the trajectory.
A point at a distance from the path and R in the normal direction n, is called the center of curvature of the trajectory. Then (17). Considering the above, formula (16) can be written: 
(18).
The total acceleration consists of two mutually perpendicular vectors: directed along the trajectory of motion and called tangential, and acceleration directed perpendicular to the trajectory along the normal, i.e. to the center of curvature of the trajectory and called normal.
We find the absolute value of full acceleration: 
(19).
Lecture 2 The movement of a material point in a circle. Angular displacement, angular velocity, angular acceleration. Relationship between linear and angular kinematic quantities. Vectors of angular velocity and acceleration.
Lecture plan
Rotational Kinematics
During rotational motion, the vector is used as a measure of the displacement of the entire body in a small time interval dt dφ elementary body rotation. Elementary turns
(denoted by or) can be considered as pseudovectors (sort of).
Angular movement is a vector quantity, the modulus of which is equal to the angle of rotation, and the direction coincides with the direction of translational motion right screw (directed along the axis of rotation so that when viewed from its end, the rotation of the body appears to be counterclockwise). The unit of angular movement is rad.
The rate of change in angular displacement over time is characterized by angular velocity
ω
... The angular velocity of a rigid body is a vector physical quantity that characterizes the rate of change in the angular displacement of a body over time and is equal to the angular displacement performed by the body per unit of time: 
Directed vector ω along the axis of rotation in the same direction as dφ (according to the rule of the right screw). The unit of angular velocity is rad / s
The rate of change in angular velocity over time is characterized by angular acceleration ε
(2).
The vector ε is directed along the axis of rotation in the same direction as dω, i.e. with accelerated rotation, with slow rotation.
The unit of angular acceleration is rad / s 2.
During dt an arbitrary point of a rigid body A move to drgoing the way ds... The figure shows that dr equal to the vector product of the angular displacement dφ by radius - point vector r : dr =[ dφ · r ] (3).
Point Linear Velocityis related to the angular velocity and radius of the trajectory by the ratio: 
In vector form, the formula for the linear velocity can be written as vector product: (4)
By the definition of a vector product its modulus is, where is the angle between the vectors and, and the direction coincides with the direction of the translational motion of the right screw when it rotates from to
Let's differentiate (4) by time:
Considering that - linear acceleration, - angular acceleration, and - linear velocity, we get:
The first vector on the right is tangential to the path of the point. It characterizes the change in the linear velocity module. Therefore, this vector is the tangential acceleration of the point: a τ =[ ε · r ] (7). The modulus of tangential acceleration is a τ = ε · r... The second vector in (6) is directed to the center of the circle and characterizes the change in the direction of the linear velocity. This vector is the normal acceleration of the point: a n =[ ω · v ] (8). Its modulus is equal to a n \u003d ω v or taking into account that v = ω· r, a n = ω 2 · r = v 2 / r (9).
Special cases of rotational motion
With uniform rotation: , hence .
Uniform rotation can be characterized by rotation period T- the time during which the point makes one complete revolution,
Rotation frequency - the number of complete revolutions made by the body during its uniform movement around the circumference, per unit of time: (11)
Unit of speed - hertz (Hz).
With uniformly accelerated rotational motion :
Lecture 3 Newton's first law. Power. The principle of independence of the acting forces. Resultant strength. Weight. Newton's second law. Pulse. Impulse conservation law. Newton's third law. Moment of momentum of a material point, moment of force, moment of inertia.
Lecture plan
Newton's first law
Newton's second law
Newton's third law
Moment of momentum of a material point, moment of force, moment of inertia
Newton's first law. Weight. Power
Newton's first law: There are such frames of reference relative to which the bodies move rectilinearly and uniformly or at rest if they are not acted upon by forces or the action of forces is compensated.
Newton's first law is fulfilled only in an inertial frame of reference and asserts the existence of an inertial frame of reference.
Inertia - this is the property of bodies to strive to keep their speed unchanged.
Inertia is called the property of bodies to prevent a change in speed under the action of an applied force.
Body mass Is a physical quantity that is a quantitative measure of inertia, it is a scalar additive quantity. Mass Additivityconsists in the fact that the mass of a system of bodies is always equal to the sum of the masses of each body separately. Weight- the basic unit of the "SI" system.
One of the forms of interaction is mechanical interaction... Mechanical interaction causes deformation of bodies, as well as a change in their speed.
PowerIs a vector quantity that is a measure of the mechanical effect on the body from other bodies, or fields, as a result of which the body acquires acceleration or changes its shape and size (deforms) Force is characterized by modulus, direction of action, point of application to the body.
Trajectory - the curve (or line) that the body describes when moving. We can speak about the trajectory only when the body is represented as a material point.
The trajectory of movement can be:
It is worth noting that if, for example, a fox runs randomly in one area, then this trajectory will be considered invisible, since it will not be clear there exactly how it moved.
The trajectory of movement in different frames of reference will be different. You can read about it here.
Way
Way is a physical quantity that shows the distance traveled by the body along the path of motion. It is denoted by L (in rare cases S).
The path is a relative value and its value depends on the selected frame of reference.
This can be seen on simple example: there is a passenger on the plane moving from tail to nose. So, its path in the reference frame associated with the aircraft will be equal to the length of this passage L1 (from tail to nose), but in the reference frame related to the Earth, the path will be equal to the sum of the lengths of the aircraft passage (L1) and the path (L2) , which the plane made relative to the Earth. Therefore, in this case, the entire path will be expressed like this:
Moving
Moving is a vector that connects the starting position of the moving point with its final position over a certain period of time.
It is designated by S. The unit of measurement is 1 meter.
When moving straight in one direction, it coincides with the trajectory and the distance traveled. In any other case, these values \u200b\u200bdo not match.
This can be easily seen with a simple example. There is a girl, and in her hands is a doll. She throws it up, and the doll travels a distance of 2 m and stops for a moment, and then starts moving down. In this case, the path will be equal to 4 m, but the displacement is 0. In this case, the doll traveled 4 m, since at first it moved up 2 m, and then the same amount down. No movement occurred in this case, since the start and end points are the same.
Section 1 MECHANICS
Chapter 1: The Basics
Mechanical movement. Trajectory. Path and movement. Speed \u200b\u200baddition
Mechanical body movementis called the change in its position in space relative to other bodies over time.
The mechanical movement of bodies studies mechanics. The section of mechanics that describes the geometric properties of motion without taking into account the masses of bodies and acting forces is called kinematics .
Mechanical movement is relative. To determine the position of a body in space, you need to know its coordinates. To determine the coordinates of a material point, first of all, select a reference body and associate a coordinate system with it.
Reference bodya body is called, relative to which the position of other bodies is determined. The reference body is chosen arbitrarily. It can be anything: Land, building, car, ship, etc.
The coordinate system, the reference body with which it is associated, and the indication of the time reference form frame of reference , relative to which the movement of the body is considered (Figure 1.1).
A body whose size, shape and structure can be neglected in the study of a given mechanical movement is called material point . A material point can be considered a body whose dimensions are much smaller than the distances characteristic of the motion considered in the problem.
Trajectory this is the line along which the body moves.
Depending on the type of trajectory, movements are divided into straight and curved
WayIs the length of the trajectory ℓ (m) ((See Figure 1.2)
The vector drawn from the initial position of the particle to its final position is called displacement this particle for a given time.
Unlike a path, displacement is not a scalar, but a vector quantity, since it shows not only how far, but also in which direction the body has moved during a given time.
Displacement vector modulus (that is, the length of the segment that connects the start and end points of the movement) can be equal to the distance traveled or less than the distance traveled. But the module of movement can never be larger than the distance traveled. For example, if the car moves from point A to point B along a curved path, then the modulus of the displacement vector is less than the distance traveled ℓ. The path and modulus of movement are equal only in one single case, when the body moves in a straight line.
Speed Is a vector quantitative characteristic of body movement
average speed Is a physical quantity equal to the ratio of the point displacement vector to the time interval
The direction of the average velocity vector coincides with the direction of the displacement vector.
Instant speed, that is, the speed at a given moment of time is a vector physical quantity equal to the limit to which the average speed tends with an infinite decrease in the time interval Δt.
The instantaneous velocity vector is directed tangentially to the trajectory of motion (Fig. 1.3).
In the SI system, speed is measured in meters per second (m / s), that is, the unit of speed is considered to be the speed of such a uniform rectilinear motion, in which a body travels a path of one meter in one second. Speed \u200b\u200bis often measured in kilometers per hour.
or 1 
Speed \u200b\u200baddition
Any mechanical phenomena are considered in any frame of reference: motion makes sense only relative to other bodies. When analyzing the motion of the same body in different frames of reference, all the kinematic characteristics of the motion (path, trajectory, displacement, speed, acceleration) are different.
For example, a passenger train travels by rail at a speed of 60 km / h. A person walks along the carriage of this train at a speed of 5 km / h. If we consider the railway stationary and take it as a reference system, then the speed of a person relative to the railway will be equal to the addition of the speeds of a train and a person, that is
60 km / h + 5 km / h \u003d 65 km / h if a person goes in the same direction as the train and
60 km / h - 5 km / h \u003d 55 km / h if a person walks against the direction of the train.
However, this is true only in this case, if the person and the train move along the same line. If a person moves at an angle, then this angle must be taken into account, and the fact that speed is a vector quantity.
Let's consider the example described above in more detail - with details and pictures.
So, in our case, the railway is a fixed frame of reference. The train that moves along this road is a moving frame of reference. The carriage on which the person is walking is part of the train. The speed of a person relative to the car (relative to the moving frame of reference) is 5 km / h. Let's designate it with a letter. The speed of the train (and hence the car) relative to a stationary frame of reference (that is, relative to the railroad) is 60 km / h. Let's designate it with a letter. In other words, the speed of a train is the speed of a moving frame of reference relative to a stationary frame of reference.
The speed of a person relative to the railroad (relative to a stationary frame of reference) is still unknown to us. Let's designate it with a letter.
Let's connect with the stationary frame of reference (Fig.1.4) the coordinate system XOY, and with the moving frame of reference - X p O p Y p. Now let's determine the speed of a person relative to the stationary frame of reference, that is, relative to the railway.
For a short time interval Δt, the following events occur:
A person moves a distance relative to the carriage
The carriage moves relative to the railway at a distance
Then, during this period of time, the movement of a person relative to the railway:
it law of addition of displacements ... In our example, the movement of a person relative to the railroad is equal to the sum of the movements of a person relative to the car and the car relative to the railroad.
Dividing both sides of the equality by a small time interval Dt, during which the movement occurred:
We get:
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