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A trajectory is a continuous line along which a material point moves in a given reference system. Depending on the shape of the trajectory, rectilinear and curvilinear motion of a material point is distinguished.
Latin Trajectorius - related to movement
Path is the length of a section of the trajectory of a material point traversed by it in a certain time.

The distance traveled is the length of the trajectory section from the start to the end point of movement.

Movement (in kinematics) is a change in the location of a physical body in space relative to the selected reference system. The vector characterizing this change is also called displacement. It has the property of additivity. The length of the segment is the displacement module, measured in meters (SI).

You can define movement as a change in the radius vector of a point: .

The displacement module coincides with the distance traveled if and only if the direction of velocity does not change during movement. In this case, the trajectory will be a straight line segment. In any other case, for example, with curvilinear motion, it follows from the triangle inequality that the path is strictly longer.

The instantaneous speed of a point is defined as the limit of the ratio of movement to the small period of time during which it was accomplished. More strictly:

Average ground speed. Average speed vector. Instant speed.

Average ground speed

Average (ground) speed is the ratio of the length of the path traveled by a body to the time during which this path was covered:

Average ground speed, unlike instantaneous speed, is not a vector quantity.

The average speed is equal to the arithmetic mean of the speeds of the body during movement only in the case when the body moved at these speeds for the same periods of time.

At the same time, if, for example, the car moved half the way at a speed of 180 km/h, and the second half at a speed of 20 km/h, then the average speed will be 36 km/h. In examples like this, the average speed is equal to the harmonic mean of all speeds on individual, equal sections of the path.

Average speed is the ratio of the length of a section of a path to the period of time during which this path is covered.

Average body speed

With uniformly accelerated motion

With uniform movement

Here we used:

Average body speed

Initial speed of the body

Body acceleration

Body movement time

The speed of a body after a certain period of time

Instantaneous speed is the first derivative of the path with respect to time =
v=(ds/dt)=s"
where the symbols d/dt or the dash at the top right of a function indicate the derivative of this function.
Otherwise, this is the speed v = s/t as t tends to zero... :)
In the absence of acceleration at the moment of measurement, the instantaneous value is equal to the average during the period of movement without acceleration Vmg. = Vavg. =S/t for this period.

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 a reference body.

In the Cartesian coordinate system, the position of point A at a given time relative 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=x(t), y=y(t), z=z(t) – kinematic equations of a material point.

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 movement, 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 rectilinear And curvilinear point movement. If the trajectory of a point is a flat curve, i.e. lies entirely in one plane, then the motion of the point is called flat.

The length of the section of the trajectory AB traversed by the material point since the start of time is called path lengthΔs is a scalar function of time: Δs=Δs(t). Unit - meter(m) – the length of the path traveled by light in a vacuum in 1/299792458 s.

IV. Vector method of specifying movement

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 a moving point to its position at a given time is called moving(increment of the radius vector of a point over the considered period of time).

Average speed vector< v> called the increment ratio Δ r radius vector of a point to the time interval Δt: (1). The direction of the average speed coincides with the direction Δ r.With an unlimited decrease in Δt, the average speed tends to a limiting value, which is called instantaneous speedv. Instantaneous speed is the speed of a body at a given moment of time and at a given point of the trajectory: (2). Instantaneous speed v is a vector quantity equal to the first derivative of the radius vector of a moving point with respect to time.

To characterize the speed of change of speed v points in mechanics, a vector physical quantity called acceleration.

Medium acceleration uneven motion in the interval from t to t+Δt is called a vector quantity equal to the ratio of the change in speed Δ v to the time interval Δt:

Instantaneous acceleration a material point at time t will be the limit of average acceleration: (4). Acceleration A is a vector quantity equal to the first derivative of speed with respect to time.

V. Coordinate method of specifying movement

The position of point M can be characterized by the radius vector r or three coordinates x, y and z: M(x,y,z). The 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),

A natural way to define movement (describing movement using trajectory parameters)

The movement is described by the formula s=s(t). Each point of the trajectory is characterized by its value s. The 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's differentiate (14). Value Δs – 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 – 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 trajectory of motion. From the definition of acceleration (16). If τ is tangent to the trajectory, then is a vector perpendicular to this tangent, i.e. directed normally. Unit vector, in the normal direction is denoted n. The value of the vector is 1/R, where R is the radius of curvature of the trajectory.

A point located at a distance from the path and R in the direction of the normal n, is called the center of curvature of the trajectory. Then (17). Taking into account 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 the total acceleration: (19).

Lecture 2 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 outline

Kinematics of rotational motion

In rotational motion, the measure of displacement of the entire body over a short period of time dt is the vector dφ elementary body rotation. Elementary turns (denoted by or) can be considered as pseudovectors (as if).

Angular movement - a vector quantity whose magnitude 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 occurring counterclockwise). The unit of angular displacement 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 time:

Directed vector ω along the axis of rotation in the same direction as dφ (according to the right screw rule). Unit of angular velocity - 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 dr, having walked the path ds. From the figure it is clear that dr equal to the vector product of the angular displacement dφ to radius – point vector r : dr =[ dφ · r ] (3).

Linear speed of a point is related to the angular velocity and radius of the trajectory by the relation:

In vector form, the formula for linear speed can be written as vector product: (4)

By definition of the vector product its module is equal to , where is the angle between the vectors and, and the direction coincides with the direction of translational motion of the right propeller as it rotates from to.

Let's differentiate (4) with respect to time:

Considering that - linear acceleration, - angular acceleration, and - linear velocity, we obtain:

The first vector on the right side is directed tangent to the trajectory of the point. It characterizes the change in linear velocity modulus. Therefore, this vector is the tangential acceleration of the point: a τ =[ ε · r ] (7). The tangential acceleration module is equal to a τ = ε · r. The second vector in (6) is directed towards the center of the circle and characterizes the change in the direction of linear velocity. This vector is the normal acceleration of the point: a n =[ ω · v ] (8). Its modulus is equal to a n =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 rotation period T- the time it takes for a point to complete one full revolution,

Rotation frequency - the number of full revolutions made by a body during its uniform motion in a circle, per unit of time: (11)

Speed ​​unit - hertz (Hz).

With uniformly accelerated rotational motion :

Lecture 3 Newton's first law. Force. The principle of independence of acting forces. Resultant force. Weight. Newton's second law. Pulse. Law of conservation of momentum. Newton's third law. Moment of impulse of a material point, moment of force, moment of inertia.

Lecture outline

Newton's first law

Newton's second law

Newton's third law

Moment of impulse of a material point, moment of force, moment of inertia

Newton's first law. Weight. Force

Newton's first law: There are reference systems relative to which bodies move rectilinearly and uniformly or are at rest if no forces act on them or the action of the forces is compensated.

Newton's first law is satisfied 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 constant.

Inertia call the property of bodies to prevent a change in speed under the influence of an applied force.

Body mass– this is a physical quantity that is a quantitative measure of inertia, it is a scalar additive quantity. Additivity of mass is 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 form of interaction is mechanical interaction. Mechanical interaction causes deformation of bodies, as well as a change in their speed.

Force– this is a vector quantity that is a measure of the mechanical impact 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 its modulus, direction of action, and point of application to the body.

Trajectory- a curve (or line) that a body describes when moving. We can talk about a 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 how exactly it moved.

The trajectory of movement in different reference systems will be different. You can read about this here.

Path

Path is a physical quantity that shows the distance traveled by a body along the trajectory of movement. Designated L (in rare cases S).

The path is a relative quantity, and its value depends on the chosen reference system.

This can be seen with a simple example: there is a passenger on an airplane who moves 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 associated with the Earth, the path will be equal to the sum of the lengths of the passage of the aircraft (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 a moving point with its final position over a certain period of time.

Denoted by S. 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 ​​do not coincide.

This is easy to see with a simple example. A girl is standing, and in her hands is a doll. She throws it up, and the doll goes a distance of 2 m and stops for a moment, and then begins to move down. In this case, the path will be equal to 4 m, but the displacement will be 0. The doll in this case covered a path of 4 m, since at first it moved up 2 m, and then down the same amount. In this case, no movement occurred, since the starting and ending points are the same.

Class: 9

Lesson objectives:

• Educational:
  – introduce the concepts of “movement”, “path”, “trajectory”.
• Developmental:
  – develop logical thinking, correct physical speech, and use appropriate terminology.
• Educational:
  – achieve high class activity, attention, and concentration of students.

Equipment:

• plastic bottle with a capacity of 0.33 liters with water and a scale;
• medical bottle with a capacity of 10 ml (or small test tube) with a scale.

Demonstrations: Determining displacement and distance traveled.

During the classes

1. Updating knowledge.

- Hello guys! Sit down! Today we will continue to study the topic “Laws of interaction and motion of bodies” and in the lesson we will get acquainted with three new concepts (terms) related to this topic. In the meantime, let's check your homework for this lesson.

2. Checking homework.

Before class, one student writes the solution to the following homework assignment on the board:

Two students are given cards with individual tasks that are completed during the oral test ex. 1 page 9 of the textbook.

1. Which coordinate system (one-dimensional, two-dimensional, three-dimensional) should be chosen to determine the position of bodies:

a) tractor in the field;
b) helicopter in the sky;
c) train
d) chess piece on the board.

2. Given the expression: S = υ 0 t + (a t 2) / 2, express: a, υ 0

1. Which coordinate system (one-dimensional, two-dimensional, three-dimensional) should be chosen to determine the position of such bodies:

a) chandelier in the room;
b) elevator;
c) submarine;
d) plane on the runway.

2. Given the expression: S = (υ 2 – υ 0 2) / 2 · a, express: υ 2, υ 0 2.

3. Study of new theoretical material.

Associated with changes in the coordinates of the body is the quantity introduced to describe the movement - MOVEMENT.

The displacement of a body (material point) is a vector connecting the initial position of the body with its subsequent position.

Movement is usually denoted by the letter . In SI, displacement is measured in meters (m).

– [m] – meter.

Displacement - magnitude vector, those. In addition to the numerical value, it also has a direction. The vector quantity is represented as segment, which begins at a certain point and ends with a point indicating the direction. Such an arrow segment is called vector.

Knowing the displacement vector means knowing its direction and magnitude. The modulus of a vector is a scalar, i.e. numerical value. Knowing the initial position and the vector of movement of the body, you can determine where the body is located.

In the process of movement, a material point occupies different positions in space relative to the chosen reference system. In this case, the moving point “describes” some line in space. Sometimes this line is visible - for example, a high-flying plane can leave a trail in the sky. A more familiar example is the mark of a piece of chalk on a blackboard.

An imaginary line in space along which a body moves is called TRAJECTORY body movements.

The trajectory of a body is a continuous line that is described by a moving body (considered as a material point) in relation to the selected reference system.

The movement in which all points body moving along the same trajectories, called progressive.

Very often the trajectory is an invisible line. Trajectory moving point can be straight or crooked line. According to the shape of the trajectory movement It happens straightforward And curvilinear.

The path length is PATH. The path is a scalar quantity and is denoted by the letter l. The path increases if the body moves. And remains unchanged if the body is at rest. Thus, the path cannot decrease over time.

The displacement module and the path can coincide in value only if the body moves along a straight line in the same direction.

What is the difference between a path and a movement? These two concepts are often confused, although in fact they are very different from each other. Let's look at these differences: ( Appendix 3) (distributed in the form of cards to each student)

1. The path is a scalar quantity and is characterized only by a numerical value.
2. Displacement is a vector quantity and is characterized by both a numerical value (module) and direction.
3. When a body moves, the path can only increase, and the displacement module can both increase and decrease.
4. If the body returns to the starting point, its displacement is zero, but the path is not zero.

4. Demonstration of experience (students perform independently in their places at their desks, the teacher, together with the students, performs a demonstration of this experience)

1. Fill a plastic bottle with a scale to the neck with water.
2. Fill the bottle with the scale with water to 1/5 of its volume.
3. Tilt the bottle so that the water comes up to the neck, but does not flow out of the bottle.
4. Quickly lower the bottle of water into the bottle (without closing it with the stopper) so that the neck of the bottle enters the water of the bottle. The bottle floats on the surface of the water in the bottle. Some of the water will spill out of the bottle.
5. Screw the bottle cap on.
6. Squeeze the sides of the bottle and lower the float to the bottom of the bottle.

1. By releasing the pressure on the walls of the bottle, make the float float to the surface. Determine the path and movement of the float:__________________________________________________________
2. Lower the float to the bottom of the bottle. Determine the path and movement of the float:________________________________________________________________________________
3. Make the float float and sink. What is the path and movement of the float in this case?_______________________________________________________________________________________

5. Exercises and questions for review.

1. Do we pay for the journey or transportation when traveling in a taxi? (Path)
2. The ball fell from a height of 3 m, bounced off the floor and was caught at a height of 1 m. Find the path and movement of the ball. (Path – 4 m, movement – ​​2 m.)

6. Lesson summary.

Review of lesson concepts:

– movement;
– trajectory;
- path.

7. Homework.

§ 2 of the textbook, questions after the paragraph, exercise 2 (p. 12) of the textbook, repeat the lesson experience at home.

Bibliography

1. Peryshkin A.V., Gutnik E.M.. Physics. 9th grade: textbook for general educational institutions - 9th ed., stereotype. – M.: Bustard, 2005.

Mechanics.

weight(kg)

Electric charge(C)

Trajectory

Distance traveled or just the path( l) -

Moving- this is a vectorS

Define and indicate the unit of measurement for speed.

Speed- vector physical quantity characterizing the speed of movement of a point and the direction of this movement. [V]=m s

Define and indicate the unit of measurement for acceleration.

Acceleration- vector physical quantity characterizing the speed of change in the magnitude and direction of velocity and equal to the increment of the velocity vector per unit time:

Define and indicate the unit of measurement for radius of curvature.

Radius of curvature- 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 on the curve. The radius of curvature is determined: R = C -1 = , [R]=1m/rad.

Define and indicate the unit of measurement of curvature

Trajectories.

Path curvature– physical quantity equal to , where is the angle between the tangents drawn at 2 points of the trajectory; - the length of the trajectory between these points. How< , тем кривизна меньше. В окружности 2 пи радиант = .

Define and indicate the unit of measurement for angular velocity.

Angular velocity- vector physical quantity characterizing the speed of change in angular position and equal to the angle of rotation per unit. time: . [w]= 1 rad/s=1s -1

Define and indicate the unit of measurement for the period.

Period(T) is a scalar physical quantity equal to the time of one full revolution of a body around its axis or the time of a full revolution of a point along a circle. where N is the number of revolutions in a time equal to t. [T]=1c.

Define and indicate the unit of frequency.

Frequency- scalar physical quantity equal to the number of revolutions per unit time: . =1/s.

Define and indicate the unit of measurement of body impulse (amount of motion).

Pulse– vector physical quantity equal to the product of mass and velocity vector. . [p]=kg m/s.

Define and indicate the unit of measurement for force impulse.

Impulse force– vector physical quantity equal to the product of force and the time of its action. [N]=N·s.

Define and indicate the unit of measurement for work.

Work of force- a scalar physical quantity characterizing the action of a force and equal to the scalar product of the force vector and the displacement vector: where is the projection of the force onto the direction of displacement, is the angle between the directions of force and displacement (velocity). [A]= =1N m.

Define and indicate the unit of measurement for power.

Power- a scalar physical quantity characterizing the speed of work and equal to the work done per unit of time: . [N]=1 W=1J/1s.

Define potential forces.

Potential or conservative forces - forces whose work when moving a body is independent of the trajectory of the body and is determined only by the initial and final positions of the body.

Define dissipative (non-potential) forces.

Non-potential forces are forces, when acting on a mechanical system, its total mechanical energy decreases, turning into other non-mechanical forms of energy.

Define leverage.

Shoulder of strength called distance between the axis and the straight line along which the force acts(distance x measured along the O axis x perpendicular to the given axis and force).

Define the moment of force about a point.

Moment of force about a certain point O- a 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= r * F= . [M] SI = 1 N m = 1 kg m 2 / s 2

Define an absolutely rigid body.

Absolutely solid body- a body whose deformations can be neglected.

Conservation of momentum.

Law of conservation of momentum:the momentum of a closed system of bodies is a constant quantity.

Mechanics.

1. Indicate the unit of measurement for the concepts: force (1 N = 1 kg m/s 2)

weight(kg)

Electric charge(C)

Define the concepts: movement, path, trajectory.

Trajectory- an imaginary line along which the body moves

Distance traveled or just the path( l) -length of the path along which the body moved

Moving- this is a vectorS, directed from the starting point to the ending point