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 in this moment time in relation to this system is characterized by three coordinates x, y and z or radius vector r – vector drawn from the origin of the coordinate system to this 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).

Vector average speed < 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.

You have already encountered the concept of a path many times. Let us now get acquainted with a new concept for you - moving, which is more informative and useful in physics than the concept of a path.

Let's say you need to transport cargo from point A to point B on the other side of the river. This can be done by car across the bridge, by boat on the river or by helicopter. In each of these cases, the path traveled by the load will be different, but the movement will be the same: from point A to point B.

By moving is a vector drawn from the initial position of a body to its final position. The displacement vector shows the distance the body has moved and the direction of movement. note that direction of movement and direction of movement are two different concepts. Let's explain this.

Consider, for example, the trajectory of a car from point A to the middle of the bridge. Let us designate the intermediate points as B1, B2, B3 (see figure). You see that on segment AB1 the car was traveling northeast (first blue arrow), on segment B1B2 - southeast (second blue arrow), and on segment B2B3 - north (third blue arrow). So, at the moment of passing the bridge (point B3), the direction of movement was characterized by the blue vector B2B3, and the direction of movement was characterized by the red vector AB3.

So, the movement of the body is vector quantity, that is, having a spatial direction and numeric value(module). Unlike movement, path is scalar quantity, that is, having only a numerical value (and no spatial direction). The path is indicated by the symbol l, movement is indicated by a symbol (important: with an arrow). Symbol s without an arrow indicate the displacement module. Note: the image of any vector in the drawing (in the form of an arrow) or its mention in the text (in the form of a word) makes the presence of an arrow above the designation optional.

Why did physics not limit itself to the concept of path, but introduced a more complex (vector) concept of displacement? Knowing the module and direction of movement, you can always say where the body will be (in relation to its initial position). Knowing the path, the position of the body cannot be determined. For example, knowing only that a tourist has walked 7 km, we cannot say anything about where he is now.

Task. While hiking on the plain, the tourist walked north 3 km, then turned east and walked another 4 km. How far was he from the starting point of the route? Draw its movement.

Solution 1 – using ruler and protractor measurements.

Displacement is a vector connecting the initial and final positions of the body. Let's draw it on checkered paper on a scale: 1 km - 1 cm (drawing on the right). Measuring the module of the constructed vector with a ruler, we get: 5 cm. According to the scale we have chosen, the module of the tourist’s movement is 5 km. But let's remember: to know a vector means to know its magnitude and direction. Therefore, using a protractor, we determine: the direction of movement of the tourist is 53° with the direction to the north (check it yourself).

Solution 2 – without using a ruler or protractor.

Since the angle between the tourist’s movements to the north and east is 90°, we apply the Pythagorean theorem and find the length of the hypotenuse, since it is also the modulus of the tourist’s movement:

As you can see, this value coincides with that obtained in the first solution. Now let’s determine the angle α between the displacement (hypotenuse) and the direction to the north (the adjacent leg of the triangle):

So, the problem was solved in two ways with matching answers.

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 verified at 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.

At straight motion in one direction coincides with the trajectory and the path 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.

Individual physical terms mixed with everyday ideas about the world look very similar. In the usual understanding, path and movement are the same thing, only one concept describes the process, and the second – the result. But if we turn to encyclopedic definitions, it becomes clear how serious the difference between them is.

Definition

Path is a movement that leads to a change in the location of an object in space. It is a scalar quantity that has no direction and denotes the total distance covered. The path can be carried out along a straight line, a curved path, in a circle or in another way.

Moving is a vector that denotes the difference between the initial and final location of a point in space after covering a certain path. A vector quantity is always positive and also has a definite direction. The path coincides with movement only if it is carried out rectilinearly and the direction does not change.

Comparison

Thus, the path is primary, movement is secondary. For the first quantity, the beginning of the movement matters; the second can do without it. The main difference between these concepts is that the path has no direction, but movement does. Hence other features characterizing the terms. Thus, the path length includes the entire distance traveled by an object in a certain time. Displacement is a vector quantity characterizing a relative change in space.

If an entrepreneur decides to go around four outlets, each of which is located at a distance of 10 kilometers from each other, and then return home, then his path will be 80 kilometers. However, the displacement will be equal to zero, since the position in space according to the results of the following has not changed. The path is always positive, since you can talk about it only after the movement has begun. For this value, what matters is the speed, which affects the total distance.

Conclusions website

1. Type. Path is a scalar quantity, displacement is a vector quantity.
2. Method of measurement. The path is calculated by the total distance traveled, the movement is calculated by the change in the location of the object in space.
3. Expression. The displacement may be equal to zero (if the movement was carried out along a closed path), but the path may not.

If we take into account physical processes in the domestic sphere, many of them seem very fine. Therefore, the concepts of path and movement are perceived as one and the same, the only difference is that the first is a description of the action, and the second is the result of the action. But if you turn to information sources for clarification, you can immediately find a significant difference between these operations.

What is the path?

A path is a movement that results in a change in the location of an object or person. This quantity is a scalar quantity, so it has no direction, but it can be used to determine the distance traveled.

The path can be executed in the following ways:

• In a straight line.
• Curvilinear.
• Round.
• Other methods are possible (for example, a zigzag trajectory).

The path can never be negative and decrease over time. The distance is measured in meters. Most often, in physics the letter is used to designate a path S, in rare cases, the letter L is used. Using a path, we cannot predict where the object we need will be at a certain point in time.

Features of movement

Displacement is the difference between the starting and ending points of the location of a person or object in space after some path has been covered.

The displacement value is always positive and also has a clear direction.

A coincidence between movement and path is possible only if the path was carried out in a straight line, and the direction did not change.

Using movement, you can calculate where a person or object was at a certain point in time.

The letter S is used to denote movement, but since movement is vector quantity then an arrow → is placed above this letter, which indicates that the movement is a vector. Unfortunately, adding to the confusion between path and movement is the fact that both concepts can also be denoted by the letter L.

What do the concepts path and movement have in common?

Despite the fact that path and movement are completely different concepts, there are certain elements that contribute to the concepts being confused:

1. Path and displacement can always only be positive quantities.
2. The same letter L can be used to indicate path and movement.

Even considering the fact that these concepts have only two common elements, their meaning is so great that it makes many people confused. Schoolchildren especially have problems when studying physics.

The main differences between the concepts of path and movement?

These concepts have a number of differences that will always help you determine what quantity is in front of you, path or movement:

1. Path is the primary concept, and movement is secondary. For example, movement determines the difference between the starting and ending points of a person’s location in space after covering a certain path. Accordingly, it is impossible to obtain the displacement value without using the path initially.
2. The beginning of the movement plays a huge role for the path, but the beginning of the movement is absolutely not necessary to determine the movement.
3. The main difference between these quantities is that the path has no direction, but movement does. For example, the path is carried out only straight forward, but movement also allows for backward movement.
4. In addition, the concepts differ in appearance. Path refers to a scalar quantity, and displacement refers to a vector quantity.
5. Calculus method. For example, a path is calculated using the total distance traveled, and displacement, in turn, is calculated using a change in the location of an object in space.
6. Path can never be zero, but movement is allowed to be zero.

Having studied these differences, you can immediately understand what the difference is between the concepts of path and movement, and never confuse them again.

Difference between path and movement with examples

In order to quickly understand the difference between path and movement, you can use certain examples:

1. The car moved 2 meters forward and 2 meters backward. The path is the sum of the total distance traveled, so it is 4 meters. And the displacement is the starting and ending point, so in this case it is equal to zero.
2. In addition, the difference between path and movement can be seen in own experience. You need to stand at the start of a 400-meter treadmill and run two laps (the second lap will end at the starting point). The result is that the path was 800 meters (400+400), and the displacement is 0, since the start and end points are the same.
3. The ball thrown upward reached a height of 15 meters and then fell to the Earth. In this case, the path will be 30 meters, since 15 meters up and 15 meters down are added. And the displacement will be equal to 0, due to the fact that the ball has returned to its original position.