What is the difference between path and distance. Moving definition. Define the moment of force about a point

With the help of this video tutorial, you can independently study the topic "Moving", which is included in school course physics for grade 9. From this lecture, students will be able to deepen their knowledge of movement. The teacher will recall the first characteristic of movement - the distance traveled, and then proceed to the definition of movement in physics.

The first characteristic of motion introduced by us earlier was the distance traveled. Recall that it is denoted by the letter S (sometimes the designation L is found) and is measured in SI in meters.

Distance traveled is a scalar quantity, i.e., a quantity that is characterized only numerical value. This means that we cannot predict where the body will be at the moment we need. We can only talk about the total distance traveled by the body (Fig. 1).

Rice. 1. Knowing only the distance traveled, it is impossible to determine the position of the body at an arbitrary point in time

To characterize the location of the body at an arbitrary moment, a quantity called displacement is introduced. Displacement is a vector quantity, that is, it is a quantity that is characterized not only by a numerical value, but also by a direction.

The movement is indicated in the same way as the distance traveled, by the letter S, but, unlike the distance traveled, an arrow is placed above the letter, thereby emphasizing that this is a vector quantity: .

What moving And distance traveled denoted by one letter is somewhat misleading, but we must clearly understand the difference between the path traveled and the movement. Note again that sometimes the path is denoted by L. This avoids confusion.

Definition

Displacement is a vector (directed line segment) that connects the starting point of the body's movement with its end point (Fig. 2).

Rice. 2. Displacement is a vector quantity

Recall that the passed path is the length of the path. This means that the path and displacement are completely different physical quantities, although sometimes there are situations when they coincide numerically.

Rice. 3. Path and displacement module are the same

On fig. 3, the simplest case is considered, when the body moves along a straight line (axis Oh). The body starts its movement from point 0 and gets to point A. In this case, we can say that the displacement modulus is equal to the distance traveled: .

An example of such a movement is an airplane flight (for example, from St. Petersburg to Moscow). If the movement was strictly rectilinear, then the displacement modulus will be equal to the distance traveled.

Rice. 4. The value of the path is greater than the displacement modulus

On fig. 4 the body moves along a curved line, i.e., the movement is curvilinear (from point A to point B). It can be seen from the figure that the displacement module (straight line) will be less than the path traveled, i.e. the length of the traveled path and the length of the displacement vector are not equal.

Rice. 5. Closed trajectory

On fig. 5 the body moves along a closed curve. It leaves point A and returns to the same point. The displacement modulus is , and distance traveled is the length of the entire curve, .

This case can be characterized by the following example. The student left the house in the morning, went to school, spent the whole day studying, besides this, he visited several other places (shop, gym, library) and returned home. Please note: as a result, the student ended up at home, which means that his displacement is 0 (Fig. 6).

Rice. 6. Student displacement is zero

When it comes to moving, it is important to remember that moving depends on the frame of reference in which the motion is considered.

Rice. 7. Determination of the modulus of displacement of the body

The body moves in a plane XOY. Point A is the initial position of the body. Her coordinates. The body moves to a point. The vector is the displacement of the body: .

You can calculate the displacement modulus as the hypotenuse of a right triangle using the Pythagorean theorem:. To find the displacement vector, it is necessary to find the angle between the axis Oh and the displacement vector.

We can choose the system arbitrarily, that is, direct the coordinate axes in the way that is convenient for us, the main thing is to consider the projections of all vectors in the future in the same chosen coordinate system.

Conclusion

In conclusion, it can be noted that we have become acquainted with an important quantity - displacement. Once again, note that the displacement and the path can only coincide in the case of rectilinear movement, without changing the direction of such movement.

Bibliography

Kikoin I.K., Kikoin A.K. Physics: textbook for grade 9 high school. - M.: Enlightenment.
Peryshkin A.V., Gutnik E.M., Physics. Grade 9: textbook for general education. institutions/A. V. Peryshkin, E. M. Gutnik. - 14th ed., stereotype. - M.: Bustard, 2009. - 300.
Sokolovich Yu.A., Bogdanova G.S.. Physics: Handbook with examples of problem solving. - 2nd edition redistribution. - X .: Vesta: Publishing house "Ranok", 2005. - 464 p.

Internet portal "vip8082p.vip8081p.beget.tech" ()
Internet portal "foxford.ru" ()

Homework

What is path and movement? What is the difference?
The motorcyclist pulled out of the garage and headed north. Drove 5 km, then turned west and drove also 5 km. How far from the garage will it be?
The minute hand has come full circle. Determine the displacement and distance traveled for the point that is at the end of the arrow (the radius of the clock is 10 cm).

« Physics - Grade 10 "

How do vector quantities differ from scalar quantities?

The line along which a point moves in space is called trajectory.

Depending on the shape of the trajectory, all movements of the point are divided into rectilinear and curvilinear.

If the path is a straight line, the movement of the point is called straightforward, and if the curve is curvilinear.

Let at some point in time the moving point occupies the position M 1 (Fig. 1.7, a). How to find its position after a certain period of time after this moment?

Suppose we know that the point is at a distance l relative to its initial position. Will we be able to uniquely determine the new position of the point in this case? Obviously not, since there are an infinite number of points that are at a distance l from the point M 1. To unambiguously determine the new position of the point, one must also know in which direction from the point M 1 a segment of length l should be laid.

Thus, if the position of a point at some point in time is known, then its new position can be found using a certain vector (Fig. 1.7, b).

The vector drawn from the initial position of a point to its final position is called displacement vector or simply moving a point

Since displacement is a vector quantity, the displacement shown in Figure (1.7, b) can be denoted

Let us show that with the vector method of specifying the motion, the displacement can be considered as a change in the radius vector of the moving point.

Let the radius vector 1 set the position of the point at time t 1 , and the radius vector 2 at time t 2 (Fig. 1.8). To find the change in the radius vector over a period of time Δt = t 2 - t 1, it is necessary to subtract the initial vector 1 from the final vector 2 . Figure 1.8 shows that the movement made by a point during the time interval Δt is a change in its radius vector during this time. Therefore, denoting the change in the radius vector through Δ , we can write: Δ = 1 - 2 .

Path s- the length of the trajectory when moving the point from position M 1 to position M 2.

The displacement modulus may not be equal to the path traveled by the point.

For example, in Figure 1.8, the length of the line connecting the points M 1 and M 2 is greater than the displacement modulus: s > |Δ|. The path is equal to the displacement only in the case of rectilinear unidirectional motion.

Body displacement Δ - vector, path s - scalar, |Δ| ≤ s.

Source: "Physics - Grade 10", 2014, textbook Myakishev, Bukhovtsev, Sotsky

Kinematics - Physics, textbook for grade 10 - Classroom physics

Physics and knowledge of the world --- What is mechanics ---

Class: 9

Lesson Objectives:

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

Equipment:

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

Demos: Determination of displacement and distance travelled.

During the classes

1. Actualization of 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, 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 performed during the oral test of exercise. 1 page 9 of the textbook.

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

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

2. An expression is given: S \u003d υ 0 t + (a t 2) / 2, express: a, υ 0

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

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

2. An expression is given: S \u003d (υ 2 - υ 0 2) / 2 a, express: υ 2, υ 0 2.

3. The study of new theoretical material.

The value introduced to describe the motion is associated with changes in body coordinates, – MOVING.

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

The 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 starts at some point and ends with a point that indicates the direction. Such an arrow segment is called vector.

- vector drawn from point M to M 1

Knowing the displacement vector means knowing its direction and module. The modulus of a vector is a scalar, i.e. numerical value. Knowing the initial position and the displacement vector of the body, it is possible to determine where the body is located.

In the process of motion, the 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 aircraft 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's motion is a continuous line, which is described by a moving body (considered as a material point) with respect to the chosen frame of reference.

The movement in which all points body moving along the same trajectories, is 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 motion happens straightforward And curvilinear.

The path length is WAY. The path is a scalar value and is denoted by the letter l. The path increases if the body moves. And remains unchanged if the body is at rest. In this way, path cannot decrease over time.

The modulus of displacement and the path can have the same value only if the body moves along a straight line in the same direction.

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

The path is a scalar value and is characterized only by a numeric value.
Displacement is a vector quantity and is characterized by both a numerical value (modulus) and a direction.
When the body moves, the path can only increase, and the displacement modulus can both increase and decrease.
If the body has returned to the starting point, its displacement is zero, and the path is not equal to zero.

	Way	moving
Definition	The length of the trajectory described by the body in a certain time	A vector connecting the initial position of the body with its subsequent position
Designation	l [m]	S [m]
The nature of physical quantities	Scalar, i.e. defined only by numeric value	Vector, i.e. defined by numerical value (modulus) and direction
The need for an introduction	Knowing the initial position of the body and the path l traveled in a time interval t, it is impossible to determine the position of the body at a given time t	Knowing the initial position of the body and S for the time interval t, the position of the body at a given time t is uniquely determined
	l = S in the case of rectilinear motion without returns

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)

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

By releasing the pressure on the walls of the bottle, achieve the ascent of the float. Determine the path and movement of the float: ______________________________________________________________
Lower the float to the bottom of the bottle. Determine the path and movement of the float:______________________________________________________________________________
Make the float float and sink. What is the path and movement of the float in this case?

5. Exercises and questions for repetition.

Do we pay for the journey or transportation when traveling in a taxi? (Way)
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 move the ball. (Path - 4 m, movement - 2 m.)

6. The result of the lesson.

Repetition of the concepts of the lesson:

– movement;
– trajectory;
- way.

7. Homework.

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

Bibliography

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

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 body of reference.

In the Cartesian coordinate system, the position of point A at a given moment of time with respect 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 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 time t 0 , as well as the laws governing the movement, determine the state of the system at all subsequent times t.

Trajectory motion 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 the point is a plane curve, i.e. lies entirely in one plane, then the movement of the point is called flat.

The length of the section of the trajectory AB traversed by a material point from the moment the time began is called path lengthΔs and is a scalar function of time: Δs=Δs(t). Unit of measurement - meter(m) is the length of the path traveled by light in vacuum in 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 ∆ r=r-r 0 , drawn from the initial position of the moving point to its position at a given moment of time is called moving(increment of the radius-vector of the point for the considered period of time).

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

To characterize the rate of change of speed v point in mechanics, a vector physical quantity is introduced, called acceleration.

Average acceleration non-uniform movement 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 the average acceleration: (4). Acceleration but is a vector quantity equal to the first derivative of the velocity with respect to time.

V. Coordinate method of motion assignment

The position of the point M can be characterized by the radius - the 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). Given (7), formula (6) can be written (8). The speed modulus can be found:(9).

Similarly for the acceleration vector:

(10),

(11),

Natural way of specifying motion (description of motion using trajectory parameters)

The movement is described by the formula s=s(t). Each point of the trajectory is characterized by its value s. Radius - the 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 value Δs is the distance between two points along the trajectory, |Δ r| is 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 (16). If τ - tangent to the trajectory, then - vector perpendicular to this tangent, i.e. directed along the normal. The unit vector, in the direction of the normal is denoted n. The value of the vector is 1/R, where R is the radius of curvature of the trajectory.

Point away from the path at a distance and R in the direction of the normal n, is called the center of curvature of the trajectory. Then (17). Given 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 is called normal.

We find the absolute value of the total acceleration: (19).

Lecture 2 Movement of a material point along a circle. Angular displacement, angular velocity, angular acceleration. Relationship between linear and angular kinematic quantities. Vectors of angular velocity and acceleration.

Lecture plan

Kinematics of rotary motion

During rotational motion, the vector dφ elementary rotation of the body. Elementary turns (denoted or) can be seen as pseudovectors (as it were).

Angular movement - vector quantity, the module 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 seems to be counterclockwise). The unit of angular displacement is rad.

The rate of change in angular displacement over time is characterized by angular velocity ω . Angular velocity solid body is a vector physical quantity that characterizes the rate of change in the angular displacement of the 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 rule of the right screw). Unit of angular velocity - rad/s

The rate of change of the angular velocity over time is characterized by angular acceleration ε

(2).

The vector ε is directed along the rotation axis in the same direction as dω, i.e. at accelerated rotation, at slow rotation.

The unit of angular acceleration is rad/s 2 .

During dt arbitrary point of the rigid body A move to dr, passing the way ds. It can be seen from the figure that dr equal to the vector product of the angular displacement dφ by radius – point vector r : dr =[ dφ · r ] (3).

Point Linear Speed is related to the angular velocity and radius of the trajectory by the relation:

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

By 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 .

Differentiate (4) with respect to time:

Considering that - linear acceleration, - angular acceleration, and - linear speed, we get:

The first vector on the right side is directed tangentially to the point trajectory. It characterizes the change in the linear velocity modulus. Therefore, this vector is the tangential acceleration of the point: a τ =[ ε · r ] (7). The tangential acceleration modulus is a τ = ε · r. The second vector in (6) is directed towards the center of the circle and characterizes the change in direction linear speed. This vector is the normal acceleration of the point: a n =[ ω · v ] (8). Its modulus is equal to a n =ω v or given that v = ω· r, a n = ω 2 · r = v 2 / r (9).

Particular cases of rotational motion

With uniform rotation: , Consequently .

Uniform rotation can be characterized rotation period T- the time it takes for a point to make one complete revolution,

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

Speed unit - hertz (Hz).

With uniformly accelerated rotational motion :

Lecture 3 Newton's first law. Strength. 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 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. Strength

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

Newton's first law is valid 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 the speed unchanged.

inertia 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 additivity consists 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 is 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.

Strength- 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 module, direction of action, point of application to the body.

Trajectory- a curve (or line) that a body describes when moving. One can speak of a trajectory only when the body is represented as a material point.

The trajectory 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.

Trajectory in different systems counting will be different. You can read about it here.

Way

Way- This is a physical quantity that shows the distance traveled by the body along the trajectory of motion. Designated L (in rare cases S).

The path is a relative value, and its value depends on the chosen frame of reference.

This can be verified on simple example: there is a passenger on the plane who makes a movement 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 did the plane relative to the Earth. Therefore, in this case the whole path would be expressed like this:

moving

moving is a vector that connects the starting position of a moving point to its final position in a certain amount of time.

Designated S. The unit of measurement is 1 meter.

At rectilinear motion in one direction coincides with the trajectory and the distance traveled. In any other case, these values do not match.

This is easy to see with a simple example. There is a girl, and in her hands is a doll. She tosses 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 4 m, but the displacement is 0. In this case, the doll traveled a path of 4 m, since at first it moved up 2 m, and then the same amount down. No movement took place in this case, since the start and end points are the same.