How to determine the magnitude of the displacement vector from a graph. Projections of the displacement vector. General methods for determining displacements

Weight is a property of a body that characterizes its inertia. Under the same influence from surrounding bodies, one body can quickly change its speed, while another, under the same conditions, can change much more slowly. It is customary to say that the second of these two bodies has greater inertia, or, in other words, the second body has greater mass.

If two bodies interact with each other, then as a result the speed of both bodies changes, i.e., in the process of interaction, both bodies acquire acceleration. The ratio of the accelerations of these two bodies turns out to be constant under any influence. In physics, it is accepted that the masses of interacting bodies are inversely proportional to the accelerations acquired by the bodies as a result of their interaction.

Force is a quantitative measure of the interaction of bodies. Force causes a change in the speed of a body. In Newtonian mechanics, forces can have a different physical nature: friction force, gravity, elastic force, etc. Force is vector quantity. The vector sum of all forces acting on a body is called resultant force.

To measure forces it is necessary to set standard of strength And comparison method other forces with this standard.

As a standard of force, we can take a spring stretched to a certain specified length. Force module F 0 with which this spring, at a fixed tension, acts on a body attached to its end is called standard of strength. The way to compare other forces with a standard is as follows: if the body, under the influence of the measured force and the reference force, remains at rest (or moves uniformly and rectilinearly), then the forces are equal in magnitude F = F 0 (Fig. 1.7.3).

If the measured force F greater (in absolute value) than the reference force, then two reference springs can be connected in parallel (Fig. 1.7.4). In this case the measured force is 2 F 0 . Forces 3 can be measured similarly F 0 , 4F 0, etc.

Measuring forces less than 2 F 0, can be performed according to the scheme shown in Fig. 1.7.5.

Reference force in International system units are called newton(N).

A force of 1 N imparts an acceleration of 1 m/s to a body weighing 1 kg 2

In practice, there is no need to compare all measured forces with a standard. To measure forces, springs calibrated as described above are used. Such calibrated springs are called dynamometers . The force is measured by the stretch of the dynamometer (Fig. 1.7.6).

Newton's laws of mechanics - three laws underlying the so-called. classical mechanics. Formulated by I. Newton (1687). First Law: “Every body continues to be maintained in its state of rest or uniform and rectilinear motion until and unless it is compelled by applied forces to change that state.” Second law: “The change in momentum is proportional to the applied driving force and occurs in the direction of the straight line along which this force acts.” Third law: “An action always has an equal and opposite reaction, otherwise, the interactions of two bodies on each other are equal and directed in opposite directions.” 1.1. Law of inertia (Newton's First Law) : a free body, which is not acted upon by forces from other bodies, is in a state of rest or uniform linear motion (the concept of speed here is applied to the center of mass of the body in the case of non-translational motion). In other words, bodies are characterized by inertia (from the Latin inertia - “inactivity”, “inertia”), that is, the phenomenon maintaining speed, if external influences on them are compensated. Reference systems in which the law of inertia is satisfied are called inertial reference systems (IRS). The law of inertia was first formulated by Galileo Galilei, who, after many experiments, concluded that for a free body to move at a constant speed, no external cause is needed. Before this, a different point of view (going back to Aristotle) was generally accepted: a free body is at rest, and to move at a constant speed it is necessary to apply a constant force. Newton subsequently formulated the law of inertia as the first of his three famous laws. Galileo's principle of relativity: in all inertial frames of reference everything physical processes proceed the same way. In a reference system brought to a state of rest or uniform rectilinear motion relative to an inertial reference system (conventionally, “at rest”), all processes proceed in exactly the same way as in a system at rest. It should be noted that the concept of an inertial reference system is an abstract model (a certain ideal object considered instead of a real object. Examples of an abstract model are an absolutely rigid body or a weightless thread), real reference systems are always associated with some object and the correspondence of the actually observed motion of bodies in such systems with the calculation results will be incomplete. 1.2 Law of motion - a mathematical formulation of how a body moves or how a more general type of motion occurs. In classical mechanics of a material point, the law of motion represents three dependences of three spatial coordinates on time, or a dependence of one vector quantity (radius vector) on time, type. The law of motion can be found, depending on the problem, either from the differential laws of mechanics or from the integral ones. Law of energy conservation - the basic law of nature, which is that the energy of a closed system is conserved over time. In other words, energy cannot arise from nothing and cannot disappear into anything; it can only move from one form to another. The law of conservation of energy is found in various branches of physics and is manifested in the conservation various types energy. For example, in classical mechanics the law is manifested in the conservation of mechanical energy (the sum of potential and kinetic energies). In thermodynamics, the law of conservation of energy is called the first law of thermodynamics and speaks of the conservation of energy in addition to thermal energy. Since the law of conservation of energy does not apply to specific quantities and phenomena, but reflects a general pattern that is applicable everywhere and always, it is more correct to call it not a law, but the principle of conservation of energy. A special case is the Law of Conservation of Mechanical Energy - the mechanical energy of a conservative mechanical system is conserved over time. Simply put, in the absence of forces such as friction (dissipative forces), mechanical energy does not arise from nothing and cannot disappear anywhere. Ek1+Ep1=Ek2+Ep2 The law of conservation of energy is an integral law. This means that it consists of the action of differential laws and is a property of their combined action. For example, it is sometimes said that the impossibility of creating a perpetual motion machine is due to the law of conservation of energy. But that's not true. In fact, in every perpetual motion machine project, one of the differential laws is triggered and it is this that makes the engine inoperative. The law of conservation of energy simply generalizes this fact. According to Noether's theorem, the law of conservation of mechanical energy is a consequence of the homogeneity of time. 1.3. Law of conservation of momentum (Law of conservation of momentum, Newton's 2nd law) states that the sum of the momenta of all bodies (or particles) of a closed system is a constant value. From Newton's laws it can be shown that when moving in empty space, momentum is conserved in time, and in the presence of interaction, the rate of its change is determined by the sum of the applied forces. In classical mechanics, the law of conservation of momentum is usually derived as a consequence of Newton's laws. However, this conservation law is also true in cases where Newtonian mechanics is not applicable (relativistic physics, quantum mechanics). Like any of the conservation laws, the law of conservation of momentum describes one of the fundamental symmetries - the homogeneity of space Newton's third law explains what happens to two interacting bodies. Let us take for example a closed system consisting of two bodies. The first body can act on the second with a certain force F12, and the second can act on the first with a force F21. How do the forces compare? Newton's third law states: the action force is equal in magnitude and opposite in direction to the reaction force. Let us emphasize that these forces are applied to different bodies, and therefore are not compensated at all. The law itself: Bodies act on each other with forces directed along the same straight line, equal in magnitude and opposite in direction: . 1.4. Inertia forces Newton's laws, strictly speaking, are valid only in inertial frames of reference. If we honestly write down the equation of motion of a body in inertial system reference, then it will differ in appearance from Newton’s second law. However, often, to simplify the consideration, a certain fictitious “force of inertia” is introduced, and then these equations of motion are rewritten in a form very similar to Newton’s second law. Mathematically, everything here is correct (correct), but from the point of view of physics, the new fictitious force cannot be considered as something real, as a result of some real interaction. Let us emphasize once again: “force of inertia” is only a convenient parameterization of how the laws of motion differ in inertial and non-inertial reference systems. 1.5. Law of viscosity Newton's law of viscosity (internal friction) - mathematical expression, connecting the internal friction stress τ (viscosity) and the change in the velocity of the medium v in space (strain rate) for fluid bodies (liquids and gases): where the value η is called the coefficient of internal friction or the dynamic coefficient of viscosity (CGS unit - poise). The kinematic viscosity coefficient is the value μ = η / ρ (CGS unit is Stokes, ρ is the density of the medium). Newton's law can be obtained analytically using methods of physical kinetics, where viscosity is usually considered simultaneously with thermal conductivity and the corresponding Fourier law for thermal conductivity. In the kinetic theory of gases, the coefficient of internal friction is calculated by the formula Where< u >is the average speed of thermal motion of molecules, λ is the average free path.

Class: 9

Lesson objectives:

Educational:
– introduce the concepts of “movement”, “path”, “trajectory”.
Developmental:
- develop logical thinking, correct physical speech, 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 starting position 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.

– vector drawn from point M to M 1

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)

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

	Path	Moving
Definition	The length of the trajectory described by a body in a certain time	A vector connecting the initial position of the body with its subsequent position
Designation	l [m]	S [m]
Nature of physical quantities	Scalar, i.e. determined only by numeric value	Vector, i.e. determined by numerical value (modulus) and direction
The need for introduction	Knowing the initial position of the body and the path l traveled over a period of time t, it is impossible to determine the position of the body at a given moment in time t	Knowing the initial position of the body and S for a period of time t, the position of the body at a given moment of 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 to the neck with water.
Fill the bottle with the 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 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.
Screw the bottle cap on.
Squeeze the sides of the bottle and lower the float to the bottom of the bottle.

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:__________________________________________________________
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 review.

Do we pay for the journey or transportation when traveling in a taxi? (Path)
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.

This term has other meanings, see Movement (meanings).

Moving(in kinematics) - a change in the position of a physical body in space over time relative to the selected reference system.

In relation to the movement of a material point moving called the vector characterizing this change. It has the property of additivity. Usually denoted by the symbol S → (\displaystyle (\vec (S))) - from Italian. s postamento (movement).

The vector modulus S → (\displaystyle (\vec (S))) is the displacement modulus, measured in meters in the International System of Units (SI); in the GHS system - in centimeters.

You can define movement as a change in the radius vector of a point: Δ r → (\displaystyle \Delta (\vec (r))) .

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:

V → = lim Δ t → 0 Δ r → Δ t = d r → d t (\displaystyle (\vec (v))=\lim \limits _(\Delta t\to 0)(\frac (\Delta (\vec (r)))(\Delta t))=(\frac (d(\vec (r)))(dt))) .

III. Trajectory, path and movement

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).

The average velocity vector v> is the ratio of the increment Δr of the radius vector of a point to the time interval Δt: (1). The direction of the average speed coincides with the direction of Δr. With an unlimited decrease in Δt, the average speed tends to a limiting value, which is called the instantaneous speed v. Instantaneous speed is the speed of a body at a given moment of time and at a given point of the trajectory: (2). Instantaneous velocity 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 are introduced vector physical quantity, called acceleration.

Medium acceleration Not uniform 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:

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) we can write (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 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

At rotational movement the measure of movement 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 is a vector quantity whose modulus equal to angle rotation, and the direction coincides with the direction of translational movement 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 . Angular velocity solid– 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). 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/s2.

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= v2 / 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 :

(13), (14) (15).

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 the inertial frame of reference and asserts the existence of the 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.

General methods for determining displacements

 1 =X 1  11 +X 2  12 +X 3  13 +…

 2 =X 1  21 +X 2  22 +X 3  23 +…

 3 =X 1  31 +X 2  32 +X 3  33 +…

Work of constant forces: A=P P, P – generalized force– any load (concentrated force, concentrated moment, distributed load),  P – generalized movement(deflection, rotation angle). The designation  mn means movement in the direction of the generalized force “m”, which is caused by the action of the generalized force “n”. Total displacement caused by several force factors:  P = P P + P Q + P M . Movements caused by a single force or a single moment:  – specific displacement . If a unit force P = 1 caused a displacement  P, then the total displacement caused by the force P will be:  P = P P. If the force factors acting on the system are designated X 1, X 2, X 3, etc. , then movement in the direction of each of them:

where X 1  11 =+ 11; X 2  12 =+ 12 ; Х i  m i =+ m i . Dimension of specific movements:

, J-joules, the dimension of work is 1J = 1Nm.

Work of external forces acting on an elastic system:

.

– the actual work under the static action of a generalized force on an elastic system is equal to half the product of the final value of the force and the final value of the corresponding displacement. The work of internal forces (elastic forces) in the case of plane bending:

,

k is a coefficient that takes into account the uneven distribution of tangential stresses over the cross-sectional area and depends on the shape of the section.

Based on the law of conservation of energy: potential energy U=A.

Work reciprocity theorem (Betley's theorem) . Two states of an elastic system:

 1

1 – movement in direction. force P 1 from the action of force P 1;

 12 – movement in direction. force P 1 from the action of force P 2;

 21 – movement in direction. force P 2 from the action of force P 1;

 22 – movement in direction. force P 2 from the action of force P 2.

A 12 =P 1  12 – work done by the force P 1 of the first state on the movement in its direction caused by the force P 2 of the second state. Similarly: A 21 =P 2  21 – work of the force P 2 of the second state on movement in its direction caused by the force P 1 of the first state. A 12 = A 21. The same result is obtained for any number of forces and moments. Work reciprocity theorem: P 1  12 = P 2  21 .

The work of the forces of the first state on displacements in their directions caused by the forces of the second state is equal to the work of the forces of the second state on displacements in their directions caused by the forces of the first state.

Theorem on the reciprocity of displacements (Maxwell's theorem) If P 1 =1 and P 2 =1, then P 1  12 =P 2  21, i.e.  12 = 21, in the general case  mn = nm.

For two unit states of an elastic system, the displacement in the direction of the first unit force caused by the second unit force is equal to the displacement in the direction of the second unit force caused by the first force.

Universal method for determining displacements (linear and rotation angles) – Mohr's method. A unit generalized force is applied to the system at the point for which the generalized displacement is sought. If the deflection is determined, then the unit force is a dimensionless concentrated force; if the angle of rotation is determined, then it is a dimensionless unit moment. When spatial system There are six components of internal effort at work. The generalized displacement is determined by the formula (Mohr's formula or integral):

The line above M, Q and N indicates that these internal forces are caused by a unit force. To calculate the integrals included in the formula, you need to multiply the diagrams of the corresponding forces. The procedure for determining the movement: 1) for a given (real or cargo) system, find the expressions M n, N n and Q n; 2) in the direction of the desired movement, a corresponding unit force (force or moment) is applied; 3) determine efforts

from the action of a single force; 4) the found expressions are substituted into the Mohr integral and integrated over the given sections. If the resulting mn >0, then the displacement coincides with the selected direction of the unit force, if

For flat design:

Usually, when determining displacements, the influence of longitudinal deformations and shear, which are caused by longitudinal N and transverse Q forces, is neglected; only displacements caused by bending are taken into account. For a flat system it will be:

.

IN

calculation of the Mohr integralVereshchagin's method . Integral

for the case when the diagram from a given load has an arbitrary outline, and from a single load it is rectilinear, it is convenient to determine it using the graph-analytical method proposed by Vereshchagin.

, where is the area of the diagram M r from the external load, y c is the ordinate of the diagram from a unit load under the center of gravity of the diagram M r. The result of multiplying diagrams is equal to the product of the area of one of the diagrams and the ordinate of another diagram, taken under the center of gravity of the area of the first diagram. The ordinate must be taken from a straight-line diagram. If both diagrams are straight, then the ordinate can be taken from any one.

P

moving:

. The calculation using this formula is carried out in sections, in each of which the straight-line diagram should be without fractures. A complex diagram M p is divided into simple ones geometric figures, for which it is easier to determine the coordinates of the centers of gravity. When multiplying two diagrams that have the form of trapezoids, it is convenient to use the formula:

. The same formula is also suitable for triangular diagrams, if you substitute the corresponding ordinate = 0.

P

Under the action of a uniformly distributed load on a simply supported beam, the diagram is constructed in the form of a convex quadratic parabola, the area of which

(for fig.

, i.e.

, x C =L/2).

D

For a “blind” seal with a uniformly distributed load, we have a concave quadratic parabola, for which

;

,

, x C = 3L/4. The same can be obtained if the diagram is represented by the difference between the area of a triangle and the area of a convex quadratic parabola:

. The "missing" area is considered negative.

Castigliano's theorem .

– the displacement of the point of application of the generalized force in the direction of its action is equal to the partial derivative of the potential energy with respect to this force. Neglecting the influence on the movement of axial and shear forces, we have potential energy:

, where

.

What is the definition of movement in physics?

Sad Roger

In physics there is movement absolute value a vector drawn from the starting point of the body’s trajectory to the final point. In this case, the shape of the path along which the movement took place (that is, the trajectory itself), as well as the size of this path, does not matter at all. Let's say, the movement of Magellan's ships - well, at least the one that eventually returned (one of three) - is equal to zero, although the distance traveled is wow.

Is Tryfon

Displacement can be viewed in two ways. 1. Change in body position in space. Moreover, regardless of the coordinates. 2. The process of movement, i.e. change in position over time. You can argue about point 1, but to do this you need to recognize the existence of absolute (initial) coordinates.

Movement is a change in the location of a certain physical body in space relative to the reference system used.

This definition is given in kinematics - a subsection of mechanics that studies the movement of bodies and the mathematical description of movement.

Displacement is the absolute value of a vector (that is, a straight line) connecting two points on a path (from point A to point B). Displacement differs from path in that it is a vector value. This means that if the object came to the same point from which it started, then the displacement is zero. But there is no way. A path is the distance an object has traveled due to its movement. To better understand, look at the picture:

What is path and movement from a physics point of view? and what is the difference between them....

very necessary) please answer)

User deleted

Alexander kalapats

Path is a scalar physical quantity that determines the length of the trajectory section traveled by the body during a given time. The path is a non-negative and non-decreasing function of time.
Displacement is a directed segment (vector) connecting the position of the body at the initial moment of time with its position at the final moment of time.
Let me explain. If you leave home, go to visit a friend, and return home, then your path will be equal to the distance between your house and your friend’s house multiplied by two (there and back), and your movement will be equal to zero, because at the final moment of time you will find yourself in the same place as at the initial moment, i.e. at home. A path is a distance, a length, i.e. a scalar quantity that has no direction. Displacement is a directed, vector quantity, and the direction is specified by a sign, i.e., displacement can be negative (If we assume that when you reach your friend’s house you have made a movement s, then when you walk from your friend to his house, you will make a movement -s , where the minus sign means that you walked in the opposite direction to the one in which you walked from the house to your friend).

Forserr33v

When we talk about moving, it's important to remember that moving depends on the frame of reference in which the motion is considered. Pay attention to the picture.

Rice. 4. Determination of the body displacement modulus

The body moves in the XOY plane. Point A is the initial position of the body. Its coordinates are A(x 1; y 1). The body moves to point B (x 2; y 2). Vector - this will be the movement of the body:

Lesson 3. Determining the coordinates of a moving body

Eryutkin Evgeniy Sergeevich

The topic of the lesson is “Determination of the coordinates of a moving body.” We have already discussed the characteristics of movement: distance traveled, speed and displacement. The main characteristic movement is the location of the bodies. To characterize it, it is necessary to use the concept of “displacement”, it is this that makes it possible to determine the location of the body at any moment in time, this is precisely the main task of mechanics.

Rice. 1. Path as the sum of many linear movements

Trajectory as a sum of displacements

In Fig. Figure 1 shows the trajectory of a body from point A to point B in the form of a curved line, which we can imagine as a set of small displacements. Moving is a vector, therefore, we can represent the entire path traveled as a set of sums of very small displacements along the curve. Each of the small movements is a straight line, all together they make up the entire trajectory. Please note: - it is the movement that determines the position of the body. We must consider any movement in specific system countdown.

Body coordinates

The drawing must be combined with the reference system for the motion of bodies. The simplest method we are considering is movement in a straight line, along one axis. To characterize the movements, we will use a method associated with a reference system - with one line; movement is linear.

Rice. 2. One-dimensional movement

In Fig. Figure 2 shows the OX axis and the case of one-dimensional motion, i.e. the body moves along a straight line, along one axis. In this case, the body moved from point A to point B, the movement was vector AB. To determine the coordinate of point A, we must do the following: lower the perpendicular to the axis, the coordinate of point A on this axis will be designated X 1, and lowering the perpendicular from point B, we obtain the coordinate of the end point - X 2. Having done this, we can talk about the projection of the vector onto the OX axis. When solving problems, we will need the projection of a vector, a scalar quantity.

Projection of a vector onto an axis

In the first case, the vector is directed along the OX axis and coincides in direction, so the projection will have a plus sign.

Rice. 3. Motion projection

with a minus sign

Example of negative projection

In Fig. Figure 3 shows another possible situation. Vector AB in this case is directed against the selected axis. In this case, the projection of the vector onto the axis will have a negative value. When calculating the projection, the vector symbol S must be placed, and the index X at the bottom: S x.

Path and displacement in linear motion

Straight-line motion is simple view movements. In this case, we can say that the modulus of the vector projection is the distance traveled. It should be noted that in this case the length of the vector modulus is equal to the distance traveled.

Rice. 4. The path traveled is the same

with displacement projection

Examples of different relative axis orientations and displacements

To finally understand the issue of vector projection onto an axis and with coordinates, let’s consider several examples:

Rice. 5. Example 1

Example 1. Motion module is equal to the displacement projection and is defined as X 2 – X 1, i.e. subtract the initial coordinate from the final coordinate.

Rice. 6. Example 2

Example 2. The second figure under the letter B is very interesting. If the body moves perpendicular to the selected axis, then the coordinate of the body on this axis does not change, and in this case the modulus of displacement along this axis is equal to 0.

Fig 7. Example 3

Example 3. If the body moves at an angle to the OX axis, then, determining the projection of the vector onto the OX axis, it is clear that the projection in its value will be less than the module of the vector S itself. By subtracting X 2 - X 1, we determine the scalar value of the projection.

Solving the problem of determining the path and movement

Let's consider the problem. Determine the location of the motor boat. The boat departed from the pier and walked along the coast straight and evenly, first for 5 km, and then in the opposite direction for another 3 km. It is necessary to determine the distance traveled and the magnitude of the displacement vector.

Topic: Laws of interaction and motion of bodies

Lesson 4. Displacement during linear uniform motion

Eryutkin Evgeniy Sergeevich

Uniform linear movement

First, let's remember the definition uniform motion. Definition: uniform motion is a motion in which a body travels equal distances in any equal intervals of time.

It should be noted that not only rectilinear, but also curvilinear movement can be uniform. Now we will look at one special case- movement along a straight line. So, uniform rectilinear motion (URM) is a motion in which a body moves along a straight line and makes equal movements in any equal intervals of time.

Speed

Important characteristic such a movement - speed. From grade 7 you know that speed is a physical quantity that characterizes the speed of movement. With uniform rectilinear motion, speed is a constant value. Speed is a vector quantity, denoted by , the unit of speed is m/s.

Rice. 1. Speed projection sign

depending on its direction

Pay attention to fig. 1. If the velocity vector is directed in the direction of the axis, then the projection of the velocity will be . If the speed is directed against the selected axis, then the projection of this vector will be negative.

Determination of speed, path and movement

Let's move on to the formula for speed calculation. Speed is defined as the ratio of movement to the time during which this movement occurred: .

We draw your attention to the fact that during rectilinear motion, the length of the displacement vector is equal to the path traveled by this body. Therefore, we can say that the displacement modulus is equal to the distance traveled. Most often you came across this formula in 7th grade and in mathematics. It is written simply: S = V * t. But it is important to understand that this is only a special case.

Equation of motion

If we remember that the projection of a vector is defined as the difference between the final coordinate and the initial coordinate, i.e. S x = x 2 – x 1, then we can obtain the law of motion for rectilinear uniform motion.

Speed graph

Please note that the velocity projection can be either negative or positive, so a plus or minus is placed here, depending on the direction of the velocity relative to the selected axis.

Rice. 2. Graph of velocity projection versus time for RPD

The graph of the projection of velocity versus time presented above is a direct characteristic of uniform motion. The horizontal axis represents time, and the vertical axis represents speed. If the velocity projection graph is located above the x-axis, then this means that the body will move along the Ox axis in the positive direction. Otherwise, the direction of movement does not coincide with the direction of the axis.

Geometric interpretation of the path

Rice. 3. Geometric meaning speed versus time graph

Topic: Laws of interaction and motion of bodies

Lesson 5. Rectilinear uniformly accelerated motion. Acceleration

Eryutkin Evgeniy Sergeevich

The topic of the lesson is “Non-uniform rectilinear motion, rectilinear uniformly accelerated motion.” To describe such a movement, we introduce an important quantity - acceleration. Let us recall that in previous lessons we discussed the issue of rectilinear uniform motion, i.e. such movement when the speed remains constant.

Uneven movement

And if the speed changes, what then? In this case, they say that the movement is uneven.

Instantaneous speed

To characterize uneven motion, a new physical quantity is introduced - instantaneous speed.

Definition: instantaneous speed is the speed of a body at a given moment or at a given point on a trajectory.

A device that shows instantaneous speed is found on any moving vehicle: in a car, train, etc. This is a device called a speedometer (from English - speed (“speed”)). Please note that instantaneous speed is defined as the ratio of movement to the time during which this movement occurred. But this definition is no different from the definition of speed with RPD that we gave earlier. For a more precise definition, it should be noted that the time interval and the corresponding displacement are taken to be very small, tending to zero. Then the speed does not have time to change much, and we can use the formula that we introduced earlier: .

Pay attention to fig. 1. x 0 and x 1 are the coordinates of the displacement vector. If this vector is very small, then the change in speed will occur quite quickly. In this case we characterize this change as a change instantaneous speed.

Rice. 1. On the issue of determining instantaneous speed

Acceleration

Thus, uneven movement It makes sense to characterize the change in speed from point to point by how quickly it happens. This change in speed is characterized by a quantity called acceleration. Acceleration is denoted by , it is a vector quantity.

Definition: Acceleration is defined as the ratio of the change in speed to the time during which the change occurred.

Acceleration is measured in m/s 2 .

In essence, the rate of change of velocity is acceleration. The acceleration projection value, since it is a vector, can be negative or positive.

It is important to note that wherever the change in velocity is directed, that is where the acceleration will be directed. This is of particular importance during curvilinear movement, when the value changes.

Topic: Laws of interaction and motion of bodies

Lesson 6. Straight line speed uniformly accelerated motion. Speed graph

Eryutkin Evgeniy Sergeevich

Acceleration

Let's remember what acceleration is. Acceleration is a physical quantity that characterizes the change in speed over a certain period of time. ,

that is, acceleration is a quantity that is determined by the change in speed over the time during which this change occurred.

Speed equation

Using the equation that determines acceleration, it is convenient to write a formula for calculating the instantaneous speed of any interval and for any moment in time:

This equation makes it possible to determine the speed at any moment of movement of a body. When working with the law of changes in speed over time, it is necessary to take into account the direction of speed in relation to the selected reference point.

Speed graph

Speed graph(velocity projection) is the law of change of velocity (velocity projection) over time for uniformly accelerated rectilinear motion, presented graphically.

Rice. 1. Graphs of the velocity projection versus time for uniformly accelerated rectilinear motion

Let's analyze various graphs.

First. Velocity projection equation: . Speed and time increase, note that on the graph there will be a straight line in the place where one of the axes is time and the other is speed. This line begins from the point, which characterizes the initial speed.

The second is the dependence for a negative value of the acceleration projection, when the movement is slow, that is, the absolute speed first decreases. In this case, the equation looks like: .

The graph begins at point and continues until point , the intersection of the time axis. At this point the speed of the body becomes equal to zero. This means that the body has stopped.

If you look closely at the speed equation, you will remember that in mathematics there was a similar function. This is the equation of a straight line, which is confirmed by the graphs we examined.

Some special cases

To finally understand the speed graph, let’s consider a special case. In the first graph, the dependence of speed on time is due to the fact that the initial speed, , is equal to zero, the projection of acceleration is greater than zero.

Writing this equation. Well, the type of graph itself is quite simple (graph 1):

Rice. 2. Various cases of uniformly accelerated motion

Two more cases uniformly accelerated motion presented in the next two graphs. The second case is a situation when the body first moved with a negative acceleration projection, and then began to accelerate in the positive direction of the OX axis.

The third case is a situation when the acceleration projection is less than zero and the body continuously moves in the direction opposite to the positive direction of the OX axis. In this case, the velocity module constantly increases, the body accelerates.

This video lesson will help users get an idea of the topic “Movement in linear uniformly accelerated motion.” During this lesson, students will be able to expand their knowledge of rectilinear uniformly accelerated motion. The teacher will tell you how to correctly determine the displacement, coordinates and speed during such a movement.

Topic: Laws of interaction and motion of bodies

Lesson 7. Displacement during rectilinear uniformly accelerated motion

Eryutkin Evgeniy Sergeevich

In previous lessons, we discussed how to determine the distance traveled during uniform linear motion. It's time to find out how to determine the coordinates of the body, the distance traveled and the displacement at . This can be done if we consider rectilinear uniformly accelerated motion as a set large quantity very small uniform movements of the body.

Galileo's experiment

The first to solve the problem of the location of a body at a certain point in time during accelerated motion was the Italian scientist Galileo Galilei. He conducted his experiments with an inclined plane. He launched a ball, a musket bullet, along the chute, and then determined the acceleration of this body. How did he do it? He knew the length inclined plane, and determined the time by the beat of his heart or pulse.

Determining movement using a speed graph

Consider the speed dependence graph uniformly accelerated linear motion from time. You know this relationship; it is a straight line: v = v 0 + at

Fig.1. Motion Definition

with uniformly accelerated linear motion

We divide the speed graph into small rectangular sections. Each section will correspond to a certain constant speed. It is necessary to determine the distance traveled during the first period of time. Let's write the formula: .

Now let's calculate the total area of all the figures we have. And the sum of the areas during uniform motion is the total distance traveled.

Please note that the speed will change from point to point, thereby we will get the path traveled by the body precisely during rectilinear uniformly accelerated motion.

Note that during rectilinear uniformly accelerated motion of a body, when speed and acceleration are directed in the same direction, the displacement module is equal to the distance traveled, therefore, when we determine the displacement module, we determine distance traveled. In this case, we can say that the displacement module will be equal to the area of the figure, limited by the graph of speed and time.

Let's use mathematical formulas to calculate the area of the indicated figure.

The area of the figure (numerically equal to the distance traveled) is equal to half the sum of the bases multiplied by the height. Note that in the figure one of the bases is the initial velocity. And the second base of the trapezoid will be the final speed, denoted by the letter, multiplied by. This means that the height of the trapezoid is the period of time during which the movement occurred.

We can write the final speed discussed in the previous lesson as the sum initial speed and the contribution due to the presence of constant acceleration in the body. The resulting expression is:

If you open the parentheses, it becomes double. We can write the following expression:

If you write each of these expressions separately, the result will be the following:

This equation was first obtained through the experiments of Galileo Galilei. Therefore, we can assume that it was this scientist who first made it possible to determine the location of the body at any moment. This is the solution to the main problem of mechanics.

Determining body coordinates

Now let's remember that the distance traveled, equal in our case movement module, is expressed by the difference:

If we substitute the expression we obtained for S into Galileo’s equation, we will write down the law according to which a body moves in rectilinear uniformly accelerated motion:

It should be remembered that velocity, its projection and acceleration can be negative.

The next stage of consideration of movement will be the study of movement along a curvilinear trajectory.

Topic: Laws of interaction and motion of bodies

Lesson 8. Movement of a body during rectilinear uniformly accelerated motion without initial velocity

Eryutkin Evgeniy Sergeevich

Rectilinear uniformly accelerated motion

Let us consider some features of the movement of a body during rectilinear uniformly accelerated motion without initial speed. The equation that describes this movement was derived by Galileo in the 16th century. It must be remembered that in case of rectilinear uniform or uneven movement, the displacement module coincides in value with the distance traveled. The formula looks like this:

S=V o t + at 2 /2,

where a is the acceleration.

Case of uniform motion

The first, simplest case is the situation when the acceleration is zero. This means that the equation above will become the equation: S = V 0 t. This equation makes it possible to find distance traveled uniform movement. S, in this case, is the modulus of the vector. It can be defined as the difference in coordinates: the final coordinate x minus the initial coordinate x 0. If we substitute this expression into the formula, we get the dependence of the coordinate on time.

The case of motion without initial speed

Let's consider the second situation. When V 0 = 0, the initial speed is 0, which means that the movement begins from a state of rest. The body was at rest, then begins to acquire and increase speed. Movement from a state of rest will be recorded without an initial speed: S = at 2 /2. If S – travel module(or the distance traveled) is designated as the difference between the initial and final coordinates (we subtract the initial coordinate from the final coordinate), then we obtain an equation of motion that makes it possible to determine the coordinate of the body for any moment in time: x = x 0 + at 2 /2.

The acceleration projection can be both negative and positive, so we can talk about the coordinate of the body, which can either increase or decrease.

Proportionality of the path to the square of time

Important principles of equations without initial velocity, i.e. when a body begins its movement from a state of rest:

S x is the distance traveled, it is proportional to t 2, i.e. square of time. If we consider equal periods of time - t 1, 2t 1, 3t 1, then we can notice the following relationships:

S 1 ~ 1 S 1 = a/2*t 1 2

S 2 ~ 4 S 2 = a/2*(2t 1) 2

S 3 ~ 9 S 3 = a/2*(3t 1) 2

If you continue, the pattern will remain.

Movements over successive periods of time

We can draw the following conclusion: the distances traveled increase in proportion to the square of the increase in time intervals. If there was one period of time, for example 1 s, then the distance traveled will be proportional to 1 2. If the second segment is 2 s, then the distance traveled will be proportional to 2 2, i.e. = 4.

If we choose a certain interval for a unit of time, then the total distances traveled by the body over subsequent equal periods of time will be related as the squares of integers.

In other words, the movements made by the body for each subsequent second will be related as odd numbers:

S 1:S 2:S 3:…:S n =1:3:5:…:(2n-1)

Rice. 1. Movement

for each second are treated as odd numbers

Considered patterns using the example of a problem

The two very important conclusions studied are characteristic only of rectilinear uniformly accelerated motion without an initial speed.

Problem: the car starts moving from a stop, i.e. from a state of rest, and in 4 s of its movement it travels 7 m. Determine the acceleration of the body and the instantaneous speed 6 s after the start of movement.

Rice. 2. Solving the problem

Solution: the car starts moving from a state of rest, therefore, the path that the car travels is calculated by the formula: S = at 2 /2. Instantaneous speed is defined as V = at. S 4 = 7 m, the distance that the car covered in 4 s of its movement. It can be expressed as the difference between the total path covered by the body in 4 s and the path covered by the body in 3 s. Using this, we obtain acceleration a = 2 m/s 2, i.e. movement is accelerated, rectilinear. To determine the instantaneous speed, i.e. speed at the end of 6 s, acceleration should be multiplied by time, i.e. for 6 s, during which the body continued to move. We get the speed v(6s) = 12 m/s.

Answer: acceleration modulus is 2 m/s 2 ; the instantaneous speed at the end of 6 s is 12 m/s.

Topic: Laws of interaction and motion of bodies

Lesson 9: Laboratory work No. 1 “Study of uniformly accelerated motion

without initial speed"

Eryutkin Evgeniy Sergeevich

Goal of the work

The purpose of the laboratory work is to determine the acceleration of the body, as well as its instantaneous speed at the end of the movement.

First time given laboratory work conducted by Galileo Galilei. It was thanks to this work that Galileo was able to experimentally establish the acceleration of free fall.

Our task is to consider and analyze how we can determine acceleration when a body moves along an inclined chute.

Equipment

Equipment: tripod with coupling and foot, an inclined groove is fixed in the foot; in the gutter there is a stop in the form of a metal cylinder. A moving body is a ball. The time counter is a metronome; if you start it, it will count the time. You will need a measuring tape to measure the distance.

Rice. 1. Tripod with coupling and foot, groove and ball

Rice. 2. Metronome, cylindrical stop

Measurement table

Let's create a table consisting of five columns, each of which must be filled out.

The first column is the number of beats of the metronome, which we use as a time counter. S – the next column is the distance covered by the body, the ball rolling down the inclined chute. Next is the travel time. The fourth column is the calculated acceleration of movement. The last column shows the instantaneous speed at the end of the ball's movement.

Required formulas

To obtain the result, use the formulas: S = at 2 /2.

From here it is easy to obtain that the acceleration will be equal to the ratio of twice the distance divided by the square of time: a = 2S/t 2.

Instantaneous speed is defined as the product of acceleration and time of movement, i.e. the period of time from the start of movement until the moment the ball collides with the cylinder: V = at.

Conducting an experiment

Let's move on to the experiment itself. To do this, you need to adjust metronome so that he makes 120 blows in one minute. Then between two metronome beats there will be a time interval of 0.5 s (half a second). We start the metronome and watch how it counts time.

Next, using a measuring tape, we determine the distance between the cylinder that makes up the stop and the starting point of movement. It is equal to 1.5 m. The distance is chosen so that the body rolling down the chute falls within a time period of at least 4 metronome beats.

Rice. 3. Setting up the experiment

Experience: a ball that is placed at the beginning of the movement and released with one of the blows gives the result - 4 blows.

Filling out the table

We record the results in a table and proceed to calculations.

The number 3 was entered in the first column. But there were 4 metronome beats?! The first blow corresponds to the zero mark, i.e. we start counting time, so the time the ball moves is the intervals between strikes, and there are only three of them.

Length the distance traveled, i.e. the length of the inclined plane is 1.5 m. Substituting these values into the equation, we obtain an acceleration equal to approximately 1.33 m/s 2 . Please note that this is an approximate calculation, accurate to the second decimal place.

The instantaneous speed at the moment of impact is approximately 1.995 m/s.

So, we have found out how we can determine the acceleration of a moving body. We draw your attention to the fact that in his experiments Galileo Galilei determined acceleration by changing the angle of inclination of the plane. We invite you to independently analyze the sources of errors when performing this work and draw conclusions.

Topic: Laws of interaction and motion of bodies

Lesson 10. Solving problems on determining acceleration, instantaneous speed and displacement in uniformly accelerated linear motion

Eryutkin Evgeniy Sergeevich

The lesson is devoted to solving problems on determining acceleration, instantaneous speed and displacement of a moving body.

Path and displacement task

Task 1 is devoted to the study of path and movement.

Condition: a body moves in a circle, passing half of it. It is necessary to determine the relationship of the traveled path to the displacement module.

Please note: the condition of the problem is given, but there is not a single number. Such problems will appear quite often in physics courses.

Rice. 1. Path and movement of the body

Let us introduce some notation. The radius of the circle along which the body moves is equal to R. When solving the problem, it is convenient to make a drawing in which we denote the circle and an arbitrary point from which the body moves, denoted by A; the body moves to point B, and S is half a circle, S is moving, connecting the starting point of movement to the ending point.

Despite the fact that there is not a single number in the problem, nevertheless, in the answer we get a very definite number (1.57).

Speed graph problem

Problem 2 will focus on velocity graphs.

Condition: two trains are moving towards each other on parallel tracks, the speed of the first train is 60 km/h, the speed of the second is 40 km/h. Below are 4 graphs, and you need to choose those that correctly depict the projection graphs of the speed of these trains.

Rice. 2. To the condition of problem 2

Rice. 3. Charts

to problem 2

The speed axis is vertical (km/h), and the time axis is horizontal (time in hours).

In the 1st graph there are two parallel straight lines, these are the modules of the speed of the body - 60 km/h and 40 km/h. If you look at the bottom chart, number 2, you will see the same thing, only in the negative area: -60 and -40. The other two charts have 60 on top and -40 on the bottom. On the 4th chart, 40 is at the top and -60 is at the bottom. What can you say about these graphs? According to the condition of the problem, two trains are traveling towards each other, along parallel tracks, so if we choose an axis associated with the direction of the speed of one of the trains, then the projection of the speed of one body will be positive, and the projection of the speed of the other will be negative (since the speed itself is directed against the selected axis) . Therefore, neither the first graph nor the second are suitable for the answer. When velocity projection has the same sign, we need to say that two trains are moving in the same direction. If we choose a reference frame associated with 1 train, then the value of 60 km/h will be positive, and the value of -40 km/h will be negative, the train is moving towards. Or vice versa, if we connect the reporting system with the second train, then one of them has a projected speed of 40 km/h, and the other -60 km/h, negative. Thus, both graphs (3 and 4) are suitable.

Answer: 3 and 4 graphs.

Problem of determining speed in uniformly slow motion

Condition: a car moves at a speed of 36 km/h, and within 10 s it brakes with an acceleration of 0.5 m/s 2. It is necessary to determine its speed at the end of braking

In this case, it is more convenient to select the OX axis and direct the initial velocity along this axis, i.e. the initial velocity vector will be directed in the same direction as the axis. The acceleration will be directed in the opposite direction, because the car is slowing down. The projection of acceleration onto the OX axis will have a minus sign. To find the instantaneous, final speed, we use the velocity projection equation. Let's write the following: V x = V 0x - at. Substituting the values, we get a final speed of 5 m/s. This means that 10 s after braking the speed will be 5 m/s. Answer: V x = 5 m/s.

The task of determining acceleration from a speed graph

The graph shows 4 dependences of speed on time, and it is necessary to determine which of these bodies has the maximum and which has the minimum acceleration.

Rice. 4. To the conditions of problem 4

To solve, you need to consider all 4 graphs in turn.

To compare accelerations, you need to determine their values. For each body, acceleration will be defined as the ratio of the change in speed to the time during which this change occurred. Below are calculations of acceleration for all four bodies:

As you can see, the acceleration modulus of the second body is minimal, and the acceleration modulus of the third body is maximum.

Answer: |a 3 | - max, |a 2 | - min.

Lesson 11. Solving problems on the topic “Rectilinear uniform and non-uniform motion”

Eryutkin Evgeniy Sergeevich

Let's look at two problems, and the solution to one of them is in two versions.

The task of determining the distance traveled during uniformly slow motion

Condition: An airplane flying at a speed of 900 km/h lands. The time until the aircraft comes to a complete stop is 25 s. It is necessary to determine the length of the runway.

Rice. 1. To the conditions of problem 1

How to determine the displacement module? (mechanics) and got the best answer

Answer from Ivan Vyazigin[newbie]
according to the Pythagorean theorem = root (16+9) = 5

Answer from Marinas[guru]
Three main ways to describe body movement
Vector method
t. O - reference body; t. A - material point (particle); - radius vector (this is a vector connecting the origin with the position of a point at an arbitrary moment in time)
Trajectory (1-2) - a line describing the movement of a body (material point A) over a period of time
Displacement () is a vector connecting the positions of a moving point at the beginning and end of a certain period of time.
Path () – length of the trajectory section.
Let's write the equation of motion of a point in vector form:
The speed of a point is the limit of the ratio of movement to the period of time during which this movement occurred, when this period of time tends to zero.
That is, instantaneous speed
Acceleration (or instantaneous acceleration) is a vector physical quantity equal to the limit of the ratio of the change in speed to the period of time during which this change occurred.
Acceleration, like the change in speed, is directed towards the concavity of the trajectory and can be decomposed into two components - tangential - tangent to the trajectory of movement - and normal - perpendicular to the trajectory.
- full acceleration;
- normal acceleration (characterizes the change in speed in direction);
- tangential acceleration (characterizes the change in speed in magnitude);
, where is the unit normal vector ()
R1 - radius of curvature.
,
Where;
Coordinate method of describing movement
With the coordinate method of describing motion, the change in the coordinates of a point over time is written in the form of functions of all three of its coordinates versus time:
kinematic levels of motion of a point)
Projections on the axis:
A natural way to describe movement

Answer from Av paap[newbie]
THX

Answer from Olga Gavrilova[active]
Why is that?

Answer from 3 answers[guru]

Hello! Here is a selection of topics with answers to your question: How to determine the displacement module? (Mechanics)