Forward movement is the meaning. Movement of bodies

Mechanics considers all possible movements material point And solid. All of them are described in several sections. For example, the question of how they move will be the prerogative of kinematics. It describes in detail the translational movement, as well as the more complex rotational movement. First, about what is simpler. Because without this it is difficult to move on to the next topics.

What assumptions does mechanics allow?

In many problems it is possible to introduce an approximation. This is due to the fact that it will not affect the result, but will simplify the course of reasoning.

The first approximation is related to body size. If the body under consideration is significantly smaller than others located in the same frame of reference, then its dimensions are neglected. And the body itself turns into a material point.

The second follows from the absence of deformation in the body during its movement. Or at least its insignificant value, which can be completely neglected.

What is the forward motion of a body?

For explanation, we will need to consider any two points inside a solid body. They need to be connected with a segment. If this segment remains parallel during movement initial position, then they say that this is a forward movement.

If the dimensions of the body are neglected and a material point is considered, then the segment is absent and it itself moves along the straight line.

Vivid examples of such a movement

The first thing you can remember is the elevator cabin. It perfectly illustrates the forward movement of the body. The elevator always moves straight up or down without any rotation.

The next example illustrating forward motion is the movement of the Ferris wheel cabin. However, this is only realistic in a situation where the slight tilt of the cabin at the beginning of each shift is not taken into account.

The third situation when we can talk about forward motion is associated with the movement of bicycle pedals. Their movement is considered relative to the frame. Here again, the assumption is introduced that a person’s feet do not swing while riding.

The list can be completed by the movement of pistons that oscillate inside the cylinders of an internal combustion engine.

Main concepts

Kinematics forward movement is that it studies and describes the movement of solid bodies and material points. However, she does not consider the reasons that force the body to do this. To describe motion, you will need coordinates to indicate its position in space. In addition, you will need knowledge about speed, and at each specific moment in time.

First, it’s worth remembering the trajectory. It is the line along which the body moved.

The first thing you need to do is enter the displacement. It represents a vector, which is denoted Latin letter r. It can connect the origin of coordinates to the position of a material point. In other cases, this vector is drawn from the initial to end point trajectories. The units of movement are meters.

The second quantity that deserves attention is the path. He equal to length the trajectory along which the body moved. The path is designated by the letter S of the Latin alphabet, which is also measured in meters.

Basic formulas

Now it's time for speed. She is also a vector. Moreover, it characterizes not only the direction of movement of the body, but also the speed of its movement. The velocity vector is always directed along a tangent line, which can be drawn to any point on the trajectory. It is designated by the letter V. Its units of measurement are m/s.
The speed at each instant of movement can be defined as the derivative of movement with respect to time. If in the problem we're talking about about uniform motion, then the following formula is valid:

• V = S: t, where t is the time of movement.

In a situation where the direction of movement changes, it is necessary to use the sum of all movements.

The next quantity is acceleration. Again a vector quantity, which is directed towards the speed with great value. It is defined as the first derivative of speed with respect to time. Accepted notation- the letter a". The dimension is indicated in m/s 2.

Formulas for each component of acceleration directed along the axes are calculated as the ratio of the change in speed along this axis to the period of time. If you do mathematical notation, then you get the following:

• a x = ∆V x: ∆t.

For projections of acceleration onto other axes, the formulas are similar.
In addition, when considering movement along a trajectory with bends, it is possible to decompose the acceleration vector into two terms:

• a = a t + a n, where a t - tangential acceleration, directed tangentially to the bend, and n is normal, which indicates the center of the curvature.

The translational motion of any rigid body is reduced to describing the movement of only one of its points. The formulas to use are:

• S = S 0 + V 0 t + (at 2) : 2.
• V = V 0 + at.

In this formula, the indices “zero” denote initial values quantities

Translational motion magnitude theorem

Its formulation is as follows: the trajectory, speed and acceleration of all points of the body are the same during its forward motion.

To prove it, you need to write down the formula for adding displacement vectors and a vector connecting two arbitrary points. The trajectories of all points are obtained by transferring them along the second vector. But it does not change its direction and magnitude over time. Therefore, it can be argued that all points of the body move along the same trajectories.

If you take the derivative with respect to time, you get the value of speed. Moreover, the expression is simplified to the extent that the velocities of the two points are equal.
The field of the second derivative with respect to time produces a result with equality of accelerations of two points.

There are five types of rigid body motion:

1. forward movement;
2. spinning around fixed axis;
3. flat movement;
4. rotation around a fixed point;
5. free movement.

The first two are called the simplest motions of a rigid body. Other types of movements can be represented as a combination of basic movements.

Definition

Translational motion is such a motion of a rigid body in which any straight line drawn in this body moves while remaining parallel to its initial direction.

Any straight motion is progressive. However, forward motion should not be confused with linear motion. When a body moves forward, the trajectories of its points can be any curved lines.

Fig.1 Translational curvilinear movement cab view wheel

Theorem

The properties of translational motion are determined by the following theorem: during translational motion, all points of the body describe identical (overlapping, coinciding) trajectories and at each moment of time have the same magnitude and direction of velocity and acceleration.

It follows from the theorem that the translational motion of a rigid body is determined by the movement of any one of its points. Consequently, the study of the translational motion of a body is reduced to the problem of the kinematics of a point.

In translational motion, the velocity $\overrightarrow (v)$ common to all points of the body is called the speed of translational motion of the body, and the acceleration $\overrightarrow (a)$ is called the acceleration of translational motion of the body. Vectors $\overrightarrow (v)$ and $\overrightarrow (a)$ can be represented as applied at any point of the body.

Note that the concept of speed and acceleration of a body makes sense only in translational motion. In all other cases, the points of the body move with different speeds and accelerations, and the terms “body speed” or “body acceleration” for these movements lose their meaning.

Rotational motion of an absolutely rigid body around a fixed axis is such a motion in which all points of the body move in planes perpendicular to a fixed straight line, called the axis of rotation, and describe circles whose centers lie on this axis.

To determine the position of a rotating body, we draw through the axis of rotation, along which we direct the axis Az, a half-plane - stationary and a half-plane embedded in the body itself and rotating with it (Fig. 2).

Figure 2. Body rotation angle

Then the position of the body at any moment of time is uniquely determined by the angle $\varphi $ taken with the appropriate sign between these half-planes, which we will call the angle of rotation of the body. We will consider the angle $\varphi $ positive if it is plotted from the fixed plane in a counterclockwise direction (for an observer looking from the positive end of the Az axis), and negative if it is clockwise. We will always measure the angle $\varphi $ in radians. To know the position of the body at any moment of time, you need to know the dependence of the angle $\varphi $ on time t, i.e. $(\mathbf \varphi )$=f(t). This equation expresses the law of rotational motion of a rigid body around a fixed axis.

When an absolutely rigid body rotates around a fixed axis, the angles of rotation of the radius vector of various points of the body are the same.

The main kinematic characteristics of the rotational motion of a rigid body are its angular velocity $\omega $ and angular acceleration $\varepsilon $.

Equations describing rotational movement, can be obtained from the equations of translational motion by making the following substitutions in the latter: displacement s --- corner displacement (angle of rotation) $\varphi $, speed u --- angular velocity$\omega $, acceleration a --- angular acceleration$\varepsilon$.

Translational motion is such a motion of a rigid body when every straight line mentally drawn in the body moves parallel to itself.

Theorem. During translational motion, all points of the body describe identical (congruent) trajectories and have geometric equal speeds and acceleration.

Proof. Let the body move forward (Fig. 91). Let us arbitrarily select two points in the body and . The vector of these points during translational motion of the body is constant vector- its direction remains constant in accordance with the definition of translational motion, its module - due to the constant distances between points of an absolutely rigid body. Therefore, for the radius vectors of the selected points at any time, the following relation holds:

This equality means that if the position of a point at some point in time becomes known, then the position of the point at this moment is found by shifting the point by vector quantity, the same at all times. Therefore, if it is known locus positions (trajectory) of the point, then the geometric locus of the positions (trajectory) of the point is obtained by shifting the trajectory of the point in the direction and magnitude of the vector. Which proves the congruence of the trajectories of points and . Since the points are chosen arbitrarily, the trajectories of all points of the body are congruent.

Differentiating the written equality successively twice in time, we are convinced of the validity of the second part of the theorem:

The speed common to all points of the body is called the speed of the body; the acceleration common to all points is the acceleration of the body. Let us immediately note that these terms make sense only in forward motion; in all other cases of body movement, individual points of the body have different speeds and acceleration.

From all that has been said, it follows that the study of the translational motion of a body comes down to the problem of the kinematics of a point. Namely, a point in the body is selected whose movement is determined most simply, and its trajectory, speed, and acceleration are determined by the methods of the point’s kinematics. The trajectories, velocities and accelerations of the remaining points are determined by simple transfer kinematic characteristics selected point.

Determine the trajectory, speed and acceleration of point M, rigidly connected to link AB of the twin-wheel mechanism (Fig. 92), if , and angle .

We notice that the link AB of the mechanism moves forward. The movement of its point A, which also serves as the end of the crank, is easily determined. Let's select this point and find its kinematic characteristics.

It is immediately clear that the trajectory of point A is a circle with a center at the point and radius . By shifting this circle so that its center is at point O, and , we obtain the trajectory of point M.

>>Physics: Movement of bodies. Forward movement

A description of the movement of a body is considered complete only when it is known how each point moves.
We paid a lot of attention to describing the movement of the point. It is for a point that the concepts of coordinates, speed, acceleration, trajectory are introduced. IN general case The task of describing the motion of bodies is complex. It is especially difficult if the bodies are noticeably deformed during movement. It is easier to describe the movement of the body, mutual arrangement parts of which do not change. Such a body is called absolutely solid. In fact, there are no absolutely solid bodies. But in cases where real bodies When moving, they deform little and can be considered absolutely solid. (Another abstract model introduced when considering motion.) However, the motion of an absolutely rigid body in the general case turns out to be very complex. Any complex movement An absolutely rigid body can be represented as the sum of two independent motions: translational and rotational.
Forward movement. The simplest movement of rigid bodies is progressive.
Progressive is the motion of a rigid body in which any segment connecting any two points of the body remains parallel to itself.
During translational motion, all points of the body make the same movements, describe the same trajectories, travel the same paths, and have equal velocities and accelerations at each moment of time. Let's show it.
Let the body move forward ( Fig.2.1). Let's connect two of its arbitrary points B And A segment. The distance does not change, since the body is absolutely rigid. When moving forward, they remain constant module and the direction of the vector. As a result, the trajectories of points B And A are the same, since they can be completely combined parallel transfer to vector.

According to Figure 2.1 moving points A And B are the same and take place at the same time. Therefore, the points A And B have the same speeds and accelerations.
It is quite obvious that to describe the translational motion of a rigid body it is enough to describe the movement of any one of its points. Only with translational motion can we talk about the speed and acceleration of the body. For any other movement of the body, its points have different speeds and acceleration, and the terms “body speed” and “body acceleration” for non-translational motion lose their meaning.
A desk drawer, the pistons of a car engine relative to the cylinders, and carriages on a straight section move approximately progressively railway, cutter lathe relative to the bed. Movement of a bicycle pedal or Ferris wheel cabin in parks ( Fig.2.2, 2.3) are also examples of translational motion.

For description forward movement of a rigid body, it is enough to write the equation of motion of one of its points.

G.Ya.Myakishev, B.B.Bukhovtsev, N.N.Sotsky, Physics 10th grade

If you have corrections or suggestions for this lesson,