Mechanical movement
Definition 1
A change in the location of a body (or its parts) relative to other bodies is called mechanical motion.
Example 1
For example, a person moving on an escalator in the subway is at rest relative to the escalator itself and moves relative to the walls of the tunnel; Mount Elbrus is at rest, conventionally the Earth, and moves with the Earth relative to the Sun.
We see that we need to indicate the point relative to which the movement is being considered; this is called the reference body. The reference point and the coordinate system to which it is connected, as well as the chosen method of measuring time, constitute the concept of reference.
The movement of a body, where all its points move equally, is called translational. To find the speed $V$ with which a body moves, you need to divide the path $S$ by the time $T$.
$ \frac(S)(T) = (V)$
The movement of a body around a certain axis is rotational. With this move, all points of the body move across the terrain, the center of which is considered to be this axis. And although the wheels make a rotational movement around their axes, at the same time, translational movement occurs along with the car body. This means that the wheel performs a rotational motion relative to the axis, and a translational motion relative to the road.
Definition 2
Oscillatory motion is a periodic movement that a body performs in turn in two opposite directions. The simplest example is a pendulum in a clock.
Translational and rotational are the simplest types of mechanical movement.
If point $X$ changes its location relative to point $Y$, then $Y$ changes its position relative to $X$. In other words, bodies move relative to each other. Mechanical motion is considered relative - to describe it you need to indicate relative to what point it is considered
Simple types of movement of a material body are uniform and rectilinear movements. It is uniform if the magnitude of the velocity vector does not change (the direction can change).
The movement is called rectilinear if the course of the velocity vector is constant (and the magnitude can change). A trajectory is a straight line on which the velocity vector is located.
We see examples of mechanical movement in everyday life. These are cars passing by, planes flying, ships sailing. We form simple examples ourselves, passing near other people. Every second our planet passes in two planes: around the Sun and its axis. And these are also examples of mechanical movement.
Varieties of movement
Translational motion is the automatic movement of a rigid body, while any stage of a straight line, clearly associated with a moving point, remains synchronous with its original position.
An important characteristic of the movement of a body is its trajectory, which represents a spatial curve, which can be shown in the form of conjugate arcs of different radii, each emanating from its center. A different position for any point of the body, which can change over time.
An elevator car or a Ferris wheel car moves progressively. Translational motion takes place in 3-dimensional space, but its main distinguishing feature - maintaining the parallelism of any segment to itself - remains in force.
We denote the period by the letter $T$. To find the rotation period, you need to divide the rotation time by the number of revolutions: $\frac(\delta t)(N) = (T)$
Rotational motion - a material point describes a circle. During the rotational process of a completely rigid body, all its points describe a circle, which are in parallel planes. The centers of these circles lie on the same straight line, perpendicular to the planes of the circles and are called the axis of rotation.
The axis of rotation can be located inside the body and behind it. The axis of rotation in the system can be movable or fixed. For example, in a reference frame connected to the Earth, the rotation axis of the generator rotor at the station is motionless.
Sometimes the axis of rotation receives a complex rotational movement - spherical, when the points of the body move along the spheres. A point moves around a fixed axis that does not pass through the center of the body or a rotating material point; such movement is called circular.
Characteristics of linear motion: displacement, speed, acceleration. They become their analogues during rotational motion: angular displacement, angular velocity, angular acceleration:
- the role of movement in the rotational process has an angle;
- the magnitude of the rotation angle per unit time is the angular velocity;
- the change in angular velocity over a period of time is angular acceleration.
Oscillatory motion
Movement in two opposite directions, oscillatory. Oscillations that occur in closed concepts are called independent or natural oscillations. Fluctuations that occur under the influence of external forces are called forced.
If we analyze the swaying according to the characteristics that change (amplitude, frequency, period, etc.), then they can be divided into damped, harmonic, increasing (as well as rectangular, complex, sawtooth).
During free oscillations in real systems, energy losses always occur. Energy is spent working to overcome the force of air resistance. The friction force reduces the amplitudes of vibrations, and they stop after some time.
Forced rocking is undamped. Therefore, it is necessary to replenish energy losses for each hour of fluctuation. To do this, it is necessary to act on the body from time to time with varying force. Forced oscillations occur with a frequency equal to changes in the external force.
The amplitude of forced oscillations reaches its greatest value when this coefficient is the same as the frequency of the oscillatory system. This is called resonance.
For example, if you periodically pull the rope in time with its vibrations, we will see an increase in the amplitude of its swing.
Definition 3
A material point is a body whose size can be neglected under certain conditions.
The car we often remember can be taken as a material point relative to the Earth. But if people move inside this car, then the size of the car can no longer be neglected.
When you solve problems in physics, the movement of a body is regarded as the movement of a material point, and such concepts as the speed of a point, the acceleration of a material body, the inertia of a material point, etc. are used.
Frame of reference
A material point moves relative to the inertia of other bodies. The body, according to the relation to which this automatic movement is considered, is called the body of reference. The reference body is chosen freely depending on the assigned tasks.
The location system is associated with the reference body, which assumes a reference point (coordinate base). The location concept has 1, 2 or 3 axes due to the condition of movement. The state of a point on a line (1 axis), plane (2 axes) or in a place (3 axes) is established in accordance with this by one, 2 or 3 coordinates.
In order to establish the position of the body in the spatial domain at any time period, it is necessary to set the start of the time count. A device for measuring time, a coordinate system, a reference point to which the coordinate system is connected - this is the reference system.
The movement of the body is considered in relation to this system. The same point, in comparison with different reference bodies in different coordinate concepts, has every chance of having completely different coordinates. The reference system also depends on the choice of motion trajectory
The types of reference systems can be varied, for example: a fixed reference system, a moving reference system, an inertial reference system, a non-inertial reference system.
If the position of a given body relative to surrounding objects changes over time, then this body moves. If the position of the body remains unchanged, then the body is at rest. The unit of time in mechanics is 1 second. By time interval we mean the number t seconds separating any two consecutive phenomena.
Observing the movement of a body, you can often see that the movements of different points of the body are different; So when a wheel rolls on a plane, the center of the wheel moves in a straight line, and a point lying on the circumference of the wheel describes a curve (cycloid); the paths traversed by these two points in the same time (per 1 revolution) are also different. Therefore, the study of body movement begins with the study of the movement of a single point.
The line described by a moving point in space is called the trajectory of this point.
The rectilinear motion of a point is a motion whose trajectory is straight line.
Curvilinear movement is movement whose trajectory is not a straight line.
Movement is determined by the direction, trajectory and distance traveled over a certain period of time (period).
Uniform motion of a point is such a motion in which the ratio of the traveled path S to the corresponding period of time remains constant for any period of time, i.e.
S/t = const(constant value).(15)
This constant ratio of path to time is called the speed of uniform motion and is denoted by the letter v. Thus, v= S/t. (16)
Solving the equation for S, we get S = vt, (17)
that is, the distance traveled by a point during uniform motion is equal to the product of speed and time. Solving the equation for t, we find that t = S/v,(18)
that is, the time during which a point travels a given path during uniform motion is equal to the ratio of this path to the speed of movement.
These equalities are the basic formulas for uniform motion. These formulas are used to determine one of the three quantities S, t, v, when the other two are known.
Speed dimension v = length / time = m/sec.
Uneven motion is the movement of a point in which the ratio of the distance traveled to the corresponding period of time is not a constant value.
With uneven movement of a point (body), they are often satisfied with finding the average speed, which characterizes the speed of movement for a given period of time, but does not give an idea of the speed of movement of the point at individual moments, i.e., the true speed.
The true speed of uneven motion is the speed at which the point is moving at the moment.
The average speed of a point is determined by formula (15).
In practice, they are often satisfied with the average speed, accepting it as true. For example, the table speed of a longitudinal planing machine is constant, with the exception of the moments of the beginning of the working and the beginning of the idle strokes, but these moments are neglected in most cases.
In a cross-planing machine, in which rotational motion is converted into translational motion by a rocker mechanism, the speed of the slide is uneven. At the beginning of the stroke it is equal to zero, then it increases to some maximum value at the moment of the vertical position of the slide, after which it begins to decrease and by the end of the stroke it becomes equal to zero again. In most cases, calculations use the average speed v cf of the slider, which is taken as the true cutting speed.
The speed of the slider of a cross-planing machine with a rocker mechanism can be characterized as uniformly variable.
Uniformly variable motion is a motion in which the speed increases or decreases by the same amount over equal periods of time.
The speed of uniformly variable motion is expressed by the formula v = v 0 + at, (19)
where v is the speed of uniformly variable movement at a given moment, m/sec;
v 0 — speed at the beginning of movement, m/sec; a - acceleration, m/sec 2.
Acceleration is the change in speed per unit time.
Acceleration a has the dimension speed / time = m / sec 2 and is expressed by the formula a = (v-v 0)/t. (20)
When v 0 = 0, a = v/t.
The path traveled during uniformly variable motion is expressed by the formula S= ((v 0 +v)/2)* t = v 0 t+(at 2)/2. (21)
Translational motion of a rigid body is such a motion in which any straight line taken on this body moves parallel to itself.
During translational motion, the speeds and accelerations of all points of the body are the same and at any point they are the speed and acceleration of the body.
Rotational motion is a motion in which all points of a certain straight line (axis) taken in this body remain motionless.
With uniform rotation at equal intervals of time, the body rotates through equal angles. Angular velocity characterizes the magnitude of rotational motion and is denoted by the letter ω (omega).
The relationship between the angular velocity ω and the number of revolutions per minute is expressed by the equation: ω = (2πn)/60 = (πn)/30 deg/sec. (22)
Rotational motion is a special case of curvilinear motion.
The speed of the rotational movement of the point is directed tangentially to the trajectory of movement and is equal in magnitude to the length of the arc traversed by the point in the corresponding period of time.
Speed of movement of a point of a rotating body expressed by the equation
v = (2πRn)/(1000*60)= (πDn)/(1000*60) m/s, (23)
where n is the number of revolutions per minute; R is the radius of the circle of rotation.
Angular acceleration characterizes the increase in angular velocity per unit time. It is denoted by the letter ε (epsilon) and expressed by the formula ε = (ω - ω 0) / t. (24)
Details Category: Mechanics Published 03/17/2014 18:55 Views: 15751Mechanical movement is considered for material point and For solid body.
Motion of a material point
Forward movement an absolutely rigid body is a mechanical movement during which any straight line segment associated with this body is always parallel to itself at any moment in time.
If you mentally connect any two points of a rigid body with a straight line, then the resulting segment will always be parallel to itself in the process of translational motion.
During translational motion, all points of the body move equally. That is, they travel the same distance in the same amount of time and move in the same direction.
Examples of translational motion: the movement of an elevator car, mechanical scales, a sled rushing down a mountain, bicycle pedals, a train platform, engine pistons relative to the cylinders.
Rotational movement
During rotational motion, all points of the physical body move in circles. All these circles lie in planes parallel to each other. And the centers of rotation of all points are located on one fixed straight line, which is called axis of rotation. Circles that are described by points lie in parallel planes. And these planes are perpendicular to the axis of rotation.
Rotational movement is very common. Thus, the movement of points on the rim of a wheel is an example of rotational movement. Rotational motion is described by a fan propeller, etc.
Rotational motion is characterized by the following physical quantities: angular velocity of rotation, period of rotation, frequency of rotation, linear speed of a point.
Angular velocity A body rotating uniformly is called a value equal to the ratio of the angle of rotation to the period of time during which this rotation occurred.
The time it takes a body to complete one full revolution is called rotation period (T).
The number of revolutions a body makes per unit time is called speed (f).
Rotation frequency and period are related to each other by the relation T = 1/f.
If a point is located at a distance R from the center of rotation, then its linear speed is determined by the formula:
Mechanical movement of a body (point) is the change in its position in space relative to other bodies over time.
Types of movements:
A) Uniform rectilinear motion of a material point: Initial conditions 
. Initial conditions 
G) Harmonic oscillatory motion. An important case of mechanical motion is oscillations, in which the parameters of a point’s motion (coordinates, speed, acceleration) are repeated at certain intervals.
ABOUT scriptures of the movement . There are various ways to describe the movement of bodies. With the coordinate method specifying the position of a body in a Cartesian coordinate system, the movement of a material point is determined by three functions expressing the dependence of coordinates on time:
x= x(t), y=y(t) And z= z(t) .
This dependence of coordinates on time is called the law of motion (or equation of motion).
With the vector method the position of a point in space is determined at any time by the radius vector r= r(t) , drawn from the origin to a point.
There is another way to determine the position of a material point in space for a given trajectory of its movement: using a curvilinear coordinate l(t) .
All three methods of describing the motion of a material point are equivalent; the choice of any of them is determined by considerations of the simplicity of the resulting equations of motion and the clarity of the description.
Under reference system understand a reference body, which is conventionally considered motionless, a coordinate system associated with the reference body, and a clock, also associated with the reference body. In kinematics, the reference system is selected in accordance with the specific conditions of the problem of describing the motion of a body.
2. Trajectory of movement. Distance traveled. Kinematic law of motion.
The line along which a certain point of the body moves is called trajectorymovement this point.
The length of the trajectory section traversed by a point during its movement is called the path traveled .
The change in radius vector over time is called kinematic law
:
In this case, the coordinates of the points will be coordinates in time: x=
x(t),
y=
y(t) Andz=
z(t).
In curvilinear motion, the path is greater than the displacement modulus, since the length of the arc is always greater than the length of the chord contracting it
The vector drawn from the initial position of the moving point to its position at a given time (increment of the radius vector of the point over the considered period of time) is called moving. The resulting displacement is equal to the vector sum of successive displacements.
During rectilinear movement, the displacement vector coincides with the corresponding section of the trajectory, and the displacement module is equal to the distance traveled.
3. Speed. Average speed. Velocity projections.
Speed
- speed of change of coordinates. When a body (material point) moves, we are interested not only in its position in the chosen reference system, but also in the law of motion, i.e., the dependence of the radius vector on time. Let the moment in time 
corresponds to the radius vector 
a moving point, and a close moment in time 
- radius vector 
.
Then in a short period of time 
the point will make a small displacement equal to 
To characterize the movement of a body, the concept is introduced average speed
his movements: 
This quantity is a vector quantity, coinciding in direction with the vector 
. With unlimited reduction Δt the average speed tends to a limiting value called instantaneous speed 
:
Velocity projections.
A) Uniform linear motion of a material point: 
Initial conditions 
B) Uniformly accelerated linear motion of a material point: 
. Initial conditions 
B) Movement of a body along a circular arc with a constant absolute speed: 
To find the coordinates of a moving body at any moment in time, you need to know the projections of the displacement vector on the coordinate axes, and therefore the displacement vector itself. What you need to know for this. The answer depends on what kind of movement the body makes.
Let's first consider the simplest type of movement - rectilinear uniform motion.
A movement in which a body makes equal movements at any equal intervals is called rectilinear uniform movement.
To find the displacement of a body in uniform rectilinear motion over a certain period of time t, you need to know what movement a body makes per unit of time, since for any other unit of time it makes the same movement.
The movement made per unit of time is called speed body movements and are designated by the letter υ . If movement in this area is denoted by , and the time period is denoted by t, then the speed can be expressed as a ratio to . Since displacement is a vector quantity, and time is a scalar quantity, then speed is also a vector quantity. The velocity vector is directed in the same way as the displacement vector.
Speed of uniform linear motion of a body is a quantity equal to the ratio of the movement of the body to the period of time during which this movement occurred:
Thus, speed shows how much movement a body makes per unit time. Therefore, to find the displacement of a body, you need to know its speed. The movement of the body is calculated by the formula:
The displacement vector is directed in the same way as the velocity vector, time t- scalar quantity.
Calculations cannot be carried out using formulas written in vector form, since a vector quantity has not only a numerical value, but also a direction. When making calculations, they use formulas that include not vectors, but their projections on the coordinate axes, since algebraic operations can be performed on projections.
Since the vectors are equal, their projections onto the axis are also equal X, from here:
Now you can get a formula for calculating the coordinates x points at any given time. We know that
From this formula it is clear that with rectilinear uniform motion, the coordinate of the body linearly depends on time, which means that with its help it is possible to describe rectilinear uniform motion.
In addition, it follows from the formula that to find the position of the body at any time during rectilinear uniform motion, you need to know the initial coordinate of the body x 0 and the projection of the velocity vector onto the axis along which the body moves.
It must be remembered that in this formula v x- projection of the velocity vector, therefore, like any projection of a vector, it can be positive and negative.
Rectilinear uniform motion is rare. More often you have to deal with movement in which the movements of the body can be different over equal periods of time. This means that the speed of the body changes somehow over time. Cars, trains, airplanes, etc., a body thrown upward, and bodies falling to the Earth move at variable speeds.
With such a movement, you cannot use a formula to calculate the displacement, since the speed changes over time and we are no longer talking about a specific speed, the value of which can be substituted into the formula. In such cases, the so-called average speed is used, which is expressed by the formula:
average speed shows the displacement that a body makes on average per unit of time.
However, using the concept of average speed, the main problem of mechanics - determining the position of a body at any moment in time - cannot be solved.
