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rectilinear uniform motionis a motion in which a body travels the same distance in equal intervals of time.
Uniform movement- this is such a movement of the body in which its speed remains constant (), that is, it moves at the same speed all the time, and acceleration or deceleration does not occur ().
Rectilinear motion- this is the movement of the body in a straight line, that is, the trajectory we get is straight.
Uniform speed rectilinear motion does not depend on time and at each point of the trajectory is directed in the same way as the movement of the body. That is, the velocity vector coincides with the displacement vector. With all this, the average speed in any period of time is equal to the initial and instantaneous speed:
Speed of uniform rectilinear motion is a physical vector quantity equal to the ratio of the displacement of the body for any period of time to the value of this interval t:
from this formula. we can easily express body movement with uniform motion:
Consider the dependence of speed and displacement on time
Since our body moves in a straight line and uniformly accelerated (), then the graph with the dependence of speed on time will look like a parallel straight line to the time axis.
depending projections of body velocity versus time there is nothing complicated. The projection of the movement of the body is numerically equal to the area of the rectangle AOBC, since the magnitude of the displacement vector is equal to the product of the velocity vector by the time during which the movement was made.
On the chart we see displacement versus time.
It can be seen from the graph that the velocity projection is equal to:
Curvilinear motion of the body
Curvilinear motion of a body definition:
Curvilinear motion is a type of mechanical motion in which the direction of velocity changes. The speed modulus can change.
Uniform body movement
Uniform body motion definition:
If a body travels equal distances in equal intervals of time, then such a movement is called. With uniform motion, the modulus of velocity is a constant value. And it can change.
Uneven body movement
Uneven body movement definition:
If a body travels different distances in equal intervals of time, then such a movement is called uneven. With uneven movement, the speed module is variable. The direction of speed can change.
Uniform body movement
Equal-variable motion of a body definition:
There is a constant value in uniformly variable motion. If at the same time the direction of the velocity does not change, then we get a rectilinear uniformly variable motion.
Uniformly accelerated motion of the body
Uniformly accelerated motion of a body definition:
Equally slow motion of the body
Uniformly slow motion of a body definition:
When we talk about the mechanical motion of a body, we can consider the concept of translational motion of a body.
Uniform movement- this is movement at a constant speed, that is, when the speed does not change (v \u003d const) and there is no acceleration or deceleration (a \u003d 0).
Rectilinear motion- this is movement in a straight line, that is, the trajectory of rectilinear movement is a straight line.
This is a movement in which the body makes the same movements for any equal intervals of time. For example, if we divide some time interval into segments of one second, then with uniform motion the body will move the same distance for each of these segments of time.
The speed of uniform rectilinear motion does not depend on time and at each point of the trajectory is directed in the same way as the movement of the body. That is, the displacement vector coincides in direction with the velocity vector. In this case, the average speed for any period of time is equal to the instantaneous speed:
vcp=vSpeed of uniform rectilinear motion is a physical vector quantity equal to the ratio of the displacement of the body for any period of time to the value of this interval t:
=/t
Thus, the speed of uniform rectilinear motion shows what movement a material point makes per unit of time.
moving with uniform rectilinear motion is determined by the formula:
Distance traveled in rectilinear motion is equal to the displacement modulus. If the positive direction of the OX axis coincides with the direction of movement, then the projection of the velocity on the OX axis is equal to the velocity and is positive:
vx = v, i.e. v > 0The projection of displacement onto the OX axis is equal to:
s = vt = x - x0where x 0 is the initial coordinate of the body, x is the final coordinate of the body (or the coordinate of the body at any time)
Motion equation, that is, the dependence of the body coordinate on time x = x(t), takes the form:
x = x0 + vtIf the positive direction of the OX axis is opposite to the direction of motion of the body, then the projection of the body velocity on the OX axis is negative, the velocity is less than zero (v< 0), и тогда уравнение движения принимает вид:
x = x0 - vtUniform rectilinear motion This is a special case of non-uniform motion.
Uneven movement- this is a movement in which a body (material point) makes unequal movements in equal intervals of time. For example, a city bus moves unevenly, since its movement consists mainly of acceleration and deceleration.
Equal-variable motion is the movement at which the speed of the body ( material point) for any equal time intervals changes equally.
Acceleration of a body in uniform motion remains constant in magnitude and direction (a = const).
Uniform motion can be uniformly accelerated or uniformly slowed down.
Uniformly accelerated motion- this is the movement of a body (material point) with a positive acceleration, that is, with such a movement, the body accelerates with a constant acceleration. When uniformly accelerated motion the modulus of the body's velocity increases with time, the direction of acceleration coincides with the direction of the speed of motion.
Uniformly slow motion- this is the movement of a body (material point) with negative acceleration, that is, with such a movement, the body slows down uniformly. With uniformly slow motion, the velocity and acceleration vectors are opposite, and the velocity modulus decreases with time.
In mechanics, any rectilinear motion is accelerated, so slow motion differs from accelerated motion only by the sign of the projection of the acceleration vector onto the selected axis of the coordinate system.
Average speed of variable motion is determined by dividing the movement of the body by the time during which this movement was made. The unit of average speed is m/s.
vcp=s/tThis is the speed of the body (material point) at a given moment of time or at a given point of the trajectory, that is, the limit to which the average speed tends to decrease with an infinite decrease in the time interval Δt:
Instantaneous velocity vector uniformly variable motion can be found as the first derivative of the displacement vector with respect to time:
= "
Velocity vector projection on the OX axis:
vx = x'this is the derivative of the coordinate with respect to time (the projections of the velocity vector onto other coordinate axes are similarly obtained).
This is the value that determines the rate of change in the speed of the body, that is, the limit to which the change in speed tends with an infinite decrease in the time interval Δt:
Acceleration vector of uniform motion can be found as the first derivative of the velocity vector with respect to time or as the second derivative of the displacement vector with respect to time:
= " = " Given that 0 is the speed of the body at the initial moment of time (initial speed), is the speed of the body at a given moment of time (final speed), t is the time interval during which the change in speed occurred, will be as follows:
From here uniform velocity formula at any given time:
0 + t Cartesian system coordinates coinciding in direction with the trajectory of the body, then the projection of the velocity vector on this axis is determined by the formula:
vx = v0x ± axtThe "-" (minus) sign in front of the projection of the acceleration vector refers to uniformly slow motion. Equations of projections of the velocity vector onto other coordinate axes are written similarly.
Since the acceleration is constant (a \u003d const) with uniformly variable motion, the acceleration graph is a straight line parallel to the 0t axis (time axis, Fig. 1.15).
Rice. 1.15. Dependence of body acceleration on time.
Speed versus time- this is linear function, whose graph is a straight line (Fig. 1.16).
Rice. 1.16. Dependence of body speed on time.
Graph of speed versus time(Fig. 1.16) shows that
In this case, the displacement is numerically equal to the area of \u200b\u200bthe figure 0abc (Fig. 1.16).
The area of a trapezoid is half the sum of the lengths of its bases times the height. The bases of the trapezoid 0abc are numerically equal:
0a = v0 bc = vThe height of the trapezoid is t. Thus, the area of the trapezoid, and hence the projection of displacement onto the OX axis, is equal to:
In the case of uniformly slow motion, the projection of acceleration is negative, and in the formula for the projection of displacement, the sign "-" (minus) is placed in front of the acceleration.
The graph of the dependence of the speed of the body on time at various accelerations is shown in Fig. 1.17. The graph of the dependence of displacement on time at v0 = 0 is shown in fig. 1.18.
Rice. 1.17. Dependence of body speed on time for various values of acceleration.
Rice. 1.18. Dependence of body displacement on time.
The speed of the body at a given time t 1 is equal to the tangent of the angle of inclination between the tangent to the graph and the time axis v \u003d tg α, and the movement is determined by the formula:
If the time of motion of the body is unknown, you can use another displacement formula by solving a system of two equations:
It will help us to derive a formula for the displacement projection:
Since the coordinate of the body at any time is determined by the sum of the initial coordinate and the displacement projection, it will look like this:
The graph of the x(t) coordinate is also a parabola (as is the displacement graph), but the vertex of the parabola generally does not coincide with the origin. For a x< 0 и х 0 = 0 ветви параболы направлены вниз (рис. 1.18).
Details Category: Mechanics Posted on 17.03.2014 18:55 Views: 16086Mechanical movement is considered for material point and for solid body.
Movement of a material point
translational movement absolutely rigid body is mechanical movement, during which any line segment associated with this body is always parallel to itself at any 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.
In translational motion, all points of the body move in the same way. That is, they cover the same distance in the same time intervals and move in the same direction.
Examples of translational motion: the movement of an elevator car, cups of mechanical scales, a sledge racing downhill, bicycle pedals, a train platform, engine pistons relative to cylinders.
rotational movement
With 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 described by points lie in parallel planes. And these planes are perpendicular to the axis of rotation.
Rotational motion is very common. Thus, the movement of points on the rim of a wheel is an example of rotational movement. The rotational motion describes the fan propeller, etc.
Rotational motion is characterized by the following physical quantities: angular speed of rotation, period of rotation, frequency of rotation, linear speed of a point.
angular velocity a body with uniform rotation is called a value equal to the ratio of the angle of rotation to the time interval during which this rotation occurred.
The time it takes a body to complete one revolution is called rotation period (T).
The number of revolutions a body makes per unit of time is called speed (f).
The rotation frequency and the period are related by the relation T = 1/f.
If the point is at a distance R from the center of rotation, then its linear velocity is determined by the formula:
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. Under the time interval is meant the number t sec separating any two consecutive phenomena.
Observing the movement of a body, one can often see that the movements various points bodies are different; so when a wheel rolls along 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 traveled by these two points in the same time (per 1 revolution) are also different. Therefore, the study of the movement of the body 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.
A rectilinear motion of a point is a motion whose trajectory is straight line.
Curvilinear motion is motion whose trajectory is not a straight line.
The movement is determined by the direction, the trajectory and the path traveled for a certain period of time (period).
The uniform motion of a point is such a motion in which the ratio of the distance traveled S to the corresponding time interval preserves constant value for any period of time, i.e.
S/t = const(constant).(15)
This constant ratio of path to time is called the speed of uniform motion and is denoted by the letter v. In this way, v = S/t. (16)
Solving the equation for S, we get S=vt, (17)
i.e., the value of the path traveled by a point in uniform motion is equal to the product of speed and time. Solving the equation for t, we find that t = S/v,(18)
i.e., the time during which a point with uniform motion passes a given path is equal to the ratio of this path to the speed of movement.
These equalities are the basic formulas for uniform motion. According to these formulas, one of the three values S, t, v is determined, when the other two are known.
Dimension of speed v = length / time = m/sec.
An uneven movement is a 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 the point at individual moments, i.e., the true speed.
The true speed of uneven movement is the speed at which the point is currently moving.
The average speed of the point is determined by the formula (15).
Almost often satisfied average speed accepting it as true. For example, the table speed of a planer is constant, with the exception of the moments of the start of the working and the beginning of idling, but these moments are neglected in most cases.
In a cross-cutting machine, in which the rotational movement is converted into translational by a rocker mechanism, the speed of the slider 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 wings, after which it begins to decrease and by the end of the stroke it becomes again zero. In most cases, the calculations use the average speed v cf of the slider, which is taken as the true cutting speed.
The slider speed of a slatted cross planer can be characterized as uniformly variable.
Uniformly variable motion is a motion in which the speed increases or decreases by the same amount over the same intervals 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 motion at a given moment, m/s;
v 0 - speed at the beginning of the movement, m / s; a - acceleration, m / s 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. (twenty)
For v 0 = 0, a = v/t.
The path traveled during uniformly variable motion is expressed by the formula S \u003d ((v 0 + v) / 2) * t \u003d v 0 t + (at 2) / 2. (21)
The translational motion of a rigid body a is such a motion in which any straight line taken on this body moves parallel to itself.
In 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 such a motion in which all points of a certain straight line (axis) taken in this body remain motionless.
With uniform rotation in equal intervals of time, the body rotates through the same angles. The angular velocity characterizes the amount 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 \u003d (2πn) / 60 \u003d (πn) / 30 deg / s. (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 time interval.
The 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 revolution.
Angular acceleration characterizes the increase in angular velocity per unit time. It is denoted by the letter ε (epsilon) and is expressed by the formula ε = (ω - ω 0) / t. (24)
