Rectilinear motion. School encyclopedia Description of body movement in mechanics
Mechanical movement is considered for material point andfor solid body.
Material point movement
Translational motion of an absolutely rigid body is a mechanical movement, during which any segment of a straight line connected with this body is always parallel to itself at any moment in time.
If you mentally connect any two points of a rigid body by a straight line, then the resulting segment will always be parallel to itself in the process of translational movement.
When moving forward, all points of the body move in the same way. That is, they travel the same distance at the same time intervals and move in the same direction.
Examples of translational motion: the movement of an elevator car, mechanical weighing cups, sledges rushing down a mountain, bicycle pedals, a train platform, engine pistons relative to 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... The circles described by the 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 a rotational movement. The 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 velocity 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 for the body to go through one full revolution is called rotation period (T).
The number of revolutions that the body makes per unit of time is called speed (f).
The frequency of rotation and the period are related to each other by the ratio T \u003d 1 / f.
If a 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 is moving. If the position of the body remains unchanged, then the body is at rest. A unit of time in mechanics is 1 sec. The interval of time means the number of t sec separating any two consecutive phenomena.
Observing the movement of a body, one can often see that the movements of various 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 (in 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 that point.
The rectilinear motion of a point is such a motion, the trajectory of which is - straight line.
Curvilinear motion is a motion whose trajectory is not a straight line.
The movement is determined by the direction, trajectory and the path traveled for a certain period of time (period).
Uniform motion of a point is a motion in which the ratio of the distance traveled S to the corresponding time interval remains constant for any time interval, i.e.
S / t \u003d const (constant). (15)
This constant ratio of path to time is called the speed of uniform motion and is denoted by the letter v. Thus, v \u003d S / t. (16)
Solving the equation for S, we get S \u003d vt, (17)
that is, the value of the path traversed by a point during uniform movement is equal to the product of speed and time. Solving the equation for t, we find that t \u003d S / v,(18)
that is, the time during which a point travels a given path with uniform movement is equal to the ratio of this path to the speed of movement.
These equalities are the basic formulas for uniform motion. These formulas determine one of the three quantities S, t, v, when the other two are known.
Dimension of speed v \u003d length / time \u003d m / s.
Uneven movement is such a movement of a point in which the ratio of the distance traveled to the corresponding time interval is not constant.
With uneven movement, points (bodies) 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 \u200b\u200bthe speed of movement of the point at certain moments, i.e., about the true speed.
The true speed of uneven movement is the speed at which the point is moving at the moment.
The average speed of movement of a point is determined by the formula (15).
Almost often they are satisfied with the average speed, taking it as true. For example, the table speed of a planer is constant, with the exception of the moments of the beginning of the working and the beginning of idle strokes, but in most cases these moments are neglected.
In a cross-planer, in which the rotary movement is converted into a translational movement by the rocker mechanism, the speed of the slider is uneven. At the beginning of the turn, 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 turn it becomes zero again. In most cases, the calculations use the average speed v cp of the slide, which is taken as the true cutting speed.
The speed of the cross-planer slider with a rocker mechanism can be characterized as uniformly variable.
Uniformly variable movement is a movement 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 \u003d 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 movement, m / sec; a - acceleration, m / s 2.
Acceleration is the change in speed per unit of time.
Acceleration a has the dimension speed / time \u003d m / sec 2 and is expressed by the formula a \u003d (v-v 0) / t. (20)
For v 0 \u003d 0, a \u003d v / t.
The path traversed with 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 is a movement in which any straight line taken on this body moves parallel to itself.
When moving forward, the speeds and accelerations of all points of the body are the same and at any point are the speed and acceleration of the body.
Rotational motion is a motion in which all points of some straight line (axis) taken in this body remain motionless.
With uniform rotation at equal intervals of time, the body rotates at 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 speed ω and the number of revolutions per minute is expressed by the equation: ω \u003d (2πn) / 60 \u003d (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 during the corresponding period of time.
The speed of movement of a point of a rotating body expressed by the equation
v \u003d (2πRn) / (1000 * 60) \u003d (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 refers to the increase in angular velocity per unit time. It is denoted by the letter ε (epsilon) and is expressed by the formula ε \u003d (ω - ω 0) / t. (24)
Characteristics of mechanical body movement:
- trajectory (the line along which the body moves),
- displacement (a directed line segment connecting the initial position of the body M1 with its subsequent position M2),
- speed (ratio of movement to movement time - for uniform movement) .
The main types of mechanical movement
Depending on the trajectory, the movement of the body is divided into:
Straight-line;
Curvilinear.
Depending on the speed, the movements are divided into:
Uniform,
Equally accelerated
Equal slow
Depending on the method of movement, movements are:
Translational
Rotational
Oscillatory
Complex movements (For example: a helical movement in which the body rotates uniformly around a certain axis and at the same time performs a uniform translational movement along this axis)
Translational motion - this is the movement of the body, in which all its points move in the same way. In forward motion, any straight line connecting any two points of the body remains parallel to itself.
Rotational movement is the movement of a body around a certain axis. With such a movement, all points of the body move along circles, the center of which is this axis.
Oscillatory motion is a periodic motion that occurs alternately in two opposite directions.
For example, a pendulum in a clock makes an oscillatory motion.
Translational and rotational movements are the simplest types of mechanical movement.
Straight-line and uniform movement such a movement is called when, for any arbitrarily small equal intervals of time, the body makes the same displacements . Let's write the mathematical expression of this definition s \u003d υ? t. This means that the displacement is determined by the formula, and the coordinate is determined by the formula .
Equally accelerated movement is the movement of a body, in which its speed for any equal time intervals increases in the same way . To characterize this movement, you need to know the speed of the body at a given moment in time or at a given point of the trajectory, i.e. . e . instantaneous speed as well as acceleration .
Instant speed is the ratio of a sufficiently small displacement on a trajectory segment adjacent to this point to a small time interval during which this displacement is performed .
υ \u003d S / t.The SI unit is m / s.
Acceleration - a value equal to the ratio of the change in speed to the time interval during which this change occurred ... α \u003d? υ / t (SI m / s2) Otherwise, acceleration is the rate at which the rate changes or the increment in rate per second α. t. Hence the formula for instantaneous speed: υ \u003d υ 0 + α.t.
The movement during this movement is determined by the formula: S \u003d υ 0 t + α. t 2/2.
Equal slow motion movement is called when the acceleration has a negative value, while the speed is uniformly slowed down.
With uniform movement around the circumference the angles of rotation of the radius for any equal time intervals will be the same . Therefore, the angular velocity ω \u003d 2πn, or ω \u003d πN / 30 ≈ 0.1N,where ω - angular speed n is the number of revolutions per second, N is the number of revolutions per minute. ω in the SI system is measured in rad / s . (1 / s) / It represents the angular velocity at which each point of the body in one second travels a path equal to its distance from the axis of rotation. During this movement, the velocity module is constant, it is directed tangentially to the trajectory and constantly changes direction (see . rice . ), so there is a centripetal acceleration .
Rotation period T \u003d 1 / n -this time , for which the body makes one complete revolution, therefore ω \u003d 2π / Т.
Linear speed during rotary motion is expressed by the formulas:
υ \u003d ωr, υ \u003d 2πrn, υ \u003d 2πr / T,where r is the distance of a point from the axis of rotation. The linear speed of points lying on the circumference of the shaft or pulley is called the peripheral speed of the shaft or pulley (in SI m / s)
With uniform motion around the circumference, the speed remains constant in magnitude, but changes in direction all the time. Any change in speed is associated with acceleration. Acceleration that changes speed in a direction is called normal or centripetal, this acceleration is perpendicular to the trajectory and directed to the center of its curvature (to the center of the circle, if the trajectory is a circle)
α p \u003d υ 2 / R or α p \u003d ω 2 R (because υ \u003d ωR Where Rcircle radius , υ - point movement speed)
Relativity of mechanical movement is the dependence of the trajectory of the body movement, the distance traveled, movement and speed on the choice reference systems.
The position of a body (point) in space can be determined relative to any other body selected for the reference body A . The reference body, the associated coordinate system and the clock make up the reference frame . The characteristics of mechanical movement are relative, t . e . they can be different in different frames of reference .
Example: two observers are watching the boat's movement: one on the shore at point O, the other on the raft at point O1 (see . rice . ). Let us mentally draw the coordinate system XOY through point O - this is a fixed frame of reference . Another system X "O" Y "we will connect with the raft - this is a moving coordinate system . With respect to the X "O" Y "(raft) system, the boat moves in time t and will move at a speed υ \u003d sboats relative to the raft / t v \u003d (s boats- sraft ) / t. In relation to the XOY (shore) system, the boat will move during the same time s boats where s boat movement relative to shore . Boat speed relative to shore or . The speed of a body relative to a stationary coordinate system is equal to the geometric sum of the speed of a body relative to a moving system and the speed of this system relative to a stationary one .
Types of reference systems can be different, for example, a stationary frame of reference, a moving frame of reference, an inertial frame of reference, a non-inertial frame of reference.
To find the coordinates of a moving body at any time, you need to know the projection 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 is making.
Let's consider first the simplest type of movement - rectilinear uniform motion.
The movement in which the body makes the same movements at any equal intervals is called rectilinear uniform movement.
To find the movement of a body in a uniform rectilinear motion for a certain period of time t, you need to know what kind of movement the 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 denote the letter υ ... If the movement in this section is designated through, and the time interval through t, then the speed can be expressed in relation to. Since displacement is a vector quantity, and time is a scalar one, the speed is also a vector quantity. The velocity vector is directed in the same way as the displacement vector.
The speed of uniform rectilinear movement bodies are called a value equal to the ratio of the movement of the body to the time interval during which this movement occurred:
Thus, the speed shows how much movement the body makes per unit of time. Therefore, to find the displacement of a body, you need to know its speed. The displacement of the body is calculated by the formula:
The displacement vector is directed in the same way as the velocity vector, time t is a scalar quantity.
According to formulas written in vector form, calculations cannot be carried out, since a vector quantity has not only a numerical value, but also a direction. In 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, then their projections on the axis are also equal X, from here:
Now you can get the formula for calculating the coordinate x points at any given time. We know that
It can be seen from this formula that with rectilinear uniform motion, the coordinate of the body linearly depends on time, which means that it can be used to describe rectilinear uniform motion.
In addition, it follows from the formula that to find the position of the body at any moment in time with 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 - the projection of the velocity vector, therefore, like any projection of the vector, it can be positive and negative.
Rectilinear uniform movement is rare. More often we have to deal with movement, in which the movements of the body can be different for equal periods of time. This means that the speed of the body somehow changes over time. Cars, trains, airplanes, etc., a body thrown up, bodies falling to the Earth move with variable speed.
With such a movement, the formula cannot be used to calculate the displacement, since the speed changes in time and we are no longer talking about some 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 what is the displacement that the body makes on average per unit of time.
However, using the concept of average speed, the main problem of mechanics - to determine the position of the body at any moment in time - cannot be solved.