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 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 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)
Mechanical movement is a change in the position of a body in space relative to other bodies.
For example, a car is moving along the road. There are people in the car. People move along with the car along the road. That is, people move in space relative to the road. But relative to the car itself, people do not move. This shows up. Next we will briefly consider main types of mechanical movement.
Forward movement- this is the movement of a body in which all its points move equally.
For example, the same car makes forward motion along the road. More precisely, only the body of the car performs translational motion, while its wheels perform rotational motion.
Rotational movement is the movement of a body around a certain axis. With such a movement, all points of the body move in circles, the center of which is this axis.
The wheels we mentioned perform rotational motion around their axes, and at the same time, the wheels perform translational motion along with the car body. That is, the wheel makes a rotational movement relative to the axis, and a translational movement relative to the road.
Oscillatory motion- This is a periodic movement that occurs alternately in two opposite directions.
For example, a pendulum in a clock performs an oscillatory motion.
Translational and rotational movements are the simplest types of mechanical movement.
Relativity of mechanical motion
All bodies in the Universe move, so there are no bodies that are at absolute rest. For the same reason, it is possible to determine whether a body is moving or not only relative to some other body.
For example, a car is moving along the road. The road is located on planet Earth. The road is still. Therefore, it is possible to measure the speed of a car relative to a stationary road. But the road is stationary relative to the Earth. However, the Earth itself revolves around the Sun. Consequently, the road along with the car also revolves around the Sun. Consequently, the car makes not only translational motion, but also rotational motion (relative to the Sun). But relative to the Earth, the car makes only translational motion. This shows relativity of mechanical motion.
Relativity of mechanical motion– this is the dependence of the trajectory of the body, the distance traveled, movement and speed on the choice reference systems.
Material point
In many cases, the size of a body can be neglected, since the dimensions of this body are small compared to the distance that this body moves, or compared to the distance between this body and other bodies. To simplify calculations, such a body can conventionally be considered a material point that has the mass of this body.
Material point is a body whose dimensions can be neglected under given conditions.
The car we have mentioned many times can be taken as a material point relative to the Earth. But if a person moves inside this car, then it is no longer possible to neglect the size of the car.
As a rule, when solving problems in physics, we consider the movement of a body as motion of a material point, and operate with such concepts as the speed of a material point, the acceleration of a material point, the momentum of a material point, the inertia of a material point, etc.
Frame of reference
A material point moves relative to other bodies. The body in relation to which this mechanical movement is considered is called the body of reference. Reference body are chosen arbitrarily depending on the tasks to be solved.
Associated with the reference body coordinate system, which is the reference point (origin). The coordinate system has 1, 2 or 3 axes depending on the driving conditions. The position of a point on a line (1 axis), plane (2 axes) or in space (3 axes) is determined by one, two or three coordinates, respectively. To determine the position of the body in space at any moment in time, it is also necessary to set the beginning of the time count.
Frame of reference is a coordinate system, a reference body with which the coordinate system is associated, and a device for measuring time. The movement of the body is considered relative to the reference system. The same body relative to different reference bodies in different coordinate systems can have completely different coordinates.
Trajectory of movement also depends on the choice of reference system.
Types of reference systems can be different, for example, a fixed reference system, a moving reference system, an inertial reference system, a non-inertial reference system.
Movements humans are very diverse, but all this diversity can be reduced to a small number of basic types of activity: ensuring posture and balance, locomotion (active movement in space over distances significantly exceeding the characteristic dimensions of the body) and voluntary movements.
Maintaining posture in humans is ensured by the same physical muscles that perform movement, and there are no specialized tonic muscles. During “postural” muscle activity, the force of their contraction is usually small, the mode is close to isometric indicators, and the duration of contraction is significant. The “postural” or postural mode of muscle work primarily involves low-threshold, slow, and fatigue-resistant motor units.
One of the main tasks of “postural” activity is maintaining the desired position of the body parts in the field of gravity (keeping the head from hanging, keeping the ankle joints from dorsiflexing when standing, etc.). “Postural” activity can also be aimed at fixing joints that do not participate in the movement being performed. In work activity, maintaining a pose is associated with overcoming external forces.
A typical example of a pose is a person standing. Maintaining balance while standing is possible if the projection of the body's center of gravity is within the support contour. Ensuring stability is achieved by the active work of many muscles of the trunk and legs, and the force developed by these muscles is small. The maximum tension when standing is developed by the muscles of the ankle joint, and the minimum tension is developed by the muscles of the knee and hip joints. In most muscles, activity is maintained at a more or less constant level. Other muscles are activated periodically. This activation is associated with small fluctuations in the body's center of gravity in both the sagittal and frontal planes that constantly occur during standing. The muscles of the lower leg counteract the deviations of the body, returning it to an upright position. Maintaining a posture is an active process that, like movement, involves feedback from receptors. Vision and the vestibular apparatus are involved in maintaining a vertical posture. Proprioception also plays an important role. Maintaining balance while standing is only a special case of “postural” activity.
Related to the concept of posture is the concept of muscle tone. The term “tone” has many meanings. At rest, muscle fibers have turgor, which determines their resistance to pressure and stretching. This constitutes that component of tone that is not associated with specific neural activation of the muscle causing its contraction. However, under natural conditions, most muscles are usually activated to some extent by the nervous system, in particular to maintain posture (“postural” tone). Another important component of tone is the reflex component, determined by the stretch reflex. In humans, it is detected by the resistance to muscle stretching during passive rotation of a limb link in the joint.
The most common form of human locomotion is walking. It refers to cyclic motor acts in which successive phases of movement are periodically repeated.
Running differs from walking in that the leg that is behind you pushes off the support before the other leg comes down on it. As a result, running has a period without support, a period of flight.
Voluntary movements in a broad sense can be called a variety of movements performed both in the process of work and in everyday life.
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 moment in 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.
Characteristics of mechanical body movement:
- trajectory (line along which the body moves),
- displacement (directed straight 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) .
Main types of mechanical movement
Depending on the trajectory, body movement is divided into:
Straight-line;
Curvilinear.
Depending on the speed, movements are divided into:
Uniform,
Uniformly accelerated
Equally slow
Depending on the method of movement, movements are:
Progressive
Rotational
Oscillatory
Complex movements (For example: a screw movement in which the body rotates uniformly around a certain axis and at the same time makes a uniform translational movement along this axis)
Forward movement - This is the movement of a body in which all its points move equally. In translational motion, any straight line connecting any two points of the body remains parallel to itself.
Rotational motion is the movement of a body around a certain axis. With such a movement, all points of the body move in 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 performs an oscillatory motion.
Translational and rotational movements are the simplest types of mechanical movement.
Straight and uniform movement is called such a movement when, for any arbitrarily small equal intervals of time, the body makes identical movements . Let us write down the mathematical expression of this definition s = v? t. This means that the displacement is determined by the formula, and the coordinate - by the formula .
Uniformly accelerated motion is the movement of a body in which its speed increases equally over any equal intervals of time . To characterize this movement, you need to know the speed of the body at a given moment in time or at a given point in the trajectory, t . e . instantaneous speed and acceleration .
Instantaneous speed- this is the ratio of a sufficiently small movement on the section of the trajectory adjacent to this point to the small period of time during which this movement occurs .
υ = S/t. The SI unit is m/s.
Acceleration is a quantity equal to the ratio of the change in speed to the period of time during which this change occurred . α = ?υ/t(SI system m/s2) Otherwise, acceleration is the rate of change of speed or the increase in speed for each second α. t. Hence the formula for instantaneous speed: υ = υ 0 + α.t.
The displacement during this movement is determined by the formula: S = υ 0 t + α . t 2 /2.
Equally slow motion motion is called when the acceleration is negative and the speed uniformly slows down.
With uniform circular motion the angles of rotation of the radius for any equal periods of time will be the same . Therefore the angular speed ω = 2πn, or ω = πN/30 ≈ 0.1N, Where ω - angular speed n - number of revolutions per second, N - number of revolutions per minute. ω in the SI system it is measured in rad/s . (1/c)/ 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 . ), therefore centripetal acceleration occurs .
Rotation period T = 1/n - this time , during which the body makes one full revolution, therefore ω = 2π/T.
Linear speed during rotational motion is expressed by the formulas:
υ = ωr, υ = 2πrn, υ = 2πr/T, where r is the distance of the point from the axis of rotation. The linear speed of points lying on the circumference of a shaft or pulley is called the peripheral speed of the shaft or pulley (in SI m/s)
With uniform motion in a circle, 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 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 = υ 2 /R or α p = ω 2 R(because υ = ωR Where R circle radius , υ - point movement speed)
Relativity of mechanical motion- this is the dependence of the trajectory of the body, the distance traveled, movement and speed on the choice reference systems.
The position of a body (point) in space can be determined relative to some other body chosen as the reference body A . The reference body, the coordinate system associated with it, and the clock constitute the reference system . The characteristics of mechanical movement are relative, t . e . they can be different in different reference systems .
Example: the movement of a boat is monitored by two observers: one on the shore at point O, the other on the raft at point O1 (see . rice . ). Let us mentally draw through the point O the XOY coordinate system - this is a fixed reference system . We will connect another X"O"Y" system to the raft - this is a moving coordinate system . Relative to the X"O"Y" system (raft), the boat moves in time t and will move at speed υ = s boats relative to raft /t v = (s boats- s raft )/t. Relative to the XOY (shore) system, the boat will move during the same time s boats where s boatsmoving the raft relative to the shore . Speed of the boat relative to the shore or . The speed of a body relative to a fixed coordinate system is equal to the geometric sum of the speed of the body relative to a moving system and the speed of this system relative to a fixed one .
Types of reference systems can be different, for example, a fixed reference system, a moving reference system, an inertial reference system, a non-inertial reference system.
Loading...
