Uniform circular motion. Circular movement. Equation of motion in a circle. Angular velocity. Normal = centripetal acceleration. Period, frequency of circulation (rotation). Relationship between linear and angular velocity Circular motion defined
Topics of the USE codifier: movement in a circle with a constant modulo speed, centripetal acceleration.
Uniform circular motion is a fairly simple example of motion with an acceleration vector that depends on time.
Let the point rotate on a circle of radius . The speed of a point is constant modulo and equal to . The speed is called linear speed points.
Period of circulation is the time for one complete revolution. For the period, we have an obvious formula:
. (1)
Frequency of circulation is the reciprocal of the period:
The frequency indicates how many complete revolutions the point makes per second. The frequency is measured in rpm (revolutions per second).
Let, for example, . This means that during the time the point makes one complete
turnover. The frequency in this case is equal to: about / s; The point makes 10 complete revolutions per second.
Angular velocity.
Consider the uniform rotation of a point in the Cartesian coordinate system. Let's place the origin of coordinates in the center of the circle (Fig. 1).
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| Rice. 1. Uniform circular motion |
Let be the initial position of the point; in other words, for , the point had coordinates . Let the point turn through an angle in time and take the position .
The ratio of the angle of rotation to time is called angular velocity point rotation:
. (2)
Angle is usually measured in radians, so angular velocity is measured in rad/s. For a time equal to the period of rotation, the point rotates through an angle. So
. (3)
Comparing formulas (1) and (3), we obtain the relationship between linear and angular velocities:
. (4)
The law of motion.
Let us now find the dependence of the coordinates of the rotating point on time. We see from Fig. 1 that
But from formula (2) we have: . Hence,
. (5)
Formulas (5) are the solution to the main problem of mechanics for the uniform motion of a point along a circle.
centripetal acceleration.
Now we are interested in the acceleration of the rotating point. It can be found by differentiating relations (5) twice:
Taking into account formulas (5), we have:
(6)
The resulting formulas (6) can be written as a single vector equality:
(7)
where is the radius vector of the rotating point.
We see that the acceleration vector is directed opposite to the radius vector, i.e., towards the center of the circle (see Fig. 1). Therefore, the acceleration of a point moving uniformly in a circle is called centripetal.
In addition, from formula (7) we obtain an expression for the modulus of centripetal acceleration:
(8)
We express the angular velocity from (4)
and substitute into (8) . Let's get one more formula for centripetal acceleration.
1. Uniform movement in a circle
2. Angular speed of rotational movement.
3.Period of rotation.
4.Frequency of rotation.
5. Relationship between linear velocity and angular velocity.
6. Centripetal acceleration.
7. Equally variable movement in a circle.
8. Angular acceleration in uniform motion in a circle.
9. Tangential acceleration.
10. The law of uniformly accelerated motion in a circle.
11. Average angular velocity in uniformly accelerated motion in a circle.
12. Formulas that establish the relationship between angular velocity, angular acceleration and the angle of rotation in uniformly accelerated motion in a circle.
1.Uniform circular motion- movement, in which a material point passes equal segments of a circular arc in equal time intervals, i.e. a point moves along a circle with a constant modulo speed. In this case, the speed is equal to the ratio of the arc of the circle passed by the point to the time of movement, i.e.
and is called the linear speed of motion in a circle.
As in curvilinear motion, the velocity vector is directed tangentially to the circle in the direction of motion (Fig.25).
2. Angular velocity in uniform circular motion is the ratio of the angle of rotation of the radius to the time of rotation:
In uniform circular motion, the angular velocity is constant. In the SI system, angular velocity is measured in (rad/s). One radian - rad is a central angle that subtends an arc of a circle with a length equal to the radius. A full angle contains a radian, i.e. in one revolution, the radius rotates by an angle of radians.
3. Rotation period- the time interval T, during which the material point makes one complete revolution. In the SI system, the period is measured in seconds.
4. Rotation frequency is the number of revolutions per second. In the SI system, the frequency is measured in hertz (1Hz = 1). One hertz is the frequency at which one revolution is made in one second. It is easy to imagine that
If in time t the point makes n revolutions around the circle, then .
Knowing the period and frequency of rotation, the angular velocity can be calculated by the formula:
5 Relationship between linear velocity and angular velocity. The length of the arc of a circle is where the central angle, expressed in radians, subtending the arc is the radius of the circle. Now we write the linear velocity in the form
It is often convenient to use formulas: or Angular velocity is often called the cyclic frequency, and the frequency is called the linear frequency.
6. centripetal acceleration. In uniform motion along a circle, the speed modulus remains unchanged, and its direction is constantly changing (Fig. 26). This means that a body moving uniformly in a circle experiences an acceleration that is directed towards the center and is called centripetal acceleration.
Let a path equal to the arc of a circle pass over a period of time. We move the vector , leaving it parallel to itself, so that its beginning coincides with the beginning of the vector at point B. The modulus of change in speed is , and the modulus of centripetal acceleration is
In Fig. 26, triangles AOB and DVS are isosceles and the angles at the vertices O and B are equal, as are the angles with mutually perpendicular sides AO and OB. This means that triangles AOB and DVS are similar. Therefore, if that is, the time interval takes on arbitrarily small values, then the arc can be approximately considered equal to the chord AB, i.e. . Therefore, we can write Considering that VD= , ОА=R we obtain Multiplying both parts of the last equality by , we will further obtain the expression for the module of centripetal acceleration in uniform motion in a circle: . Given that we get two frequently used formulas:
So, in uniform motion along a circle, the centripetal acceleration is constant in absolute value.
It is easy to figure out that in the limit at , angle . This means that the angles at the base of the DS of the ICE triangle tend to the value , and the velocity change vector becomes perpendicular to the velocity vector , i.e. directed along the radius towards the center of the circle.
7. Uniform circular motion- movement in a circle, in which for equal intervals of time the angular velocity changes by the same amount.
8. Angular acceleration in uniform circular motion is the ratio of the change in the angular velocity to the time interval during which this change occurred, i.e.
where the initial value of the angular velocity, the final value of the angular velocity, angular acceleration, in the SI system is measured in. From the last equality we obtain formulas for calculating the angular velocity
And if .
Multiplying both parts of these equalities by and taking into account that , is the tangential acceleration, i.e. acceleration directed tangentially to the circle, we obtain formulas for calculating the linear velocity:
And if .
9. Tangential acceleration is numerically equal to the change in velocity per unit time and is directed along the tangent to the circle. If >0, >0, then the motion is uniformly accelerated. If<0 и <0 – движение.
10. Law of uniformly accelerated motion in a circle. The path traveled along the circle in time in uniformly accelerated motion is calculated by the formula:
Substituting here , , reducing by , we obtain the law of uniformly accelerated motion in a circle:
Or if .
If the movement is uniformly slowed down, i.e.<0, то
11.Full acceleration in uniformly accelerated circular motion. In uniformly accelerated motion in a circle, the centripetal acceleration increases with time, because due to tangential acceleration, the linear speed increases. Very often centripetal acceleration is called normal and denoted as . Since the total acceleration at the moment is determined by the Pythagorean theorem (Fig. 27).
12. Average angular velocity in uniformly accelerated motion in a circle. The average linear speed in uniformly accelerated motion in a circle is equal to . Substituting here and and reducing by we get
If , then .
12. Formulas that establish the relationship between angular velocity, angular acceleration and the angle of rotation in uniformly accelerated motion in a circle.
Substituting into the formula the quantities , , , ,
and reducing by , we get
Lecture - 4. Dynamics.
1. Dynamics
2. Interaction of bodies.
3. Inertia. The principle of inertia.
4. Newton's first law.
5. Free material point.
6. Inertial frame of reference.
7. Non-inertial frame of reference.
8. Galileo's principle of relativity.
9. Galilean transformations.
11. Addition of forces.
13. Density of substances.
14. Center of mass.
15. Newton's second law.
16. Unit of measurement of force.
17. Newton's third law
1. Dynamics there is a branch of mechanics that studies mechanical motion, depending on the forces that cause a change in this motion.
2.Body interactions. Bodies can interact both with direct contact and at a distance through a special type of matter called the physical field.
For example, all bodies are attracted to each other and this attraction is carried out by means of a gravitational field, and the forces of attraction are called gravitational.
Bodies that carry an electric charge interact through an electric field. Electric currents interact through a magnetic field. These forces are called electromagnetic.
Elementary particles interact through nuclear fields and these forces are called nuclear.
3.Inertia. In the IV century. BC e. The Greek philosopher Aristotle argued that the cause of the movement of a body is a force acting from another body or bodies. At the same time, according to the movement of Aristotle, a constant force imparts a constant speed to the body, and with the termination of the force, the movement stops.
In the 16th century Italian physicist Galileo Galilei, conducting experiments with bodies rolling down an inclined plane and with falling bodies, showed that a constant force (in this case, the weight of the body) imparts acceleration to the body.
So, on the basis of experiments, Galileo showed that the force is the cause of the acceleration of bodies. Let us present Galileo's reasoning. Let a very smooth ball roll on a smooth horizontal plane. If nothing interferes with the ball, then it can roll indefinitely. If, on the way of the ball, a thin layer of sand is poured, then it will stop very soon, because. the friction force of the sand acted on it.
So Galileo came to the formulation of the principle of inertia, according to which a material body maintains a state of rest or uniform rectilinear motion if external forces do not act on it. Often this property of matter is called inertia, and the movement of a body without external influences is called inertia.
4. Newton's first law. In 1687, based on Galileo's principle of inertia, Newton formulated the first law of dynamics - Newton's first law:
A material point (body) is in a state of rest or uniform rectilinear motion if no other bodies act on it, or the forces acting from other bodies are balanced, i.e. compensated.
5.Free material point- a material point, which is not affected by other bodies. Sometimes they say - an isolated material point.
6. Inertial Reference System (ISO)- a reference system, relative to which an isolated material point moves in a straight line and uniformly, or is at rest.
Any frame of reference that moves uniformly and rectilinearly relative to the ISO is inertial,
Here is one more formulation of Newton's first law: There are frames of reference, relative to which a free material point moves in a straight line and uniformly, or is at rest. Such frames of reference are called inertial. Often Newton's first law is called the law of inertia.
Newton's first law can also be given the following formulation: any material body resists a change in its speed. This property of matter is called inertia.
We encounter the manifestation of this law every day in urban transport. When the bus picks up speed sharply, we are pressed against the back of the seat. When the bus slows down, then our body skids in the direction of the bus.
7. Non-inertial frame of reference - a frame of reference that moves non-uniformly relative to the ISO.
A body that, relative to ISO, is at rest or in uniform rectilinear motion. Relative to a non-inertial frame of reference, it moves non-uniformly.
Any rotating frame of reference is a non-inertial frame of reference, since in this system, the body experiences centripetal acceleration.
There are no bodies in nature and technology that could serve as ISO. For example, the Earth rotates around its axis and any body on its surface experiences centripetal acceleration. However, for fairly short periods of time, the reference system associated with the Earth's surface can be considered, in some approximation, the ISO.
8.Galileo's principle of relativity. ISO can be salt you like a lot. Therefore, the question arises: how do the same mechanical phenomena look in different ISOs? Is it possible, using mechanical phenomena, to detect the movement of the IFR in which they are observed.
The answer to these questions is given by the principle of relativity of classical mechanics, discovered by Galileo.
The meaning of the principle of relativity of classical mechanics is the statement: all mechanical phenomena proceed in exactly the same way in all inertial frames of reference.
This principle can also be formulated as follows: all laws of classical mechanics are expressed by the same mathematical formulas. In other words, no mechanical experiments will help us detect the movement of the ISO. This means that trying to detect the movement of the ISO is meaningless.
We encountered the manifestation of the principle of relativity while traveling in trains. At the moment when our train stops at the station, and the train that was standing on the neighboring track slowly starts moving, then in the first moments it seems to us that our train is moving. But it also happens the other way around, when our train is gradually picking up speed, it seems to us that the neighboring train started moving.
In the above example, the principle of relativity manifests itself within small time intervals. With an increase in speed, we begin to feel shocks and rocking of the car, i.e., our frame of reference becomes non-inertial.
So, the attempt to detect the movement of the ISO is meaningless. Therefore, it is absolutely indifferent which IFR is considered fixed and which one is moving.
9. Galilean transformations. Let two IFRs and move relative to each other with a speed . In accordance with the principle of relativity, we can assume that the IFR K is motionless, and the IFR moves relatively at a speed of . For simplicity, we assume that the corresponding coordinate axes of the systems and are parallel, and the axes and coincide. Let the systems coincide at the start time and the motion occurs along the axes and , i.e. (Fig.28)
11. Addition of forces. If two forces are applied to a particle, then the resulting force is equal to their vector, i.e. diagonals of a parallelogram built on vectors and (Fig. 29).
The same rule when decomposing a given force into two components of the force. To do this, on the vector of a given force, as on a diagonal, a parallelogram is built, the sides of which coincide with the direction of the components of the forces applied to the given particle.
If several forces are applied to the particle, then the resulting force is equal to the geometric sum of all forces:
12.Weight. Experience has shown that the ratio of the modulus of force to the modulus of acceleration, which this force imparts to a body, is a constant value for a given body and is called the mass of the body:
From the last equality it follows that the greater the mass of the body, the greater force must be applied to change its speed. Therefore, the greater the mass of the body, the more inert it is, i.e. mass is a measure of the inertia of bodies. The mass defined in this way is called the inertial mass.
In the SI system, mass is measured in kilograms (kg). One kilogram is the mass of distilled water in the volume of one cubic decimeter taken at a temperature
13. Matter density- the mass of a substance contained in a unit volume or the ratio of the mass of a body to its volume
Density is measured in () in the SI system. Knowing the density of the body and its volume, you can calculate its mass using the formula. Knowing the density and mass of the body, its volume is calculated by the formula.
14.Center of mass- a point of the body that has the property that if the direction of the force passes through this point, the body moves translationally. If the direction of action does not pass through the center of mass, then the body moves while simultaneously rotating around its center of mass.
15. Newton's second law. In ISO, the sum of forces acting on a body is equal to the product of the body's mass and the acceleration imparted to it by this force
16.Force unit. In the SI system, force is measured in newtons. One newton (n) is the force that, acting on a body with a mass of one kilogram, imparts an acceleration to it. So .
17. Newton's third law. The forces with which two bodies act on each other are equal in magnitude, opposite in direction and act along one straight line connecting these bodies.
Uniform circular motion is the simplest example. For example, the end of the clock hand moves along the dial along the circle. The speed of a body in a circle is called line speed.
With a uniform motion of the body along a circle, the module of the velocity of the body does not change over time, that is, v = const, and only the direction of the velocity vector changes in this case (a r = 0), and the change in the velocity vector in the direction is characterized by a value called centripetal acceleration() a n or a CA. At each point, the centripetal acceleration vector is directed to the center of the circle along the radius.
The module of centripetal acceleration is equal to
a CS \u003d v 2 / R
Where v is the linear speed, R is the radius of the circle
Rice. 1.22. The movement of the body in a circle.
When describing the motion of a body in a circle, use radius turning angle is the angle φ by which the radius drawn from the center of the circle to the point where the moving body is at that moment rotates in time t. The rotation angle is measured in radians. equal to the angle between two radii of the circle, the length of the arc between which is equal to the radius of the circle (Fig. 1.23). That is, if l = R, then
1 radian= l / R
Because circumference is equal to
l = 2πR
360 o \u003d 2πR / R \u003d 2π rad.
Hence
1 rad. \u003d 57.2958 about \u003d 57 about 18 '
Angular velocity uniform motion of the body in a circle is the value ω, equal to the ratio of the angle of rotation of the radius φ to the time interval during which this rotation is made:
ω = φ / t
The unit of measure for angular velocity is radians per second [rad/s]. The linear velocity modulus is determined by the ratio of the distance traveled l to the time interval t:
v= l / t
Line speed with uniform motion along a circle, it is directed tangentially at a given point on the circle. When the point moves, the length l of the circular arc traversed by the point is related to the angle of rotation φ by the expression
l = Rφ
where R is the radius of the circle.
Then, in the case of uniform motion of the point, the linear and angular velocities are related by the relation:
v = l / t = Rφ / t = Rω or v = Rω
Rice. 1.23. Radian.
Period of circulation- this is the period of time T, during which the body (point) makes one revolution around the circumference. Frequency of circulation- this is the reciprocal of the circulation period - the number of revolutions per unit time (per second). The frequency of circulation is denoted by the letter n.
n=1/T
For one period, the angle of rotation φ of the point is 2π rad, therefore 2π = ωT, whence
T = 2π / ω
That is, the angular velocity is
ω = 2π / T = 2πn
centripetal acceleration can be expressed in terms of the period T and the frequency of revolution n:
a CS = (4π 2 R) / T 2 = 4π 2 Rn 2
Since the linear speed uniformly changes direction, then the movement along the circle cannot be called uniform, it is uniformly accelerated.
Angular velocity
Pick a point on the circle 1 . Let's build a radius. For a unit of time, the point will move to the point 2 . In this case, the radius describes the angle. The angular velocity is numerically equal to the angle of rotation of the radius per unit time.
Period and frequency
Rotation period T is the time it takes the body to make one revolution.
RPM is the number of revolutions per second.
The frequency and period are related by the relationship
Relationship with angular velocity
Line speed
Each point on the circle moves at some speed. This speed is called linear. The direction of the linear velocity vector always coincides with the tangent to the circle. For example, sparks from under a grinder move, repeating the direction of instantaneous speed.
Consider a point on a circle that makes one revolution, the time that is spent - this is the period T. The path traveled by a point is the circumference of a circle.
centripetal acceleration
When moving along a circle, the acceleration vector is always perpendicular to the velocity vector, directed to the center of the circle.
Using the previous formulas, we can derive the following relations
Points lying on the same straight line emanating from the center of the circle (for example, these can be points that lie on the wheel spoke) will have the same angular velocities, period and frequency. That is, they will rotate in the same way, but with different linear speeds. The farther the point is from the center, the faster it will move.
The law of addition of velocities is also valid for rotational motion. If the motion of a body or frame of reference is not uniform, then the law applies to instantaneous velocities. For example, the speed of a person walking along the edge of a rotating carousel is equal to the vector sum of the linear speed of rotation of the edge of the carousel and the speed of the person.
The Earth participates in two main rotational movements: daily (around its axis) and orbital (around the Sun). The period of rotation of the Earth around the Sun is 1 year or 365 days. The Earth rotates around its axis from west to east, the period of this rotation is 1 day or 24 hours. Latitude is the angle between the plane of the equator and the direction from the center of the Earth to a point on its surface.
According to Newton's second law, the cause of any acceleration is a force. If a moving body experiences centripetal acceleration, then the nature of the forces that cause this acceleration may be different. For example, if a body moves in a circle on a rope tied to it, then the acting force is the elastic force.
If a body lying on a disk rotates along with the disk around its axis, then such a force is the force of friction. If the force ceases to act, then the body will continue to move in a straight line
Consider the movement of a point on a circle from A to B. The linear velocity is equal to v A and v B respectively. Acceleration is the change in speed per unit of time. Let's find the difference of vectors.
Among the various types of curvilinear motion, of particular interest is uniform motion of a body in a circle. This is the simplest form of curvilinear motion. At the same time, any complex curvilinear motion of a body in a sufficiently small section of its trajectory can be approximately considered as uniform motion along a circle.
Such a movement is made by points of rotating wheels, turbine rotors, artificial satellites rotating in orbits, etc. With uniform motion in a circle, the numerical value of the speed remains constant. However, the direction of the velocity during such a movement is constantly changing.
The speed of the body at any point of the curvilinear trajectory is directed tangentially to the trajectory at this point. This can be seen by observing the work of a disc-shaped grindstone: pressing the end of a steel rod to a rotating stone, you can see hot particles coming off the stone. These particles fly at the same speed that they had at the moment of separation from the stone. The direction of the sparks always coincides with the tangent to the circle at the point where the rod touches the stone. Sprays from the wheels of a skidding car also move tangentially to the circle.
Thus, the instantaneous velocity of the body at different points of the curvilinear trajectory has different directions, while the modulus of velocity can either be the same everywhere or change from point to point. But even if the modulus of speed does not change, it still cannot be considered constant. After all, speed is a vector quantity, and for vector quantities, the modulus and direction are equally important. So curvilinear motion is always accelerated, even if the modulus of speed is constant.
Curvilinear motion can change the speed modulus and its direction. Curvilinear motion, in which the modulus of speed remains constant, is called uniform curvilinear motion. Acceleration during such movement is associated only with a change in the direction of the velocity vector.
Both the modulus and the direction of acceleration must depend on the shape of the curved trajectory. However, it is not necessary to consider each of its myriad forms. Representing each section as a separate circle with a certain radius, the problem of finding acceleration in a curvilinear uniform motion will be reduced to finding acceleration in a uniform motion of a body around a circle.
Uniform motion in a circle is characterized by a period and frequency of circulation.
The time it takes for a body to make one revolution is called circulation period.
With uniform motion in a circle, the period of revolution is determined by dividing the distance traveled, i.e., the circumference of the circle by the speed of movement:
The reciprocal of a period is called circulation frequency, denoted by the letter ν . Number of revolutions per unit time ν called circulation frequency:
Due to the continuous change in the direction of speed, a body moving in a circle has an acceleration that characterizes the speed of change in its direction, the numerical value of the speed in this case does not change.
When a body moves uniformly along a circle, the acceleration at any point in it is always directed perpendicular to the speed of movement along the radius of the circle to its center and is called centripetal acceleration.
To find its value, consider the ratio of the change in the velocity vector to the time interval for which this change occurred. Since the angle is very small, we have
