Rotation frequency formula through radius. Motion of a point in a circle. Rotational movement of the body, formulas

Among various types curvilinear movement is of particular interest uniform movement of a body in a circle. This is the simplest type of curvilinear movement. At the same time, any complex curvilinear motion of a body in a sufficiently small portion of its trajectory can be approximately considered as uniform motion in a circle.

Such movement is performed by the points of rotating wheels, turbine rotors, artificial satellites rotating in orbits, etc. When uniform motion along the circumference the numerical value of the speed remains constant. However, the direction of speed during such movement continuously changes.

The speed of movement of a body at any point on a curvilinear trajectory is directed tangentially to the trajectory at that point. You can verify this by observing the operation of a disk-shaped sharpener: pressing the end of a steel rod against a rotating stone, you can see hot particles coming off the stone. These particles fly at the speed they had at the moment they left the stone. The direction of the sparks always coincides with the tangent to the circle at the point where the rod touches the stone. The splashes from the wheels of a skidding car also move tangentially to the circle.

Thus, the instantaneous velocity of a body at different points of a curvilinear trajectory has different directions, while the magnitude of the velocity can either be the same everywhere or vary from point to point. But even if the speed module does not change, it still cannot be considered constant. After all, speed is a vector quantity, and for vector quantities, modulus and direction are equally important. That's why curvilinear motion is always accelerated, even if the speed module is constant.

During curvilinear motion, the velocity module and its direction may change. Curvilinear motion in which the velocity modulus remains constant is called uniform curvilinear movement. Acceleration during such movement is associated only with a change in the direction of the velocity vector.

Both the magnitude and direction of acceleration must depend on the shape of the curved trajectory. However, there is no need to consider each of its countless forms. Having imagined each section as a separate circle with a certain radius, the problem of finding acceleration during curvilinear uniform motion will be reduced to finding acceleration during uniform motion of a body in a circle.

Uniform circular motion is characterized by the period and frequency of revolution.

The time it takes 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 by the speed of movement:

The reciprocal of the period is called frequency of circulation, denoted by the letter ν . Number of revolutions per unit time ν called frequency of circulation:

Due to the continuous change in the direction of speed, a body moving in a circle has an acceleration, which 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 around a circle, the acceleration at any point 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 during which this change occurred. Since the angle is very small, we have.

Since linear speed uniformly changes direction, the circular motion cannot be called uniform, it is uniformly accelerated.

Angular velocity

Let's choose a point on the circle 1 . Let's construct the radius. In a unit of time, the point will move to point 2 . In this case, the radius describes the angle. Angular velocity is numerically equal to the angle of rotation of the radius per unit time.

Period and frequency

Rotation period T- this is the time during which the body makes one revolution.

Rotation frequency is the number of revolutions per second.

Frequency and period are interrelated by the relationship

Relationship with angular velocity

Linear speed

Each point on the circle moves at a certain 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 grinding machine move, repeating the direction of instantaneous speed.

Consider a point on a circle that makes one revolution, the time spent is the period T. The path that a point travels is the circumference.

Centripetal acceleration

When moving in a circle, the acceleration vector is always perpendicular to the velocity vector, directed towards the center of the circle.

Using the previous formulas, we can derive the following relationships

Points lying on the same straight line emanating from the center of the circle (for example, these could be points that lie on the spokes of a wheel) will have the same angular velocities, period and frequency. That is, they will rotate the same way, but with different linear speeds. The further a point is from the center, the faster it will move.

The law of addition of speeds 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 speeds. 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: diurnal (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 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 with the disk around its axis, then such a force is the friction force. If the force stops its action, 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 speed is equal to v A And v B respectively. Acceleration is the change in speed per unit time. Let's find the difference between the vectors.

Rotational motion around a fixed axis is another special case of motion solid.
Rotational movement of a rigid body around a fixed axis it is called such a movement in which all points of the body describe circles, the centers of which are on the same straight line, called the axis of rotation, while the planes to which these circles belong are perpendicular rotation axis (Fig.2.4).

In technology, this type of motion occurs very often: for example, the rotation of the shafts of engines and generators, turbines and aircraft propellers.
Angular velocity . Each point of a body rotating around an axis passing through the point ABOUT, moves in a circle, and different points travel different paths over time. So, , therefore the modulus of the point velocity A more than a point IN (Fig.2.5). But the radii of the circles rotate through the same angle over time. Angle - the angle between the axis OH and radius vector, which determines the position of point A (see Fig. 2.5).

Let the body rotate uniformly, i.e., rotate through equal angles at any equal intervals of time. The speed of rotation of a body depends on the angle of rotation of the radius vector, which determines the position of one of the points of the rigid body for a given period of time; it is characterized angular velocity . For example, if one body rotates through an angle every second, and the other through an angle, then we say that the first body rotates 2 times faster than the second.
Angular velocity of a body during uniform rotation is a quantity equal to the ratio of the angle of rotation of the body to the period of time during which this rotation occurred.
We will denote the angular velocity by the Greek letter ω (omega). Then by definition

Angular velocity is expressed in radians per second (rad/s).
For example, the angular velocity of the Earth's rotation around its axis is 0.0000727 rad/s, and that of a grinding disk is about 140 rad/s 1 .
Angular velocity can be expressed through rotation speed , i.e. the number of full revolutions in 1s. If a body makes (Greek letter “nu”) revolutions in 1 s, then the time of one revolution is equal to seconds. This time is called rotation period and denoted by the letter T. Thus, the relationship between frequency and rotation period can be represented as:

A complete rotation of the body corresponds to an angle. Therefore, according to formula (2.1)

If during uniform rotation the angular velocity is known and at the initial moment of time the angle of rotation is , then the angle of rotation of the body during time t according to equation (2.1) is equal to:

If , then , or .
Angular velocity takes positive values ​​if the angle between the radius vector, which determines the position of one of the points of the rigid body, and the axis OH increases, and negative when it decreases.
Thus, we can describe the position of the points of a rotating body at any time.
Relationship between linear and angular velocities. The speed of a point moving in a circle is often called linear speed , to emphasize its difference from angular velocity.
We have already noted that when a rigid body rotates, its different points have unequal linear velocities, but the angular velocity is the same for all points.
There is a relationship between the linear speed of any point of a rotating body and its angular speed. Let's install it. A point lying on a circle of radius R, per revolution will go the way. Since the time of one revolution of a body is a period T, then the modulus of the linear velocity of the point can be found as follows:

Topics of the Unified State Examination codifier: motion in a circle with a constant absolute speed, centripetal acceleration.

Uniform movement around a circle - This is a fairly simple example of motion with an acceleration vector that depends on time.

Let the point rotate along a circle of radius . The speed of the point is constant in absolute value and equal to . Speed ​​is called linear speed points.

Circulation period - this is the time of one full revolution. For the period we have an obvious formula:

. (1)

Frequency is the reciprocal of the period:

Frequency shows how many full revolutions a point makes per second. The frequency is measured in rps (revolutions per second).

Let, for example, . This means that during the time the point makes one complete
turnover The frequency is then equal to: r/s; per second the point makes 10 full revolutions.

Angular velocity.

Let's consider the uniform rotation of a point in a Cartesian coordinate system. Let's place the origin of coordinates at the center of the circle (Fig. 1).

Rice. 1. Uniform movement in a circle

Let be the initial position of the point; in other words, at the point had coordinates . Let the point turn through an angle and take position .

The ratio of the angle of rotation to time is called angular velocity point rotation:

. (2)

Angle is typically measured in radians, so angular velocity is measured in rad/s. In a time equal to the rotation period, the point rotates through an angle. That's why

. (3)

Comparing formulas (1) and (3), we obtain the relationship between linear and angular velocities:

. (4)

Law of motion.

Let us now find the dependence of the coordinates of the rotating point on time. We see from Fig.

1 that

. (5)

But from formula (2) we have: . Hence,

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:

(6)

Taking into account formulas (5) we have:

(7)

The resulting formulas (6) can be written as one vector equality:

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 around a circle is called

In addition, from formula (7) we obtain an expression for the modulus of centripetal acceleration:

(8)

Let us express the angular velocity from (4)

and substitute it into (8). Let's get another formula for centripetal acceleration.

1.Uniform movement in a circle

2. Angular speed of rotational motion.

3. Rotation period.

4. Rotation speed.

5. Relationship between linear speed and angular speed.

6.Centripetal acceleration.

7. Equally alternating movement in a circle.

8. Angular acceleration in uniform circular motion.

9.Tangential acceleration.

10. Law of uniformly accelerated motion in a circle.

11. Average angular velocity in uniformly accelerated motion in a circle.

12. Formulas establishing the relationship between angular velocity, angular acceleration and angle of rotation in uniformly accelerated motion in a circle.

1.Uniform movement around a circle- a movement in which material point in equal intervals of time passes equal segments of an arc of a circle, i.e. the point moves in a circle with a constant absolute speed. In this case, the speed is equal to the ratio of the arc of a circle traversed by the point to the time of movement, i.e.

and is called the linear speed of movement 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– ratio of the radius rotation angle to the rotation time:

In uniform circular motion, the angular velocity is constant. In the SI system, angular velocity is measured in (rad/s). One radian - a rad is the central angle subtending an arc of a circle with a length equal to the radius. A full angle contains radians, i.e. per revolution the radius rotates by an angle of radians.

3. Rotation period– time interval T during which a material point makes one full revolution. In the SI system, the period is measured in seconds.

4. Rotation frequency– the number of revolutions made in one second. In the SI system, frequency is measured in hertz (1Hz = 1). One hertz is the frequency at which one revolution is completed in one second. It's easy to imagine that

If during time t a point makes n revolutions around a circle then .

Knowing the period and frequency of rotation, the angular velocity can be calculated using the formula:

5 Relationship between linear speed and angular speed. The length of an arc of a circle is equal to where is the central angle, expressed in radians, the radius of the circle subtending the arc. Now we write the linear speed in the form

It is often convenient to use the formulas: or Angular velocity is often called cyclic frequency, and frequency is called linear frequency.

6. Centripetal acceleration. In uniform motion around a circle, the velocity module remains unchanged, but its direction continuously changes (Fig. 26). This means that a body moving uniformly in a circle experiences acceleration, which is directed towards the center and is called centripetal acceleration.

Let a distance travel equal to an arc of a circle in a period of time. Let's 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 equal to , and the modulus of centripetal acceleration is equal

In Fig. 26, the 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 the triangles AOB and DVS are similar. Therefore, if, that is, the time interval takes arbitrarily small values, then the arc can be approximately considered equal to the chord AB, i.e. . Therefore, we can write Considering that VD = , OA = R we obtain Multiplying both sides of the last equality by , we further obtain the expression for the modulus of centripetal acceleration in uniform motion in a circle: . Considering that we get two frequently used formulas:

So, in uniform motion around a circle, the centripetal acceleration is constant in magnitude.

It is easy to understand 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 speed change vector becomes perpendicular to the speed vector, i.e. directed radially towards the center of the circle.

7. Equally alternating circular motion– circular motion in which the angular velocity changes by the same amount over equal time intervals.

8. Angular acceleration in uniform circular motion– the ratio of the change in angular velocity to the time interval during which this change occurred, i.e.

where the initial value of angular velocity, the final value of 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 sides 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 linear speed:

And if .

9. Tangential acceleration numerically equal to the change in speed per unit time and 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 around a circle in time in uniformly accelerated motion is calculated by the formula:

Substituting , , and reducing by , we obtain the law of uniformly accelerated motion in a circle:

Or if .

If the movement is uniformly slow, i.e.<0, то

11.Total acceleration in uniformly accelerated circular motion. In uniformly accelerated motion in a circle, centripetal acceleration increases over time, because Due to tangential acceleration, linear speed increases. Very often, centripetal acceleration is called normal and is denoted as. Since the total acceleration at a given 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 establishing the relationship between angular velocity, angular acceleration and angle of rotation in uniformly accelerated motion in a circle.

Substituting the quantities , , , , into the formula

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 reference system.

7. Non-inertial reference system.

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 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.Interactions of bodies. Bodies can interact both in direct contact and at a distance through a special type of matter called a physical field.

For example, all bodies are attracted to each other and this attraction is carried out through the gravitational field, and the forces of attraction are called gravitational.

Bodies carrying 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 4th century. BC e. The Greek philosopher Aristotle argued that the cause of the movement of a body is the force acting from another body or bodies. At the same time, according to Aristotle’s movement, a constant force imparts a constant speed to the body and, with the cessation of the action of the force, the movement ceases.

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 a body) imparts acceleration to the body.

So, based on experiments, Galileo showed that force is the cause of the acceleration of bodies. Let us present Galileo's reasoning. Let a very smooth ball roll along a smooth horizontal plane. If nothing interferes with the ball, then it can roll for as long as desired. If a thin layer of sand is poured on the path of the ball, it will stop very soon, because it was affected by the frictional force of the sand.

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 no external forces act on it. This property of matter is often called inertia, and the movement of a body without external influences is called motion by 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 linear motion if other bodies do not act on it, or the forces acting from other bodies are balanced, i.e. compensated.

5.Free material point- a material point that is not affected by other bodies. Sometimes they say - an isolated material point.

6. Inertial reference system (IRS)– a reference system relative to which an isolated material point moves rectilinearly and uniformly, or is at rest.

Any reference system that moves uniformly and rectilinearly relative to the ISO is inertial,

Let us give another formulation of Newton's first law: There are reference systems relative to which a free material point moves rectilinearly and uniformly, or is at rest. Such reference systems are called inertial. Newton's first law is often called the law of inertia.

Newton's first law can also be given the following formulation: every material body resists a change in its speed. This property of matter is called inertia.

We encounter manifestations of this law every day in urban transport. When the bus suddenly picks up speed, we are pressed against the back of the seat. When the bus slows down, our body skids in the direction of the bus.

7. Non-inertial reference system – a reference system that moves unevenly relative to the ISO.

A body that, relative to the ISO, is in a state of rest or uniform linear motion. It moves unevenly relative to a non-inertial reference frame.

Any rotating reference system is a non-inertial reference system, because in this system the body experiences centripetal acceleration.

There are no bodies in nature or technology that could serve as ISOs. 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, to some approximation, be considered ISO.

8.Galileo's principle of relativity. ISO can be as much salt as you like. Therefore, the question arises: what do the same mechanical phenomena look like in different ISOs? Is it possible, using mechanical phenomena, to detect the movement of the ISO 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 exactly the same way in all inertial frames of reference.

This principle can 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 ISO movement is pointless.

We encountered the manifestation of the principle of relativity while traveling on trains. At the moment when our train is standing at the station, and the train standing on the adjacent track slowly begins to move, 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 smoothly picks up speed, it seems to us that the neighboring train has started moving.

In the above example, the principle of relativity manifests itself over small time intervals. As the speed increases, we begin to feel shocks and swaying of the car, i.e. our reference system becomes non-inertial.

So, trying to detect ISO movement is pointless. Consequently, it is absolutely indifferent which ISO is considered stationary and which is moving.

9. Galilean transformations. Let two ISOs move relative to each other with a speed. In accordance with the principle of relativity, we can assume that the ISO K is stationary, and the ISO moves relatively at a speed. 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 moment of beginning and the movement occurs along the axes and , i.e. (Fig.28)