Rotation speed. Rotational motion How to derive the formula for centripetal acceleration

Angular velocity

Let's choose a point on the circle 1 2

Period and frequency

Rotation period T

Relationship with angular velocity

Linear speed

Earth Rotation

vA And vB

There is a vector difference . Since we get

Movement along a cycloid*

The number of repetitions of any events or their occurrence in one timer unit is called frequency. This physical quantity is measured in hertz – Hz (Hz). It is denoted by the letters ν, f, F, and is the ratio of the number of repeating events to the period of time during which they occurred.

When an object rotates around its center, we can talk about such a physical quantity as the frequency of rotation, formula:

N – number of revolutions around an axis or in a circle,
t is the time during which they were completed.

In the SI system it is denoted as – s-1 (s-1) and is referred to as revolutions per second (rps). Other units of rotation are also used. When describing the rotation of planets around the Sun, they speak of revolutions in hours. Jupiter rotates once every 9.92 hours, while the Earth and Moon rotate every 24 hours.

Rated rotation speed

Before defining this concept, it is necessary to determine what the nominal operating mode of a device is. This is the order of operation of the device in which the greatest efficiency and reliability of the process are achieved over a long period of time. Based on this, the nominal rotation speed is the number of revolutions per minute when operating in nominal mode. The time required for one revolution is 1/v seconds. It is called the rotation period T. This means that the relationship between the period of revolution and frequency has the form:

FYI. The rotational speed of the asynchronous motor shaft is 3000 rpm, this is the rated rotation speed of the output shaft shank at the nominal operating mode of the electric motor.

How to find or find out the rotation frequencies of various mechanisms? For this, a device called a tachometer is used.

Angular velocity

When a body moves in a circle, not all its points move at the same speed relative to the axis of rotation. If we take the blades of an ordinary household fan that rotate around a shaft, then the point located closer to the shaft has a rotation speed greater than the marked point on the edge of the blade. This means they have different linear rotation speeds. At the same time, the angular velocity of all points is the same.

Angular velocity is the change in angle per unit time, not distance. It is denoted by the letter of the Greek alphabet – ω and has a unit of measure: radians per second (rad/s). In other words, angular velocity is a vector tied to the axis of rotation of the object.

The formula for calculating the relationship between rotation angle and time interval is:

ω – angular velocity (rad/s);
∆ϕ – change in the angle of deflection when turning (rad.);
∆t – time spent on deviation (s).

The designation of angular velocity is used when studying the laws of rotation. It is used to describe the motion of all rotating bodies.

Angular velocity in specific cases

In practice, they rarely work with angular velocity values. It is needed in the design development of rotating mechanisms: gearboxes, gearboxes, etc.

You can calculate it using the formula. To do this, use the connection between angular velocity and rotational speed.

π – number equal to 3.14;
ν – rotation speed, (rpm).

As an example, the angular velocity and rotational speed of the wheel disk when moving a walk-behind tractor can be considered. It is often necessary to reduce or increase the speed of the mechanism. To do this, a device in the form of a gearbox is used, with the help of which the speed of rotation of the wheels is reduced. At a maximum speed of 10 km/h, the wheel makes about 60 rpm. After converting minutes to seconds, this value is 1 rpm. After substituting the data into the formula, the result will be:

ω = 2*π*ν = 2*3.14*1 = 6.28 rad/s.

FYI. A reduction in angular velocity is often required in order to increase the torque or tractive effort of mechanisms.

How to determine angular velocity

The principle of determining angular velocity depends on how the circular motion occurs. If uniform, then the formula is used:

If not, then you will have to calculate the values of the instantaneous or average angular velocity.

The quantity we are talking about is a vector quantity, and Maxwell’s rule is used to determine its direction. In common parlance - the gimlet rule. The velocity vector has the same direction as the translational movement of a screw with a right-hand thread.

Let's look at an example of how to determine the angular velocity, knowing that the angle of rotation of a disk with a radius of 0.5 m varies according to the law ϕ = 6*t:

ω = ϕ / t = 6 * t / t = 6 s-1

The vector ω changes due to rotation in space of the rotation axis and when the value of the angular velocity modulus changes.

Rotation angle and period of revolution

Consider point A on an object rotating around its axis. When circulating over a certain period of time, it will change its position on the circle line by a certain angle. This is the rotation angle. It is measured in radians, because the unit is a segment of a circle equal to the radius. Another value for measuring the angle of rotation is a degree.

When, as a result of the rotation, point A returns to its original place, it means that it has completed a full rotation. If its movement is repeated n times, then we speak of a certain number of revolutions. Based on this, you can consider 1/2, 1/4 turn and so on. A striking practical example of this is the path that a cutter takes when milling a part fixed in the center of the machine spindle.

Attention! The rotation angle has a direction. It is negative when the rotation occurs clockwise and positive when it rotates counterclockwise.

If a body moves uniformly around a circle, we can talk about a constant angular velocity during movement, ω = const.

In this case, the following characteristics are used:

period of revolution – T, this is the time required for a full revolution of a point in a circular motion;
circulation frequency – ν, this is the total number of revolutions that a point makes along a circular path in a unit time interval.

Interesting. According to known data, Jupiter revolves around the Sun every 12 years. When the Earth makes almost 12 revolutions around the Sun during this time. The exact value of the round giant's orbital period is 11.86 Earth years.

Cyclic speed (reversal)

A scalar quantity that measures the frequency of rotational motion is called cyclic speed. This is the angular frequency, which is not equal to the angular velocity vector itself, but to its magnitude. It is also called radial or circular frequency.

Cyclic rotation frequency is the number of body revolutions in 2*π seconds.

For AC electric motors, this frequency is asynchronous. Their rotor speed lags behind the rotation speed of the stator magnetic field. The value that determines this lag is called slip - S. During the sliding process, the shaft rotates because an electric current arises in the rotor. Slip is permissible up to a certain value, exceeding which leads to overheating of the asynchronous machine, and its windings may burn out.

The design of this type of motor differs from the design of DC machines, where a current-carrying frame rotates in the field of permanent magnets. The armature contained a large number of frames, and many electromagnets formed the basis of the stator. In three-phase AC machines the opposite is true.

When an asynchronous motor operates, the stator has a rotating magnetic field. It always depends on the parameters:

mains frequency;
number of pole pairs.

The rotation speed of the rotor is in direct relation to the speed of the stator magnetic field. The field is created by three windings, which are located at an angle of 120 degrees relative to each other.

Transition from angular to linear velocity

There is a difference between the linear velocity of a point and the angular velocity. When comparing the quantities in the expressions describing the rules of rotation, you can see the commonality between these two concepts. Any point B belonging to a circle with radius R makes a path equal to 2*π*R. At the same time, it makes one revolution. Considering that the time required for this is the period T, the modular value of the linear speed of point B is found by the following action:

ν = 2*π*R / Т = 2*π*R* ν.

Since ω = 2*π*ν, it turns out:

Consequently, the linear speed of point B is greater, the further away the point is from the center of rotation.

FYI. If we consider cities at the latitude of St. Petersburg as such a point, their linear speed relative to the earth’s axis is 233 m/s. For objects on the equator – 465 m/s.

The numerical value of the acceleration vector of point B, moving uniformly, is expressed through R and angular velocity, thus:

a = ν2/ R, substituting here ν = ω* R, we get: a = ν2/ R = ω2* R.

This means that the larger the radius of the circle along which point B moves, the greater the value of its acceleration in absolute value. The farther a point of a rigid body is located from the axis of rotation, the greater the acceleration it has.

Therefore, it is possible to calculate accelerations, velocity modules of the required points of bodies and their positions at any time.

Understanding and ability to use calculations and not get confused in definitions will help in practice to calculate linear and angular velocities, as well as freely move from one quantity to another when making calculations.

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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 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.

Earth Rotation

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.

Connection to Newton's second law

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

How to derive the formula for centripetal acceleration

Consider the movement of a point on a circle from A to B. The linear speed is equal to vA And vB respectively. Acceleration is the change in speed per unit time. Let's find the difference between the vectors.

There is a vector difference . Since we get

Movement along a cycloid*

In the reference frame associated with the wheel, the point rotates uniformly along a circle of radius R with a speed that changes only in direction. The centripetal acceleration of a point is directed radially towards the center of the circle.

Now let's move to a stationary system connected to the earth. The total acceleration of point A will remain the same both in magnitude and direction, since when moving from one inertial reference system to another, the acceleration does not change. From the point of view of a stationary observer, the trajectory of point A is no longer a circle, but a more complex curve (cycloid), along which the point moves unevenly.

Instantaneous speed is determined by the formula

When designing equipment, it is necessary to know the speed of the electric motor. To calculate the rotation speed, there are special formulas that are different for AC and DC motors.

Synchronous and asynchronous electric machines

There are three types of AC motors: synchronous, the angular speed of the rotor coincides with the angular frequency of the stator magnetic field; asynchronous - in them the rotation of the rotor lags behind the rotation of the field; commutator motors, the design and operating principle of which are similar to DC motors.

Synchronous speed

The rotation speed of an AC electric machine depends on the angular frequency of the stator magnetic field. This speed is called synchronous. In synchronous motors, the shaft rotates at the same speed, which is an advantage of these electric machines.

To do this, the rotor of high-power machines has a winding to which a constant voltage is applied, creating a magnetic field. In low power devices, permanent magnets are inserted into the rotor, or there are pronounced poles.

Slip

In asynchronous machines, the number of shaft revolutions is less than the synchronous angular frequency. This difference is called the “S” slip. Due to sliding, an electric current is induced in the rotor and the shaft rotates. The larger S, the higher the torque and the lower the speed. However, if the slip exceeds a certain value, the electric motor stops, begins to overheat and may fail. The rotation speed of such devices is calculated using the formula in the figure below, where:

n – number of revolutions per minute,
f – network frequency,
p – number of pole pairs,
s – slip.

There are two types of such devices:

With squirrel-cage rotor. The winding in it is cast from aluminum during the manufacturing process;
With wound rotor. The windings are made of wire and are connected to additional resistances.

Speed adjustment

During operation, it becomes necessary to adjust the speed of electrical machines. This is done in three ways:

Increasing additional resistance in the rotor circuit of electric motors with a wound rotor. If it is necessary to greatly reduce the speed, it is possible to connect not three, but two resistances;
Connecting additional resistances in the stator circuit. It is used to start high-power electrical machines and to regulate the speed of small electric motors. For example, the speed of a table fan can be reduced by connecting an incandescent lamp or capacitor in series with it. The same result is achieved by reducing the supply voltage;
Changing the network frequency. Suitable for synchronous and asynchronous motors.

Attention! The rotation speed of commutator electric motors operating from an alternating current network does not depend on the frequency of the network.

DC motors

In addition to AC machines, there are electric motors connected to a DC network. The speed of such devices is calculated using completely different formulas.

Rated rotation speed

The speed of a DC machine is calculated using the formula in the figure below, where:

n – number of revolutions per minute,
U – network voltage,
Rya and Iya – armature resistance and current,
Ce – motor constant (depending on the type of electric machine),
Ф – stator magnetic field.

These data correspond to the nominal values of the parameters of the electric machine, the voltage on the field winding and the armature or the torque on the motor shaft. Changing them allows you to adjust the rotation speed. It is very difficult to determine the magnetic flux in a real motor, so calculations are made using the current flowing through the field winding or armature voltage.

The speed of commutator AC motors can be found using the same formula.

Speed adjustment

Adjustment of the speed of an electric motor operating from a DC network is possible within a wide range. It is possible in two ranges:

Up from nominal. To do this, the magnetic flux is reduced using additional resistances or a voltage regulator;
Down from par. To do this, it is necessary to reduce the voltage on the armature of the electric motor or connect a resistance in series with it. In addition to reducing the speed, this is done when starting the electric motor.

Knowing what formulas are used to calculate the rotation speed of an electric motor is necessary when designing and setting up equipment.

Video

Sometimes questions from mathematics and physics come up in relation to cars. In particular, one such issue is angular velocity. It relates both to the operation of mechanisms and to cornering. Let's figure out how to determine this value, how it is measured, and what formulas need to be used here.

How to determine angular velocity: what is this quantity?

From a physical and mathematical point of view, this quantity can be defined as follows: these are data that show how quickly a certain point rotates around the center of the circle along which it moves.

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This seemingly purely theoretical value has considerable practical significance when operating a car. Here are just a few examples:

It is necessary to correctly correlate the movements with which the wheels rotate when turning. The angular speed of a car wheel moving along the inner part of the trajectory should be less than that of the outer one.
You need to calculate how fast the crankshaft rotates in the car.
Finally, the car itself, when going through a turn, also has a certain value of motion parameters - and in practice, the stability of the car on the highway and the likelihood of capsizing depend on them.

Formula for the time it takes for a point to rotate around a circle of a given radius

In order to calculate angular velocity, the following formula is used:

ω = ∆φ /∆t

ω (read “omega”) is the actual calculated value.
∆φ (read “delta phi”) – rotation angle, the difference between the angular position of a point at the first and last moment of measurement.
∆t
(read “delta te”) – the time during which this very shift occurred. More precisely, since “delta”, it means the difference between the time values at the moment when the measurement was started and when it was completed.

The above formula for angular velocity applies only in general cases. Where we are talking about uniformly rotating objects or the relationship between the movement of a point on the surface of a part, the radius and the time of rotation, it is necessary to use other relationships and methods. In particular, a rotation frequency formula will be needed here.

Angular velocity is measured in a variety of units. In theory, rad/s (radians per second) or degrees per second are often used. However, this value means little in practice and can only be used in design work. In practice, it is measured more in revolutions per second (or minute, if we are talking about slow processes). In this regard, it is close to the rotational speed.

Rotation angle and period of revolution

Much more commonly used than rotation angle is rotation rate, which measures how many rotations an object makes in a given period of time. The fact is that the radian used for calculations is the angle in a circle when the length of the arc is equal to the radius. Accordingly, there are 2 π radians in a whole circle. The number π is irrational, and it cannot be reduced to either a decimal or a simple fraction. Therefore, if uniform rotation occurs, it is easier to count it in frequency. It is measured in rpm - revolutions per minute.

If the matter concerns not a long period of time, but only the period during which one revolution occurs, then the concept of circulation period is used here. It shows how quickly one circular movement is made. The unit of measurement here will be the second.

The relationship between angular velocity and rotation frequency or rotation period is shown by the following formula:

ω = 2 π / T = 2 π *f,

ω – angular velocity in rad/s;
T – circulation period;
f – rotation frequency.

You can get any of these three quantities from another using the rule of proportions, without forgetting to convert the dimensions into one format (in minutes or seconds)

What is the angular velocity in specific cases?

Let's give an example of a calculation based on the above formulas. Let's say we have a car. When driving at 100 km/h, its wheel, as practice shows, makes an average of 600 revolutions per minute (f = 600 rpm). Let's calculate the angular velocity.

Since it is impossible to accurately express π in decimal fractions, the result will be approximately 62.83 rad/s.

Relationship between angular and linear speeds

In practice, it is often necessary to check not only the speed with which the angular position of a rotating point changes, but also its speed in relation to linear motion. In the example above, calculations were made for a wheel - but the wheel moves along the road and either rotates under the influence of the speed of the car, or it itself provides this speed. This means that each point on the surface of the wheel, in addition to the angular one, will also have a linear speed.

The easiest way to calculate it is through the radius. Since the speed depends on time (which will be the period of revolution) and the distance traveled (which will be the circumference), then, taking into account the above formulas, the angular and linear speed will be related as follows:

V – linear speed;
R – radius.

From the formula it is obvious that the larger the radius, the higher the value of this speed. In relation to the wheel, the point on the outer surface of the tread will move with the highest speed (R is maximum), but exactly in the center of the hub the linear speed will be zero.

Acceleration, moment and their connection with mass

In addition to the above values, there are several other issues associated with rotation. Considering how many rotating parts of different weights there are in a car, their practical importance cannot be ignored.

Even rotation is important. But there is not a single part that rotates evenly all the time. The number of revolutions of any rotating component, from the crankshaft to the wheel, always eventually rises and then falls. And the value that shows how much the revolutions have increased is called angular acceleration. Since it is a derivative of angular velocity, it is measured in radians per second squared (like linear acceleration - in meters per second squared).

Another aspect is associated with movement and its change in time - angular momentum. If up to this point we could only consider purely mathematical features of movement, then here we need to take into account the fact that each part has a mass that is distributed around its axis. It is determined by the ratio of the initial position of the point, taking into account the direction of movement - and momentum, that is, the product of mass and speed. Knowing the moment of impulse arising during rotation, it is possible to determine what load will fall on each part when it interacts with another

Hinge as an example of impulse transmission

A typical example of how all the above data is applied is the constant velocity joint (CV joint). This part is used primarily on front-wheel drive cars, where it is important not only to ensure different rates of rotation of the wheels when turning, but also to control them and transfer the impulse from the engine to them.

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The design of this unit is precisely intended to:

compare with each other how quickly the wheels rotate;
ensure rotation at the moment of turning;
guarantee the independence of the rear suspension.

As a result, all the formulas given above are taken into account in the operation of the CV joint.

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

Angular velocity

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

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

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

One of the most common types of movement in nature and technology is rotation. This type of movement of bodies in space is characterized by a set of physical quantities. An important characteristic of any rotation is frequency. The rotation speed formula can be found if you know certain quantities and parameters.

What is rotation?

In physics, it is understood as such a movement of a material point around a certain axis, at which its distance to this axis remains constant. This is called the radius of rotation.

Examples of this movement in nature are the rotation of planets around the Sun and around their own axis. In technology, rotation is represented by the movement of shafts, gears, car or bicycle wheels, and the movement of windmill blades.

Physical quantities describing rotation

For the numerical description of rotation in physics, a number of characteristics were introduced. Let's list them and characterize them.

First of all, this is the angle of rotation, denoted by θ. Since a complete circle is characterized by a central angle of 2*pi radians, then, knowing the amount θ by which the rotating body turned over a certain period of time, we can determine the number of revolutions during this time. In addition, the angle θ allows you to calculate the linear path traversed by the body along the curved circle. The corresponding formulas for the number of revolutions n and the distance traveled L have the form:

Where r is the radius of the circle or radius of rotation.

The next characteristic of the type of movement under consideration is angular velocity. It is usually denoted by the letter ω. It is measured in radians per second, that is, it shows the angle in radians through which a rotating body turns in one second. For the angular velocity in the case of uniform rotation, the formula is valid:

Angular frequency, period and angular velocity

It was already noted above that an important property of any rotational motion is the time it takes to complete one revolution. This time is called the rotation period. It is designated by the letter T and measured in seconds. The formula for period T can be written in terms of angular velocity ω. The corresponding expression looks like:

The reciprocal of the period is called frequency. It is measured in hertz (Hz). For circular motion, it is convenient to use not the frequency itself, but its angular analogue. Let's denote it f. The formula for angular rotation frequency f is:

Comparing the last two formulas, we arrive at the following equality:

This equality means the following:

the formulas for angular frequency and angular velocity coincide, therefore these quantities are numerically equal to each other;
Like speed, frequency shows the angle in radians a body rotates in one second.

The only difference between these quantities is that the angular frequency is a scalar quantity, while the speed is a vector.

Linear rotation speed, frequency and angular frequency

In technology, for some rotating structures, for example, gears and shafts, their operating frequencies μ and linear speeds v are known. However, each of these characteristics can be used to determine the angular or cyclic frequency.

It was noted above that the frequency μ is measured in hertz. It shows the number of revolutions of a rotating body in one second. The formula for it takes the form:

If we compare this expression with the corresponding equality for f, then the formula for finding the rotation frequency f through μ describing it will look like:

This formula is intuitive because μ shows the number of revolutions per unit of time, and f reflects the same value, only represented in radians.

Linear speed v is related to angular speed ω by the following equality:

Since the absolute values of f and ω are equal, from the last expression it is easy to obtain the corresponding formula for the cyclic rotation frequency. Let's write it down:

Where r is the radius of rotation. Note that the speed v increases linearly with increasing radius r, and the ratio of these quantities is a constant. The last conclusion means that if you measure the cyclic frequency of rotation at any point in the cross section of a rotating massive object, then it will be the same everywhere.

The task of determining the cyclic speed of a shaft

Angular frequencies contain useful information because they allow one to calculate important physical properties such as angular momentum or angular velocity. Let's solve this problem: it is known that the operating speed of the shaft is 1500 rpm. What is the cyclic frequency for this shaft?

From the units of measurement given in the condition, it is clear that the usual frequency μ is given. Therefore, the formula for the cyclic shaft rotation speed has the form:

Before using it, you should convert the figure indicated in the condition to standard units of measurement, that is, to reciprocal seconds. Since the shaft makes 1500 revolutions per minute, then in a second it will make 60 times fewer revolutions, that is, 25. That is, its rotation frequency is 25 Hz. Substituting this number into the formula written above, we obtain the value of the cyclic frequency: f = 157 rad/s.