Uniform movement around a circle with a radius. Angular velocity

Usually, when we talk about movement, we imagine an object moving in a straight line. The speed of such movement is usually called linear, and the calculation of its average value is simple: it is enough to find the ratio of the distance traveled to the time during which it was covered by the body. If an object moves in a circle, then in this case it is not linear that is determined, but What is this quantity and how is it calculated? This is exactly what will be discussed in this article.

Angular velocity: concept and formula

When moving along a circle, the speed of its movement can be characterized by the magnitude of the angle of rotation of the radius that connects the moving object to the center of this circle. It is clear that this value is constantly changing depending on time. The speed with which this process occurs is nothing more than angular velocity. In other words, this is the ratio of the deviation of the radius vector of an object to the period of time it took the object to make such a turn. Angular velocity formula (1) can be written as follows:

w = φ / t, where:

φ - radius rotation angle,

t - rotation time period.

Units of measurement

In the International System of Common Units (SI), radians are used to characterize turns. Therefore, 1 rad/s is the basic unit used in angular velocity calculations. At the same time, no one prohibits the use of degrees (recall that one radian is equal to 180/pi, or 57˚18’). Also, angular velocity can be expressed in the number of revolutions per minute or per second. If movement around the circle occurs uniformly, then this value can be found using formula (2):

where n is the rotation speed.

Otherwise, in the same way as for ordinary speed, the average or instantaneous angular speed is calculated. It should be noted that the quantity under consideration is a vector one. To determine its direction, it is usually used, which is often used in physics. The angular velocity vector is directed in the same direction as the screw with a right-hand thread. In other words, it is directed along the axis around which the body rotates, in the direction from which the rotation is seen to occur counterclockwise.

Calculation examples

Suppose you need to determine what the linear and angular speed of a wheel are, if it is known that its diameter is equal to one meter, and the angle of rotation changes in accordance with the law φ = 7t. Let's use our first formula:

w = φ / t = 7t / t = 7 s -1 .

This will be the desired angular velocity. Now let's move on to searching for the speed of movement that is familiar to us. As is known, v = s/t. Considering that s in our case is the wheels (l = 2π*r), and 2π is one full revolution, we get the following:

v = 2π*r / t = w * r = 7 * 0.5 = 3.5 m/s

Here's another puzzle on this topic. It is known that at the equator it is 6370 kilometers. It is required to determine the linear and angular speed of movement of points located on this parallel, which arises as a result of the rotation of our planet around its axis. In this case, we need the second formula:

w = 2π*n = 2*3.14 *(1/(24*3600)) = 7.268 *10 -5 rad/s.

It remains to find out what the linear speed is equal to: v = w*r = 7.268 * 10 -5 * 6370 * 1000 = 463 m/s.

Circular motion is the simplest case of curvilinear motion of a body. When a body moves around a certain point, along with the displacement vector it is convenient to enter the angular displacement ∆ φ (angle of rotation relative to the center of the circle), measured in radians.

Knowing the angular displacement, you can calculate the length of the circular arc (path) that the body has traversed.

∆ l = R ∆ φ

If the angle of rotation is small, then ∆ l ≈ ∆ s.

Let us illustrate what has been said:

Angular velocity

With curvilinear motion, the concept of angular velocity ω is introduced, that is, the rate of change in the angle of rotation.

Definition. Angular velocity

The angular velocity at a given point of the trajectory is the limit of the ratio of the angular displacement ∆ φ to the time interval ∆ t during which it occurred. ∆ t → 0 .

ω = ∆ φ ∆ t , ∆ t → 0 .

The unit of measurement for angular velocity is radian per second (r a d s).

There is a relationship between the angular and linear speeds of a body when moving in a circle. Formula for finding angular velocity:

With uniform motion in a circle, the velocities v and ω remain unchanged. Only the direction of the linear velocity vector changes.

In this case, uniform motion in a circle affects the body by centripetal, or normal acceleration, directed along the radius of the circle to its center.

a n = ∆ v → ∆ t , ∆ t → 0

The modulus of centripetal acceleration can be calculated using the formula:

a n = v 2 R = ω 2 R

Let us prove these relations.

Let's consider how the vector v → changes over a short period of time ∆ t. ∆ v → = v B → - v A → .

At points A and B, the velocity vector is directed tangentially to the circle, while the velocity modules at both points are the same.

By definition of acceleration:

a → = ∆ v → ∆ t , ∆ t → 0

Let's look at the picture:

Triangles OAB and BCD are similar. It follows from this that O A A B = B C C D .

If the value of the angle ∆ φ is small, the distance A B = ∆ s ≈ v · ∆ t. Taking into account that O A = R and C D = ∆ v for the similar triangles considered above, we obtain:

R v ∆ t = v ∆ v or ∆ v ∆ t = v 2 R

When ∆ φ → 0, the direction of the vector ∆ v → = v B → - v A → approaches the direction to the center of the circle. Assuming that ∆ t → 0, we obtain:

a → = a n → = ∆ v → ∆ t ; ∆ t → 0 ; a n → = v 2 R .

With uniform motion around a circle, the acceleration modulus remains constant, and the direction of the vector changes with time, maintaining orientation to the center of the circle. That is why this acceleration is called centripetal: the vector at any moment of time is directed towards the center of the circle.

Writing centripetal acceleration in vector form looks like this:

a n → = - ω 2 R → .

Here R → is the radius vector of a point on a circle with its origin at its center.

In general, acceleration when moving in a circle consists of two components - normal and tangential.

Let us consider the case when a body moves unevenly around a circle. Let us introduce the concept of tangential (tangential) acceleration. Its direction coincides with the direction of the linear velocity of the body and at each point of the circle is directed tangent to it.

a τ = ∆ v τ ∆ t ; ∆ t → 0

Here ∆ v τ = v 2 - v 1 - change in velocity module over the interval ∆ t

The direction of the total acceleration is determined by the vector sum of the normal and tangential accelerations.

Circular motion in a plane can be described using two coordinates: x and y. At each moment of time, the speed of the body can be decomposed into components v x and v y.

If the motion is uniform, the quantities v x and v y as well as the corresponding coordinates will change in time according to a harmonic law with a period T = 2 π R v = 2 π ω

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Among the various types of curvilinear motion, of particular interest is 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. With uniform motion in a circle, 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.

Rotational motion around a fixed axis is another special case of rigid body motion.
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 1s, 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, will cover the distance in one revolution. 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:

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 must 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 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 itself provides it with 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.