>> Physics: Period and frequency of revolution
Uniform movement around a circle is characterized by the period and frequency of revolution.
Period of circulation- this is the time during which one revolution is completed.
If, for example, in a time t = 4 s the body, moving in a circle, has made n = 2 revolutions, then it is easy to figure out that one revolution lasted 2 s. This is the period of circulation. It is designated by the letter T and is determined by the formula:
So, to find the period of circulation, it is necessary to divide the time during which n revolutions are completed by the number of revolutions.
Another characteristic of uniform circular motion is the frequency of revolution.
Call frequency is the number of revolutions made in 1 s. If, for example, in time t = 2 s the body has completed n = 10 revolutions, then it is easy to figure out that in 1 s it managed to complete 5 revolutions. This number also expresses the frequency of circulation. It is denoted by a Greek letter V(read: nude) and is determined by the formula:
So, to find the frequency of revolution, it is necessary to divide the number of revolutions by the time during which they occurred.
The unit of the frequency of revolution in SI is the frequency of revolution, at which for each second the body makes one revolution. This unit is designated as follows: 1 / s or s -1 (read: second to the minus first power). Previously, this unit was called "revolution per second", but now this name is considered obsolete.
Comparing formulas (6.1) and (6.2), it can be noted that the period and frequency are mutually inverse quantities. That's why
Formulas (6.1) and (6.3) allow us to find the period of revolution T, if the number n and the time of revolutions t or the frequency of revolution are known V... However, it can also be found in the case when none of these quantities is known. Instead, it is enough to know the speed of the body. V and the radius of the circle r along which it moves.
To derive a new formula, recall that the orbital period is the time during which the body makes one revolution, that is, it travels a path equal to the circumference ( l env = 2 NS r, where NS≈3,14 is the number "pi", known from the course of mathematics). But we know that with uniform movement, time is found by dividing the distance traveled by the speed of movement. Thus,
So, to find the period of revolution of a body, it is necessary to divide the length of the circle along which it moves by the speed of its movement.
??? 1. What is the circulation period? 2. How can you find the period of circulation, knowing the time and the number of revolutions? 3. What is the circulation frequency? 4. How is the unit of frequency indicated? 5. How can you find the frequency of circulation, knowing the time and number of revolutions? 6. How are the period and frequency of circulation related? 7. How can you find the period of revolution, knowing the radius of the circle and the speed of the body?
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Collection of abstracts of lessons in physics, abstracts on a topic from the school curriculum. Calendar thematic planning. physics grade 8 online, books and textbooks on physics. The student should prepare for the lesson.
Lesson content lesson outline support frame lesson presentation accelerative methods interactive technologies Practice tasks and exercises self-test workshops, trainings, cases, quests home assignments discussion questions rhetorical questions from students Illustrations audio, video clips and multimedia photos, pictures, charts, tables, schemes humor, jokes, jokes, comics parables, sayings, crosswords, quotes Supplements abstracts articles chips for the curious cheat sheets textbooks basic and additional vocabulary of terms others Improving textbooks and lessonsbug fixes in the tutorial updating a fragment in the textbook elements of innovation in the lesson replacing obsolete knowledge with new ones For teachers only perfect lessons calendar plan for the year methodological recommendations of the discussion program Integrated lessonsSince the linear speed uniformly changes direction, the movement around the circle cannot be called uniform, it is uniformly accelerated.
Angular velocity
Choose a point on the circle 1 ... Let's build a radius. In 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 of time.
Period and frequency
Rotation period T- this is the time during which the body makes one revolution.
Rotation speed is the number of revolutions per second.
Frequency and period are interrelated by the ratio
Angular Velocity Relationship
Linear Velocity
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 a grinder move in the same direction as the instantaneous speed.
Consider a point on a circle that makes one revolution, the time it takes is a period T... The path that the point overcomes is the circumference.
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, the following relations can be derived
Points lying on one straight line outgoing from the center of the circle (for example, these can be points that lie on the spoke of a wheel) will have the same angular velocity, period and frequency. That is, they will rotate in the same way, but with different linear speeds. The further the point is from the center, the faster it will move.
The law of addition of velocities is also valid for rotary motion. If the movement of a body or a frame of reference is not uniform, then the law is applied for 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 person's movement speed.
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 equatorial plane and the direction from the center of the Earth to a point on its surface.
According to Newton's second law, force is the cause of any acceleration. If a moving body experiences centripetal acceleration, then the nature of the forces that cause this acceleration can 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 ceases to act, then the body will 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 - the change in speed per unit of time. Let's find the difference in vectors.
Rotational motion around a fixed axis is another special case of the motion of a rigid body.
Rotational motion of a rigid body around a fixed axis
its movement is called such in which all points of the body describe circles, the centers of which are on one straight line, called the axis of rotation, while the planes to which these circles belong are perpendicular axis of rotation
(Figure 2.4).
In technology, this type of movement is very common: for example, the rotation of the shafts of engines and generators, turbines and aircraft propellers.
Angular velocity
... Each point of a body rotating around the axis passing through a point O, moves in a circle, and different points pass different paths in time. So, therefore, the modulus of the speed of the point A more than the point V (fig.2.5). But the radii of the circles rotate by the same angle over time. Angle - the angle between the axis OH and a radius vector defining the position of point A (see Fig. 2.5).
Let the body rotate uniformly, that is, for any equal intervals of time it rotates at the same angles. The speed of rotation of the 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 one rotates through an angle, then we say that the first body rotates 2 times faster than the second.
The angular velocity of the body with uniform rotation
is called a value equal to the ratio of the angle of rotation of the body to the time interval during which this rotation occurred.
We will denote the angular velocity by the Greek letter ω
(omega). Then by definition
The angular velocity is expressed in radians per second (rad / s).
For example, the angular velocity of the Earth's rotation around the axis is 0.0000727 rad / s, and the grinding disk is about 140 rad / s 1.
The angular velocity can be expressed in terms of rotational speed
, i.e. the number of full revolutions in 1 s. If the 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:
An angle corresponds to a full rotation of the body. Therefore, according to formula (2.1)
If, with uniform rotation, the angular velocity is known and at the initial moment of time the angle of rotation, then the angle of rotation of the body during the time t according to equation (2.1) is equal to:
If, then, or 
.
The angular velocity takes on positive values if the angle between the radius vector defining 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 the rotating body at any time.
Relationship between linear and angular velocities.
The speed of a point moving in a circle is often called linear velocity
to emphasize its difference from the 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 velocity of any point of a rotating body and its angular velocity. Let's install it. A point on a circle with a radius R, in one revolution will pass the way. Since the time of one revolution of the body is a period T, then the modulus of the linear velocity of a 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 should be used here.
How to determine the angular velocity: what is this value?
From a physical and mathematical point of view, this value can be defined as follows: this is data that shows how quickly a point rotates around the center of the circle along which it moves.
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This seemingly purely theoretical value is of considerable practical importance 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.
- It is required to calculate how fast the crankshaft rotates in the car.
- Finally, the car itself, passing a turn, also has a certain value of the motion parameters - and in practice, the stability of the car on the track and the probability of overturning depend on them.
The formula for the time it takes a point to rotate around a circle of a given radius
In order to calculate the angular velocity, the following formula is used:
ω = ∆φ / ∆t
- ω (read "omega") - actually calculated value.
- ∆φ (read "delta phi") is the angle of rotation, the difference between the angular position of a point at the first and last moment of measurement.
- ∆t
(reads "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 finished.
The above formula for the angular velocity applies only in general cases. Where we are talking about uniformly rotating objects or about the relationship between the movement of a point on the surface of the part, the radius and time of rotation, other relationships and methods are required. In particular, the formula for the rotational speed will already be needed here.
Angular velocity is measured in a variety of units. In theory, rad / s (radians per second) or degrees per second is often used. However, this value means little in practice and can only be used in design work. In practice, however, 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 orbital period
Much more often than rotation angle, rotation frequency is used, which indicates how many revolutions 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, the whole circle contains 2 π radians. The number π is irrational, and it cannot be reduced to either a decimal or a simple fraction. Therefore, in the event that a uniform rotation occurs, it is easier to read it in frequency. It is measured in rpm - revolutions per minute.
If the matter concerns not a long period of time, but only that for which one revolution occurs, then the concept of the period of circulation is used here. It shows how quickly one circular motion is made. The unit of measurement here will be the second.
The relationship between the angular velocity and the frequency of rotation or the period of revolution is shown by the following formulas:
ω = 2 π / T = 2 π * f,
- ω - angular velocity in rad / s;
- T is the circulation period;
- f - rotation frequency.
You can get any of these three values from another using the rule of proportions, without forgetting to translate 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 you have a car. When driving at 100 km / h, as practice shows, its wheel 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 of angular and linear velocities
In practice, it is often necessary to check not only the speed with which the angular position of the rotating point changes, but also its speed as applied to linear motion. In the above example, calculations were made for a wheel - but the wheel moves along the road and either rotates under the action 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 velocity.
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 is the circumference), then, taking into account the above formulas, the angular and linear speed will be related as follows:
- V - linear speed;
- R is the radius.
It is obvious from the formula that the larger the radius, the higher the value of this speed. With regard to the wheel, a point on the outer surface of the tread (R is maximum) will move with the highest speed, 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 factors associated with rotation. Considering how many rotating parts of different weights are in the car, their practical importance cannot be ignored.
Uniform rotation is important. But there is not a single part that rotates evenly all the time. The number of revolutions of any rotating assembly, from the crankshaft to the wheel, always eventually rises and then falls. And the value that shows how much the revolutions have grown 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 - the moment of impulse. If up to this point we could only consider purely mathematical features of motion, then here it is already necessary to take into account the fact that each part has a mass that is distributed around the axis. It is determined by the ratio of the initial position of a point, taking into account the direction of movement - and momentum, that is, the product of mass and speed. Knowing the angular momentum 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 momentum transfer
A typical example of how all of the above data is applied is the constant velocity joint (CV joint). This part is used primarily on front-wheel drive vehicles, where it is important not only to ensure a different rate of rotation of the wheels when cornering, but also at the same time their controllability and transmission of an impulse from the engine to them.
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The design of this unit is just intended to:
- equalize with each other how fast the wheels rotate;
- provide 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 work of the SHRUS.