The laws governing the motion of a body in a circle are similar to the laws of translational motion. The equations describing rotational motion can be derived from the equations of translational motion by making the following substitutions in the latter:
If:
moving s- angular movement (angle of rotation) ?
,
speed u- angular velocity ?
,
acceleration a- angular acceleration ?
Rotation angle
In all equations of rotational motion, angles are specified in radians, abbreviated (glad).
If
?
- angular displacement in radians,
s- length of the arc enclosed
between the sides of the rotation angle,
r- radius,
then by definition of radian
Relationship between angle units
Please note: The name of the unit radian (rad) is usually indicated in formulas only in cases where it can be confused with a degree. Since a radian is equal to the ratio of the lengths of two segments
(1rad = 1m/ 1m = 1), it has no dimension.
The relationship between angular velocity, angular displacement and time for all types of circular motion is clearly visible on the angular velocity graph (dependence ?
from t). 
Therefore, the graph can determine what angular velocity a body has at a given moment in time and at what angle it has turned since the beginning of its movement (it is characterized by the area under the curve).
In addition, to represent the relationships between these quantities, use a graph of angular displacement (dependence ? from t) and graph of angular acceleration (dependence ? from t).
Speed
A characteristic of all types of rotation is the number of revolutions n or an equivalent characteristic - frequency f. Both quantities characterize the number of revolutions per unit time.
SI unit of frequency (or number of revolutions)
In engineering, the number of revolutions is usually measured in revolutions per minute (rpm) = 1/min.
Thus, the reciprocal of the number of revolutions is the duration of one revolution.
If
n- number of revolutions,
f- frequency,
T- duration of one revolution, period,
?
- angular movement,
N- total number of revolutions,
t- time, duration of rotation,
?
- angular frequency,
That
Period
Angular movement
The angular movement is equal to the product of the total number of revolutions by 2?:
Angular velocity
From the formula for one revolution it follows:
Please note:
the formulas are valid for all types of rotational motion - both for uniform motion and for accelerated motion. These can include constant values, average values, start and end values, and any instantaneous values.
contrary to its name, the number of revolutions n- this is not a number, but a physical quantity.
it is necessary to differentiate the number of revolutions n and full speed N.
Uniform movement of a body in a circle
A body is said to move uniformly in a circle if its angular velocity is constant, i.e. the body rotates through the same angle at equal intervals of time.
?
- angular velocity (constant over time t)
?
- angular movement
t- turning time ?
Since the area of the rectangle on the angular velocity graph corresponds to the angular displacement, we have:
Constant angular velocity- is the ratio of angular movement (angle of rotation) to the time spent on this movement.
SI unit of angular velocity:
Uniformly accelerated motion in a circle without initial angular velocity
The body begins to move from a state of rest, and its angular velocity uniformly increases.
?
- instantaneous angular velocity of the body at the moment of time t
?
- angular acceleration, permanent over time t
?
t, (?
in radians)
t- time
Since the angular displacement on the velocity graph is equal to the area of the triangle, we have:
Since the rotation of the body begins from a state of rest, the change in angular velocity?? equal to the angular velocity achieved as a result of acceleration?. Therefore, the formula takes the following form:
Uniformly accelerated motion in a circle with an initial angular velocity
The initial speed of the body is equal to ?0 at the moment t= 0, changes uniformly by the amount ?? . (The angular acceleration is constant.)
?0
- initial angular velocity
?
- final angular velocity
?
- angular movement of the body over time t in radians
t- time
?
- angular acceleration is constant over time t
Since on the velocity graph the angular displacement corresponds to the area of the trapezoid under the velocity curve, we have:
Since the area of a trapezoid is equal to the sum of the areas of the triangle and rectangle forming it, we obtain:
Combining the formulas we get
After the transformation we get an expression that does not contain time:
Unevenly accelerated movement of a body in a circle
The motion of a body in a circle will be unevenly accelerated if the change in angular velocity is not proportional to time, that is, if the angular acceleration does not remain constant. In this case, both angular velocity and angular acceleration are functions of time.
Relationship between quantities ? , ? And ? presented in the corresponding graphs.
Instantaneous angular velocity
The instantaneous angular velocity is the first derivative of the function ? = ? (t) by time.
Please note:
1)
to calculate the instantaneous angular velocity ?
, it is necessary to know the dependence of angular displacement on time.
2)
the formula for angular displacement for uniform motion of a body in a circle and the formula for angular displacement for uniformly accelerated motion in a circle without initial angular velocity are special cases of formula (2), respectively for ?
= 0 and ?
= const.
From the formulas it follows:
Integrating both sides of the expression, we get
Angular displacement is the time integral of the angular velocity.
Please note:
To calculate angular displacement? it is necessary to know the dependence of angular velocity on time.
Average angular velocity
Average angular velocity for a certain time interval
The average number of revolutions is determined similarly to the formula:
Rotational movement of the body, formulas
In addition, these quantities are related in a certain way to the angular displacement ? , angular velocity ? and angular acceleration ? .
Note: The formulas are valid for constant, instantaneous and average quantities, in all cases of body motion in a circle.
Vector quantities characterizing the rotational motion of a body
Definition: If a body participates simultaneously in several rotational movements, then the resulting angular velocity is determined by the rule of vector (geometric) addition:
The magnitude of the resulting angular velocity is determined by analogy with the formula (Addition of movements):
or, if the axes of rotation are perpendicular to each other
Note: The resulting angular acceleration is determined in a similar way. Graphically, the resultant can be found as the diagonal of a parallelogram of velocities or accelerations.
Rotational movement around a fixed axis is another special case movement 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 various points take 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 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 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:
Revolutions per minute
Car tachometer (indicator of engine revolutions per minute)
Revolutions per minute(designation rpm, 1/min, min −1, the English designation is also often used rpm) - unit of rotation speed: the number of full revolutions made around a fixed axis. Used to measure the rotation speed of mechanical components.
The unit is also used revolutions per second(symbol r/s or s −1). RPM is converted to RPM by dividing by 60. The reverse conversion is RPM multiplied by 60.
1 rpm = 1/min = 1/(60s) = 1/60 r/s ≈ 0.01667 r/s
Another physical quantity is associated with this concept: angular velocity; in the SI system it is measured in radians per second (rad s −1):
1 rpm = 2π rad min −1 = 2π/60 rad s −1 = 0.1047 rad s −1 ≈ 1/10 rad s −1
Examples
See also
Notes
Wikimedia Foundation. 2010.
See what “Revolutions per minute” is in other dictionaries:
revolutions per minute- Unit of measurement used to characterize centrifugation parameters by rotor rotation speed (along with the g indicator, gravity acceleration). [Arefyev V.A., Lisovenko L.A. English-Russian explanatory dictionary of genetic terms 1995... ... Technical Translator's Guide
Rpm (round per minute) revolutions per minute. A unit of measurement used to characterize centrifugation parameters by rotor rotation speed (along with the g indicator, gravity acceleration). (Source: “English-Russian explanatory dictionary... ... Molecular biology and genetics. Explanatory dictionary.
Non-system units rotation speed. Designation rpm 1 rpm = 1 min 116.667 s 1 ... Big Encyclopedic Polytechnic Dictionary
Turns, m. 1. Full circle of rotation, circular turn. Wheel revolution. The shaft makes 20 revolutions per minute. || Moving back and forth, returning to the starting place. Speed up the turnover of wagons. 2. A separate stage, a completed process in a sequential... ... Ushakov's Explanatory Dictionary
- (Revolution) on fleet vessels, in relation to the operation of the main machine, a full rotation (860°) of the propeller shaft rotated by this machine. To have so many revolutions is an order in a machine, requiring that the propeller shaft give a specified number of revolutions per minute ... Marine Dictionary
This term has other meanings, see Verso. Revolution (cycle, circle) is a unit of measurement of angle or phase of oscillation. When measuring angle, the name "revolution" is usually used, and when measuring phase, "cycle". One revolution is equal to... ... Wikipedia
Noun, m., used. compare often Morphology: (no) what? turnover, what? turn around, (I see) what? turnover, what? in turn, about what? about turnover; pl. What? revs, (no) what? revolutions, what? revolutions, (I see) what? rpm, what? revs, about what? about revolutions 1... Dmitriev's Explanatory Dictionary
turnover- A; m. see also. negotiable, turnover 1) a) Full circle of rotation; circular turn. Revolution/t of wheel. Number of revolutions per minute. Turn the key two turns... Dictionary of many expressions
A; m. 1. Full circle of rotation; circular turn. O. wheels. Number of revolutions per minute. Turn the key two turns. // Special Turning from one side to the other, reverse. Plowing with formation turnover. // plural: revolutions, ov. Specialist. decomposition ABOUT… … Encyclopedic Dictionary
number of circular divisions per minute- 3.1 dial division per minute: The speed of rotation of the stirrer used in this method. Note One complete revolution of the stirrer (360°) is divided into 100 divisions. The turnover rate is characterized by speed... ... Dictionary-reference book of terms of normative and technical documentation
The director of a company, who only has indicators of profit and overall profitability before his eyes, cannot always understand how to adjust them in the right direction. In order to have all the control levers in your hands, it is absolutely necessary to also calculate the turnover working capital.
The picture of the use of working capital consists of four main indicators:
- Duration of turnover (determined in days);
- How many times do working capital turn over in the reporting period;
- How much working capital is there per unit? products sold;
- Load factor of funds in circulation.
Let's consider the calculation of this data using the example of an ordinary enterprise, as well as the calculation of a number of important coefficients for understanding the significance of turnover indicators in the overall picture of the company's success.
Turnover ratio
The main formula determining the rate of turnover of working capital is as follows:
Cob is the turnover ratio. It shows how many turnovers of working capital were made during a specific period of time. Other designations in this formula: Vp - volume of product sales for the reporting period;
Osr is the average balance of working capital for the reporting period.
Most often, the indicator is calculated for the year, but absolutely any period needed for analysis can be selected. This coefficient is the rate of turnover of working capital. For example, the annual turnover of a mini-store of mobile phones was 4,800,000 rubles. The average balance in circulation was RUB 357,600. We get the turnover ratio:
4800000 / 357600 = 13.4 revolutions.
Duration of turnover
It also matters how many days one revolution lasts. This is one of the most important indicators, which shows how many days later the company will see the funds invested in turnover in the form of cash proceeds and will be able to use them. Based on this, you can plan both making payments and expanding your turnover. The duration is calculated as follows:
T is the number of days in the analyzed period.
Let's calculate this indicator for the above digital example. Since the enterprise is a trading enterprise, it has minimum quantity weekends - 5 days a year, for calculation we use the figure of 360 working days.
Let's calculate how many days later the company could see the money invested in turnover in the form of revenue:
357,600 x 360 / 4,800,000 = 27 days.
As you can see, the turnover of funds is short; the management of the enterprise can plan payments and use of funds to expand trade almost monthly.
To calculate the turnover of working capital, the profitability indicator is also important. To calculate it, you need to calculate the ratio of profit to the average annual balance of working capital.
The enterprise's profit for the analyzed year amounted to 1,640,000 rubles, the average annual balance was 34,080,000 rubles. Accordingly, the profitability of working capital in this example is only 5%.
Load factor of funds in circulation
And one more indicator necessary to assess the speed of turnover of working capital is the load factor of funds in circulation. The coefficient shows how much working capital is advanced per 1 ruble. revenue. This is the working capital intensity, which shows how much working capital must be spent for the company to receive 1 ruble of revenue. It is calculated like this:
Where Kz is the load factor of funds in circulation, kopecks;
100 - conversion of rubles to kopecks.
This is the opposite of the turnover ratio. The smaller it is, the better the use of working capital. In our case, this coefficient is equal to:
(357,600 / 4,800,000) x 100 = 7.45 kopecks.
This indicator is an important confirmation that working capital is used very rationally. The calculation of all these indicators is mandatory for an enterprise that seeks to influence operational efficiency using all possible economic levers.
In Forecast NOW! can be calculated
- Turnover in monetary and natural units both for a specific product and for a group of products, and by section - for example, by suppliers
- Dynamics of changes in turnover in any necessary sections
An example of calculating the turnover rate by product groups:
Assessing the dynamics of changes in turnover by product/group of products is also very important. In this case, it is important to correlate the turnover schedule with the service level schedule (how much we satisfied consumer demand in the previous period).
For example, if turnover and the level of service are declining, then this is an unhealthy situation - you need to study this group of products more carefully.
If turnover increases, but the level of service decreases, then the increase in turnover is most likely due to smaller purchases and an increase in shortages. The opposite situation is also possible - turnover decreases, but in this calculation the level of service - customer demand is ensured by large purchases of goods.
In these two situations, it is necessary to evaluate the dynamics of profit and profitability - if these indicators grow, then the changes taking place are beneficial for the company; if they fall, it is necessary to take action.
In Forecast NOW! It’s easy to assess the dynamics of turnover, level of service, profit and profitability - just carry out the necessary analysis.
Example:
Since August, there has been an increase in turnover with a decrease in the level of service - it is necessary to evaluate the dynamics of profitability and profit:
Profitability and profit have been falling since August, we can conclude that the dynamics of changes are negative