One of the types of motion common 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 the frequency. The formula for the rotational speed can be found by knowing certain quantities and parameters.
What is rotation?
In physics, it is understood as such a displacement of a material point around a certain axis at which its distance to this axis remains constant. It is called the radius of rotation.
Examples of this movement in nature are the rotation of the planets around the sun and around its own axis. In technology, rotation is represented by the movement of shafts, gears, wheels of a car or bicycle, the movement of the blades of windmills.
Physical quantities describing rotation
A number of characteristics were introduced for the numerical description of rotation in physics. We list them and characterize them.
First of all, this is the rotation angle, denoted by θ. Since the full circle is characterized by a central angle of 2 * pi radians, then, knowing the value of θ by which the rotating body rotated in a certain period of time, we can determine the number of revolutions during this time. In addition, the angle θ makes it possible to calculate the linear path traveled by the body along a circle curve. The corresponding formulas for the number of revolutions n and the distance traveled L have the form:
n = θ / (2 * pi);
L = θ * r.
Where r is the radius of the circle or the radius of rotation.
The next characteristic of the type of motion under consideration is the angular velocity. It is usually denoted by the letter ω. It is measured in radians per second, that is, it shows the value of the angle in radians by which the rotating body rotates in one second. For the angular velocity in the case of uniform rotation, the formula is valid:
ω = θ / t
Angular frequency, period and angular velocity
It has already been noted above that an important property of any rotational motion is the time during which one revolution takes place. This time is called the rotation period. It is denoted by the letter T and measured in seconds. The formula for the period T can be written in terms of the angular velocity ω. The corresponding expression is:
T = 2 * pi / ω
The inverse of the period is called the frequency. It is measured in hertz (Hz). For circular motion, it is convenient to use not the frequency itself, but its angular analogue. Denote it by f. The formula for the angular rotation frequency f has the form:
f = 2 * pi / T
To calculate the angular frequency, it is necessary to know the period of rotational motion.
Comparing the last two formulas, we arrive at the following equality:
f = ω
This equality means the following:
- the formulas of the angular frequency and angular velocity coincide; therefore, these quantities are numerically equal to each other;
- as well as speed, the frequency shows at what angle in radians the body rotates in one second.
The difference between these quantities is the only one: the angular frequency is a scalar quantity, while the velocity is a vector.
Linear rotation speed, frequency and angular frequency
In the technique 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:
μ = 1 / T
If we compare this expression with the corresponding equality for f, then the formula how to find the frequency of rotation f through μ describing will look like:
f = 2 * pi * μ
This formula is intuitive, since μ shows the number of revolutions per unit of time, and f reflects the same value only represented in radians.
The linear velocity v is related to the angular velocity ω by the following equality:
v = ω * r
Since the moduli of f and ω are equal, it is easy to obtain the corresponding formula for the cyclic rotation frequency from the last expression. We write it down:
f = v / r
Where r is the radius of rotation. We note that the velocity v increases linearly with increasing radius r, and the ratio of these quantities is constant. The last conclusion means that if we 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 frequency of rotation of the shaft
Angular rotational speeds contain useful information because they allow you to calculate important physical characteristics such as angular momentum or angular velocity. We will solve this problem: it is known that the working shaft speed is 1500 rpm. What is the cyclic frequency for this shaft?
From the units given in the condition, it is clear that the usual frequency μ is given. Therefore, the formula for the cyclic shaft rotation frequency has the form:
f = 2 * pi * μ
Before using it, you should translate the number indicated in the condition to the standard units of measurement, that is, to the reverse seconds. Since the shaft makes 1500 revolutions per minute, it will make 60 times less revolutions per second, that is 25. That is, its rotation frequency is 25 Hz. Substituting this number in the above formula, we obtain the value of the cyclic frequency: f = 157 rad / s.