Why for a time equal to the period of revolution, the angle of rotation of the radius-vector is 2π? Where did pi come from in this particular formula?

Question

Mathematics

Where did π come from in the angular velocity formula?
How is this formula derived?

Answer 1

I'll add one more answer - maybe the author of the question will mark it as a solution. I noticed that he has a problem with the concept of "radian". An angle of one radian is an angle of 57 and a quarter (approximately) degrees. Where it came from is an angle whose radius is equal to the length of the arc. In this case, the linear measure is reduced, and the angle remains unchanged, no matter what radius you take. Therefore, the radius is not included in the formulas.
The value of this angle was taken for a reason, but from mathematical formulas, according to which it turns out that a full circle (360 degrees) contains 6.28 ... (and so on) of these same radians. I remember that this conclusion was given to us in the first year in the Fundamentals of Higher Mathematics. And in electrical/radio engineering applications of higher mathematics, this same "two pi radians" is included almost everywhere.

Answer 2

Angular velocity is the ratio of the angle between the initial position of the body on the circle, the center of the circle and the final position to the time of movement along the circle from the initial position to the final position.
w=a/t
Let us introduce the notion of a period of rotation. This is the time T it takes for the body to make a complete revolution of 360 degrees or 2P radiant.
Then we calculate the angular velocity
W=a/t=2P/T.
Let us introduce the concept of frequency
V=1/T
Then
W=2PV,
Ch.t.d.
Verification: in half a period (double frequency) the body passes half the circle
W=2P/2*2V=2PV
Scientists don't lie.

Answer 3

Damn summer, what is the angular velocity?
You count or imagine an angle of 6.283185307179586476925286766559 radians, maybe it will become clearer.

Answer 4

Definition of π: the ratio of the circumference (P) to the diameter. That is, P=2πR.
Definition of a radian: An arc equal to a radius. From the previous formula, a full turn = 2π radians.
Angular velocity is measured in angular units per second. Due to the convenient conversion from angular to linear units and vice versa, radians are widely used in science, which are 2π in a full turn.
If T = seconds/rev, then T/2π = seconds/radians, and 2π/T = radians/second. That's all.
Do you want degrees? You are welcome! T/360 = seconds/degree, and 360/T = degrees/second.