Imagine a point on the coordinate plane. Two rays emanating from it form an angle. Its value can be determined both in radians and in degrees. Now, at some distance from the center point, we mentally draw a circle. The measure of the angle, expressed in radians, in this case is the mathematical ratio of the length of the arc L, separated by two rays, to the distance between the center point and the circle line (R), that is:
Fi = l / r
If we now imagine the described system as material, then we can apply to it not only the concept of angle and radius, but also centripetal acceleration, rotation, etc. Most of them describe the behavior of a point located on a rotating circle. By the way, a solid disk can also be represented by a set of circles, the difference of which is only in the distance from the center.
One of the characteristics of such a rotating system is the period of revolution. It indicates the value of time for which a point on an arbitrary circle returns to its original position or, which is also true, will turn 360 degrees. At a constant rotation speed, the correspondence T = (2 * 3.1416) / Ug (hereinafter Ug is the angle) is satisfied.
Speed indicates the number of full revolutions performed in 1 second. At a constant speed, we obtain v = 1 / T.
The angular velocity depends on time and the so-called angle of rotation. That is, if we take an arbitrary point A on the circle as the reference point, then when the system rotates, this point shifts to A1 in time t, forming an angle between the radii of the A-center and A1-center. Knowing the time and angle, you can calculate the angular velocity.
And since there is a circle, movement and speed, then there is a centripetal acceleration. It is one of the components describing the movement of a material point in the case of curvilinear motion. The terms "normal" and "centripetal acceleration" are identical. The difference is that the second is used to describe the movement in a circle when the acceleration vector is directed toward the center of the system. Therefore, it is always necessary to know exactly how the body (point) moves and its centripetal acceleration. Its definition is as follows: it is the rate of change of speed, the vector of which is directed perpendicular to the direction of the instantaneous velocity vector and changes the direction of the latter. The encyclopedia indicates that Huygens was engaged in the study of this issue. The centripetal acceleration formula proposed by him looks like:
Acs = (v * v) / r,
where r is the radius of curvature of the path traveled; v is the speed of movement.
The formula by which centripetal acceleration is calculated still causes heated debate among enthusiasts. For example, a curious theory has recently been voiced.
Huygens, considering the system, proceeded from the fact that the body moves in a circle of radius R with a speed v measured at the starting point A. Since the inertia vector is directed along the tangent to the circle, we get a trajectory in the form of a straight AB. However, a centripetal force keeps the body in a circle at point C. If we designate the center behind O and draw the lines AB, BO (the sum of BS and CO), as well as AO, we get a triangle. In accordance with the law of Pythagoras:
OA = CO;
AB = t * v;
BS = (a * (t * t)) / 2, where a is the acceleration; t is time (a * t * t is speed).
If we now use the Pythagorean formula, then:
R2 + t2 + v2 = R2 + (a * t2 * 2 * R) / 2+ (a * t2 / 2) 2, where R is the radius, and alphanumeric writing without the multiplication sign is the degree.
Huygens admitted that, since the time t is short, it can be ignored in the calculations. Transforming the previous formula, she came to the famous Acs = (v * v) / r.
However, since time is squared, a progression arises: the greater t, the higher the error. For example, for 0.9 it turns out to be unaccounted for almost the final value of 20%.
The concept of centripetal acceleration is important for modern science, but it is obvious that it is too early to put an end to this issue.