Centripetal acceleration

**Proportional or inversely proportional to radius**

It is undeniable that the material of circular motion in the first grades of high school is more difficult for students compared to rectilinear motion and there is some relief for students after they "get out" of this material. I tried to answer why this is so. The micro-methodology of teaching physics that deals with details in teaching reveals several conceptual difficulties that make this material "difficult"? In general, circular motion is not intuitive, so ways of interpretation are suggested that facilitate the acquisition of this material. It is also a fact that in this part of the material there is a conceptually new mathematical apparatus that should be gradually introduced.

In textbooks, as an illustration of centripetal acceleration (and force), photos of a carousel from a Luna park are usually included. On that device, the acceleration is greater if we are further from the center of rotation. And then the text provides a formula of inverse proportionality with the radius, which says that the acceleration is greater if we are closer to the center. The problem is that angular velocity is not yet introduced in the 1^{st} grade, but only in the 3^{rd} when describing vibrations. To get to the concept of angular velocity, one must gradually master the concept of a circle, the length (extent) of a circular path, explain the number π, introduce the concept of period, i.e. the time of one round of a circle, learn what peripheral or tangential velocity is, introduce a new measure of angle - radians, and only then learn what angular velocity is.

For an observer in a stationary system, centrifugal acceleration and centrifugal force do not exist. The truth is that centrifugal and centripetal forces are of the same amount and opposite directions and act on the same object but **in different systems**, so they are not subject to Newton's 1^{st} law and do not cancel each other out. If we were to claim that the centripetal and centrifugal forces cancel each other out, it would mean that the total force on the object is zero, so according to Newton's 1^{st} law, the object should be at rest or moving in a straight line, but it does not because it is rotating.

The ease of introducing concepts, which the teacher considers trivial, makes physics difficult to learn, and even hated by students.

Why physics is the subject with the most bad grades (next to mathematics). Why doesn't anyone want to study physics after their own experience with physics in primary and high school? Perhaps precisely because it is "not a problem" to pronounce textbook material.

However, a teacher's good intentions and subject-matter knowledge alone — along with his determination to do his job as well as possible — are insufficient for successful teaching. Because teaching is different from lecturing, successful teaching calls for more than that. Teaching is something else. It is, first and foremost, an interpretation of the material modified to suit particular intentions.
Teaching, therefore, cannot be satisfied with simply talking about the topic, "lecturing", but requires a kind of work strategy whose goal is to bring the student into a situation of discovering new knowledge about himself and the world. In class, definitions are not taught, but the discovery of new insights is organized. Lecture needs to be prepared, to determine what to do and how to proceed. To prepare for lesson processing, it's important to have a basic understanding of how to interpret the content and how to implement it in class.

Teacher professional development meetings have little or no impact on teaching. From my own experience (I've given workshops and lectures at over 80 meetings), I've noticed that few people use any of this in their teaching.
That's natural.

The teacher always teaches the best he knows, and he believes that this is beneficial. It will be difficult to adjust the "style" and approach. Any failure in teaching (negative grades, lack of interest, boredom, etc.) is typically attributed to factors other than the teacher himself (a large number of students in the class, low salary, no support from classmasters and principals, lack of equipment for experiments and laboratories, etc.).

No advisor who signs certificates of attendance at a professional meeting inquires as to how many of those topics listened to for hours were used in class. The meeting is finished; individuals presented their statistically processed findings in PowerPoint, and that is it. Only a few teachers who question their teaching seek better solutions.