USAPhO 2024, Първи кръг — Задача 25
2024 Exam · 1 т.
Указания
25 QUESTIONS - 75 MINUTES
DO NOT OPEN THIS TEST UNTIL YOU ARE TOLD TO BEGIN
- Use throughout, unless otherwise specified.
- You may write in this question booklet and the scratch paper provided by the proctor.
- This test has 25 multiple choice questions. Select the best response to each question, and use a No.2 pencil to completely fill the box corresponding to your choice. If you change an answer, completely erase the previous mark. Only use the boxes numbered 1 through 25 on the answer sheet.
- All questions are equally weighted, but are not necessarily equally difficult.
- You will receive one point for each correct answer, and zero points for each incorrect or blank answer. There is no additional penalty for incorrect answers.
- You may use a hand-held calculator. Its memory must be cleared of data and programs. You may use only the basic functions found on a simple scientific calculator. Calculators may not be shared. Cell phones may not be used during the exam or while the exam papers are present. You may not use any external references, such as books or formula sheets.
- The question booklet, answer sheet and scratch paper will be collected at the end of this exam.
- To maintain exam security, do not communicate any information about the questions or their solutions until after February 24, 2024.
DO NOT OPEN THIS TEST UNTIL YOU ARE TOLD TO BEGIN
We acknowledge the following people for their contributions to this year’s exams (in alphabetical order):
Tengiz Bibilashvili, Orhun Ciftcioglu, Kellan Colburn, Natalie LeBaron, Brian Skinner, Elena Yudovina and Kevin Zhou
Условие
A yo-yo consists of two massive uniform disks of radius connected by a thin axle. A thick string is wrapped many times around the axle, so that the end of the string is initially a distance from the axle. Then, the end of the string is held in place and the yo-yo is dropped from rest. Assume that energy losses are negligible, and that the string has negligible mass and always remains vertical. Below, we show a cross-section of the yo-yo partway through its descent.
Between the moment the yo-yo is released and the moment the string completely unwinds, which of the following is true regarding the yo-yo’s acceleration?
(A) It is always zero.
(B) It points downward, but decreases in magnitude over time.
(C) It points downward and has constant magnitude.
(D) It points downward, but increases in magnitude over time.
(E) None of the above.
Отговори
- None of the above.
Решение
As the string unwinds, the radius of the string remaining about the axle decreases. Specifically, if the yo-yo has fallen a distance , and the string has total length , then
This change in affects how the energy of the yo-yo is divided between translational and rotational motion. If the discs have total mass , conservation of energy gives
where but . Solving for gives
which has a maximum at around . Intuitively, in the beginning the yo-yo has roughly constant acceleration downward, but as the string unwinds, more of the energy goes into the yo-yo’s rotational motion. In fact, the translational speed of the yo-yo goes to zero when the string completely unwinds, at which point it’s just spinning in place! Therefore, during the latter part of its fall, the yo-yo has upward acceleration, so the answer is “none of the above”.
This remains true regardless of the details of the yo-yo, because it only depends on the fact that the radius starts equal to and then becomes very small, since the axle is thin. In particular, changing how depends on , or the moment of inertia of the disc, won’t change the answer.
Оригинал в Архива: 2024_F-ma_Exam.pdf · официални решения: 2024_F-ma_Solutions.pdf