USAPhO 2021, Първи кръг — Задача 21
2021 Exam · 1 т.
Указания
25 QUESTIONS - 75 MINUTES
DO NOT OPEN THIS TEST UNTIL YOU ARE TOLD TO BEGIN
- Use N/kg throughout this contest.
- You may write in this booklet of questions. However, you will not receive any credit for anything written in this booklet. You may only use the scratch paper provided by the proctor.
- This test contains 25 multiple choice questions. Select the answer that provides the best response to each question. Please be sure to use a No.2 pencil and completely fill the box corresponding to your choice. If you change an answer, the previous mark must be completely erased. Only the boxes preceded by numbers 1 through 25 are to be used on the answer sheet.
- All questions are equally weighted, but are not necessarily of the same level of difficulty.
- Correct answers will be awarded one point; incorrect answers or leaving an answer blank will be awarded zero points. There is no additional penalty for incorrect answers.
- A hand-held calculator may be used. 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 tables, books, or collections of formulas.
- The question booklet, the answer sheet and the scratch paper will be collected at the end of this exam.
- In order to maintain exam security, do not communicate any information about the questions (or their answers or solutions) on this contest until after February 1, 2021.
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):
JiaJia Dong, Mark Eichenlaub, Abi Krishnan, Kye Shi, Brian Skinner, Mike Winer and Kevin Zhou
Условие
A solid sphere sits at the top of a ramp of height inclined at angle to the horizontal. Both the static and kinetic coefficients of friction between the sphere and incline are . The sphere is released from rest at the top of the incline. For which of the following values of is the total translational plus rotational kinetic energy of the sphere greatest when it reaches the bottom of the incline?
(A)
(B)
(C)
(D)
(E) The mechanical energy is the same for all choices.
Отговори
- A
Решение
As long as the sphere rolls without slipping, it converts all its gravitational potential energy into kinetic energy, so its mechanical energy stays constant. That’s what happens when the slope is very shallow. When the slope is steep, the acceleration is greater, and the torque required to roll without slipping is greater than what friction can provide. At these slopes, the sphere slips and dissipates mechanical energy, so its mechanical energy at the bottom of the slope is lower.
When rolling without slipping, a solid sphere has acceleration down an incline of .
The critical transition slope occurs when the friction force required to provide angular acceleration for the ball exceeds the maximum friction of . This occurs when , or about when . This means that case (a) is rolling without slipping, while (b), (c), and (d) all have slipping and less kinetic energy at the bottom of the slope.
Оригинал в Архива: 2021_Fma_exam.pdf · официални решения: 2021_Fma_Solutions.pdf