USAPhO 2021, Първи кръг — Задача 10
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 uniform solid circular disk of mass is on a flat, frictionless horizontal table. The center of mass of the disk is at rest and the disk is spinning with angular frequency . A stone, modeled as a point object also of mass , is placed on the edge of the disk, with zero initial velocity relative to the table. A rim built into the disk constrains the stone to slide, with friction, along the disk’s edge. After the stone stops sliding with respect to the disk, what is the angular frequency of rotation of the disk and stone together?
(A)
(B)
(C)
(D)
(E)
Отговори
- C
Решение
The center of mass of the stone-disk system is initially stationary, so it remains stationary. This means the final motion of the system is purely rotation about its center of mass.
The disk and stone conserve angular momentum. If the angular momentum of the disk about its center is , the initial angular momentum is
We need to calculate the angular momentum of the stone-disk system about its center of mass.
The center of mass is half way between the center of the disk and the stone, so if the radius of the disk is , and its mass , the moment of inertia of the stone is
By the parallel axis theorem, the angular momentum of the disk is
Using , the angular momentum of the stone-disk system is
By angular momentum conservation, doubling the moment of inertia cuts the angular frequency in half, so the final angular frequency is
Оригинал в Архива: 2021_Fma_exam.pdf · официални решения: 2021_Fma_Solutions.pdf