USAPhO 2023, Първи кръг — Задача 22
2023 Exam · 1 т.
Указания
2023 Exam
25 QUESTIONS - 75 MINUTES
INSTRUCTIONS
DO NOT OPEN THIS TEST UNTIL YOU ARE TOLD TO BEGIN
- Use 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, answer sheet and 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 25, 2023.
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, Kellan Colburn, Samuel Gebretsadkan, Abi Krishnan, Natalie LeBaron, Kye Shi, Mike Winer, and Kevin Zhou
Условие
Two springs with different spring constants are connected in three ways, as shown.
In the second case, the springs are connected to opposite ends of a string, which runs under a massless frictionless pulley. In each case, the two springs act like a single spring with an effective spring constant , , or . Which of the following is correct?
- (A)
- (B)
- (C)
- (D)
- (E)
Отговори
- A
Решение
Let the springs have spring constants and . In the first case, the forces of the springs simply add, so . In the third case, the springs adjust so they exert an equal force. If the total displacement is , then and . Solving for the force gives an effective spring constant
The second case is a bit less familiar, but it is actually similar to the third case. Again, we have the constraint that the forces exerted by the springs are equal, but the difference is that (1) pulling a distance now increases the length of the spring system by , and (2) the forces from each spring add together. Each doubles the resulting force, so
and so clearly . We also have since the arithmetic mean is always greater than the geometric mean.
Оригинал в Архива: 2023_F-ma_Exam.pdf · официални решения: 2023_F-ma_Solutions_v3.pdf