USAPhO 2023, Първи кръг — Задача 16

2023 F=maF = ma Exam · 1 т.

Указания

2023 F=maF = ma Exam

25 QUESTIONS - 75 MINUTES

INSTRUCTIONS

DO NOT OPEN THIS TEST UNTIL YOU ARE TOLD TO BEGIN

  • Use g=10 N/kgg = 10\ \mathrm{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, 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

A bead attached to a string of length ℓ=10 m\ell = 10\ \mathrm{m} is released from a very small angle θ0\theta_0 to the vertical. A wall is placed in the path of the bead such that the bead collides elastically with the wall when the string is at an angle θ0/2\theta_0/2 to the vertical, as shown.

A pendulum of length ℓ hanging from the ceiling near a wall on the left; the bead m is pulled out to the right at angle θ0 from the dotted vertical. A dotted line from the pivot to the wall's base shows the string at the collision position on the left.

What is the time interval between the bead’s collisions with the wall?

  • (A) 2π3 s\dfrac{2\pi}{3}\ \mathrm{s}
  • (B) 3π4 s\dfrac{3\pi}{4}\ \mathrm{s}
  • (C) 4π3 s\dfrac{4\pi}{3}\ \mathrm{s}
  • (D) 3π2 s\dfrac{3\pi}{2}\ \mathrm{s}
  • (E) 2π s2\pi\ \mathrm{s}

Отговори

Решение

Оригинал в Архива: 2023_F-ma_Exam.pdf · официални решения: 2023_F-ma_Solutions_v3.pdf