USAPhO 2024, Първи кръг — Задача 10

2024 F=maF = ma Exam · 1 т.

Указания

25 QUESTIONS - 75 MINUTES

DO NOT OPEN THIS TEST UNTIL YOU ARE TOLD TO BEGIN

  • Use g=10 N/kgg = 10\ \mathrm{N/kg} 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 block of mass mm is connected to the walls of a frictionless box by two massless springs with relaxed lengths ℓ\ell and 2ℓ2\ell, and spring constants kk and 2k2k respectively. The length of the box is 3ℓ3\ell. The system rotates with a constant angular velocity ω\omega about one of its walls.

An open box rotating with angular velocity ω about a vertical axis through its left wall; inside, a block m is attached to the left wall by a spring labelled ℓ, k and to the right wall by a spring labelled 2ℓ, 2k.

Suppose the block stays at a constant distance rr from the axis of rotation, without touching either of the walls. What is the value of rr?

(A) 2kℓ2k−mω2\dfrac{2k\ell}{2k - m\omega^2}

(B) 2kℓ2k+mω2\dfrac{2k\ell}{2k + m\omega^2}

(C) 2kℓ3k+mω2\dfrac{2k\ell}{3k + m\omega^2}

(D) 3kℓ3k−mω2\dfrac{3k\ell}{3k - m\omega^2}

(E) 3kℓ3k+mω2\dfrac{3k\ell}{3k + m\omega^2}

Отговори

Решение

Оригинал в Архива: 2024_F-ma_Exam.pdf · официални решения: 2024_F-ma_Solutions.pdf