USAPhO 2026, Първи кръг — Задача 18

2026 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 13, 2026.

DO NOT OPEN THIS TEST UNTIL YOU ARE TOLD TO BEGIN

We acknowledge The US Physics Team coaches and other contributors to this year’s exams:

Tengiz Bibilashvili, Eddie Chen, Orhun Ciftcioglu, Kellan Colburn, Evan Erickson, Harry Kroft, Natalie LeBaron, Mai Qi, Elena Yudovina

Условие

Three springs and a block of mass mm are connected as shown in the figure below. The spring constants are k1=kk_1 = k, k2=k/5k_2 = k/5, and k3=k/3k_3 = k/3. All pulleys and ropes are massless, and there is no friction anywhere in the system. Find the ratio of the period of oscillations of this system, TT, to the period T0T_0 of a simple mass–spring system with mass mm and spring constant kk. The value of TT0\frac{T}{T_0} is:

Between a ceiling bar and a floor bar: a rope from the ceiling passes through spring k_1 down around a lower movable pulley that carries block m, then up through spring k_2 over a fixed pulley hanging from the ceiling, and down through spring k_3 to the floor.

The value of TT0\frac{T}{T_0} is:

(A) 23\frac{2}{3}

(B) 32\frac{3}{2}

(C) 23\frac{\sqrt{2}}{\sqrt{3}}

(D) 76\frac{\sqrt{7}}{6}

(E) 67\frac{\sqrt{6}}{7}

Отговори

Решение

Оригинал в Архива: 2026_FMA_exam.pdf · официални решения: 2026_FMA_solutions.pdf