USAPhO 2026, Първи кръг — Задача 10
2026 Exam · 1 т.
Указания
25 QUESTIONS - 75 MINUTES
DO NOT OPEN THIS TEST UNTIL YOU ARE TOLD TO BEGIN
- Use 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
Consider the following three solid objects, each of mass and uniform density:
(i) a half-ball of radius ;
(ii) a cylinder of radius and height ;
(iii) a cylinder of radius and height .
For each object, consider the gravitational acceleration at the center of its flat circular face. Let , , and denote these accelerations for the half-ball, the cylinder of height , and the cylinder of height , respectively. Which of the following is correct?
(A)
(B)
(C)
(D)
(E)
Отговори
- C
Решение
In all three cases, symmetry ensures that the gravitational acceleration at the specified point is perpendicular to the flat circular face.
Starting from the half-ball, imagine continuously deforming it into a cylinder of radius and height while keeping the total mass fixed. During this deformation, mass elements move farther from the observation point and spread farther from the symmetry axis, reducing the perpendicular component of the gravitational field. Thus,
Next, deform the cylinder of height into a cylinder of the same radius but height . This moves mass farther away along the symmetry axis and further reduces the gravitational acceleration at the center of the base:
Therefore,
Оригинал в Архива: 2026_FMA_exam.pdf · официални решения: 2026_FMA_solutions.pdf