USAPhO 2024, Първи кръг — Задача 22
2024 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 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 spherical shell is made from a thin sheet of material with a mass per area of . Consider two points, and , which are close to each other, but just inside and outside the sphere, respectively. If the accelerations due to gravity at these points are and , respectively, what is the value of ?
(A)
(B)
(C)
(D)
(E)
Отговори
- 4\pi G\sigma
Решение
Suppose the shell has radius . By the shell theorem, the gravitational field vanishes inside, so . The gravitational field just outside has magnitude where is the total mass of the shell. Thus, the unknown variable cancels out, and the answer is .
Note that this is actually a universal statement. Above, we found that it didn’t depend on the sphere’s radius, but more generally, it is true regardless of the shell’s shape. The reason is that is solely determined by the mass near and , and if one zooms in enough to a patch of any surface, then it always looks like an infinite plane, so the answer doesn’t depend on the shape of the surface away from that patch.
Оригинал в Архива: 2024_F-ma_Exam.pdf · официални решения: 2024_F-ma_Solutions.pdf