USAPhO 2015, Полуфинал — Задача 3. Question A3

USA Physics Olympiad Exam — Part A / Part B · 25 т.

Внимание: Бележка към темата

Original question and matching key disagree at A4: the question uses Tmax and X=C; repeated key statement uses400K and says X need not coincide with C, but actual worked key graph/argument still place C/X/Y together. Both native versions are explicitly retained. B1 question says three independent parts although only a/b exist; repeated key says two. Original mathematical/sign/symbol defects remain printed, with evidence notes.

Important Instructions for the Exam Supervisor — DO NOT DISTRIBUTE THIS PAGE

  • This examination consists of two parts.

  • Part A has four questions and is allowed 90 minutes.

  • Part B has two questions and is allowed 90 minutes.

  • The first page that follows is a cover sheet. Examinees may keep the cover sheet for both parts of the exam.

  • The parts are then identified by the center header on each page. Examinees are only allowed to do one part at a time, and may not work on other parts, even if they have time remaining.

  • Allow 90 minutes to complete Part A. Do not let students look at Part B. Collect the answers to Part A before allowing the examinee to begin Part B. Examinees are allowed a 10 to 15 minutes break between parts A and B.

  • Allow 90 minutes to complete Part B. Do not let students go back to Part A.

  • Ideally the test supervisor will divide the question paper into 4 parts: the cover sheet (page 2), Part A (pages 3-7), Part B (pages 9-11), and several answer sheets for one of the questions in Part A (pages 13-13). Examinees should be provided parts A and B individually, although they may keep the cover sheet. The answer sheets should be printed single sided!

  • The supervisor must collect all examination questions, including the cover sheet, at the end of the exam, as well as any scratch paper used by the examinees. Examinees may not take the exam questions. The examination questions may be returned to the students after April 15, 2015.

  • Examinees are allowed calculators, but they may not use symbolic math, programming, or graphic features of these calculators. Calculators may not be shared and their memory must be cleared of data and programs. Cell phones, PDA’s or cameras may not be used during the exam or while the exam papers are present. Examinees may not use any tables, books, or collections of formulas.

USA Physics Olympiad Exam — Instructions

DO NOT OPEN THIS TEST UNTIL YOU ARE TOLD TO BEGIN - Work Part A first. You have 90 minutes to complete all four problems. Each question is worth 25 points. Do not look at Part B during this time. - After you have completed Part A you may take a break. - Then work Part B. You have 90 minutes to complete both problems. Each question is worth 50 points. Do not look at Part A during this time. - Show all your work. Partial credit will be given. Do not write on the back of any page. Do not write anything that you wish graded on the question sheets. - Start each question on a new sheet of paper. Put your AAPT ID number, your name, the question number and the page number/total pages for this problem, in the upper right hand corner of each page. For example,

AAPT ID # Doe, Jamie A1 - 1/3

  • 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, PDA’s or cameras 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. - Questions with the same point value are not necessarily of the same difficulty. - In order to maintain exam security, do not communicate any information about the questions (or their answers/solutions) on this contest until after April 15, 2015.
Possibly Useful Information. You may use this sheet for both parts of the exam.
InformationValue
gg9.8 N/kg9.8\ \mathrm{N/kg}
k=1/(4πε0)k=1/(4\pi\varepsilon_0)8.99×109 N⋅m2/C28.99\times10^9\ \mathrm{N\cdot m^2/C^2}
cc3.00×108 m/s3.00\times10^8\ \mathrm{m/s}
NAN_A6.02×1023 (mol)−16.02\times10^{23}\ (\mathrm{mol})^{-1}
σ\sigma5.67×10−8 J/(s⋅m2⋅K4)5.67\times10^{-8}\ \mathrm{J/(s\cdot m^2\cdot K^4)}
1 eV1\ \mathrm{eV}1.602×10−19 J1.602\times10^{-19}\ \mathrm J
mem_e9.109×10−31 kg=0.511 MeV/c29.109\times10^{-31}\ \mathrm{kg}=0.511\ \mathrm{MeV}/c^2
sin⁡θ\sin\theta≈θ−16θ3\approx\theta-\frac16\theta^3 for ∣θ∣≪1\vert \theta\vert \ll1
GG6.67×10−11 N⋅m2/kg26.67\times10^{-11}\ \mathrm{N\cdot m^2/kg^2}
km=μ0/(4π)k_m=\mu_0/(4\pi)10−7 T⋅m/A10^{-7}\ \mathrm{T\cdot m/A}
kBk_B1.38×10−23 J/K1.38\times10^{-23}\ \mathrm{J/K}
R=NAkBR=N_Ak_B8.31 J/(mol⋅K)8.31\ \mathrm{J/(mol\cdot K)}
ee1.602×10−19 C1.602\times10^{-19}\ \mathrm C
hh6.63×10−34 J⋅s=4.14×10−15 eV⋅s6.63\times10^{-34}\ \mathrm{J\cdot s}=4.14\times10^{-15}\ \mathrm{eV\cdot s}
(1+x)n(1+x)^n≈1+nx\approx1+nx for ∣x∣≪1\vert x\vert \ll1
cos⁡θ\cos\theta≈1−12θ2\approx1-\frac12\theta^2 for ∣θ∣≪1\vert \theta\vert \ll1

Условие

A large block of mass mbm_b is located on a horizontal frictionless surface. A second block of mass mtm_t is located on top of the first block; the coefficient of friction (both static and kinetic) between the two blocks is given by μ\mu. All surfaces are horizontal; all motion is effectively one dimensional. A spring with spring constant kk is connected to the top block only; the spring obeys Hooke's Law equally in both extension and compression. Assume that the top block never falls off of the bottom block; you may assume that the bottom block is very, very long. The top block is moved a distance AA away from the equilibrium position and then released from rest.

Complete native stacked-block/spring diagram, mt/mb labels, wall and horizontal frictionless surface.

a. Depending on the value of AA, the motion can be divided into two types: motion that experiences no frictional energy losses and motion that does. Find the value AcA_c that divides the two motion types. Write your answer in terms of any or all of μ\mu, the acceleration of gravity gg, the masses mtm_t and mbm_b, and the spring constant kk.

b. Consider now the scenario A≫AcA\gg A_c. In this scenario the amplitude of the oscillation of the top block as measured against the original equilibrium position will change with time. Determine the magnitude of the change in amplitude, ΔA\Delta A, after one complete oscillation, as a function of any or all of AA, μ\mu, gg, and the angular frequency of oscillation of the top block ωt\omega_t.

c. Assume still that A≫AcA\gg A_c. What is the maximum speed of the bottom block during the first complete oscillation cycle of the upper block?

Решение

Оригинал в Архива: USAPhO-2015.pdf · официални решения: USAPhO-2015-Solutions.pdf