USAPhO 2003, Полуфинал — Задача 2. A2

AAPT / AIP United States Physics Team 2003 — 2003 Semi-Final Exam

Указания

DO NOT OPEN THIS TEST UNTIL YOU ARE TOLD TO BEGIN

  • Work Part A first. You have 90 minutes to complete all four problems.

  • After you have completed Part A, you may take a break.

  • Then work Part B. You have 90 minutes to complete both problems.

  • Show all your work. Partial credit will be given.

  • Start each question on a new sheet of paper. Be sure to put your name in the upper right-hand corner of each page, along with the question number and the page number/total pages for this problem. For example,

    Doe, Jamie
    A1 – 1/3

  • A hand-held calculator may be used. Its memory must be cleared of data and programs. It may not be used in graphing mode. Calculators may not be shared. You may not use any tables, books, or collections of formulas. You may use a ruler or straight edge.

  • Questions with the same point value are not necessarily of the same difficulty.

  • Good luck!

Possibly Useful Information
Gravitational field at the Earth’s surfaceg=9.8 N/kgg = 9.8\ \mathrm{N/kg}
Newton’s gravitational constantG=6.67×10−11 N⋅m2/kg2G = 6.67 \times 10^{-11}\ \mathrm{N{\cdot}m^2/kg^2}
Coulomb’s constantk=1/4πε0=9.0×109 N⋅m2/C2k = 1/4\pi\varepsilon_0 = 9.0 \times 10^{9}\ \mathrm{N{\cdot}m^2/C^2}
Biot-Savart constantkm=μ0/4π=10−7 T⋅m/Ak_\mathrm{m} = \mu_0/4\pi = 10^{-7}\ \mathrm{T{\cdot}m/A}
Speed of light in a vacuumc=3.0×108 m/sc = 3.0 \times 10^{8}\ \mathrm{m/s}
Boltzmann’s constantkB=1.38×10−23 J/Kk_\mathrm{B} = 1.38 \times 10^{-23}\ \mathrm{J/K}
Avogadro’s numberNA=6.02×1023 (mol)−1N_\mathrm{A} = 6.02 \times 10^{23}\ \mathrm{(mol)^{-1}}
Ideal gas constantR=NAkB=8.31 J/(mol⋅K)R = N_\mathrm{A}k_\mathrm{B} = 8.31\ \mathrm{J/(mol{\cdot}K)}
Stefan-Boltzmann constantσ=5.67×10−8 J/(s⋅m2⋅K4)\sigma = 5.67 \times 10^{-8}\ \mathrm{J/(s{\cdot}m^2{\cdot}K^4)}
Elementary chargee=1.6×10−19 Ce = 1.6 \times 10^{-19}\ \mathrm{C}
1 electron volt1 eV=1.6×10−19 J1\ \mathrm{eV} = 1.6 \times 10^{-19}\ \mathrm{J}
Planck’s constanth=6.63×10−34 J⋅s=4.14×10−15 eV⋅sh = 6.63 \times 10^{-34}\ \mathrm{J{\cdot}s} = 4.14 \times 10^{-15}\ \mathrm{eV{\cdot}s}
Electron massm=9.1×10−31 kg=0.51MeV/c2m = 9.1 \times 10^{-31}\ \mathrm{kg} = 0.51\mathrm{MeV/c^2}
Binomial expansion(1+x)n≈1+nx(1 + x)^n \approx 1 + nx for ∣x∣≪1\lvert x\rvert \ll 1
Small angle approximationssin⁡θ≈θ\sin\theta \approx \theta; cos⁡θ≈1−12θ2\cos\theta \approx 1 - \tfrac{1}{2}\theta^2

Условие

All of the numbered boxes represent resistors. Each one has resistance RR.

Resistor network in the shape of a square standing on a corner with vertices A (left), B (top), C (right) and D (bottom). Resistors 1 (A–B), 3 (B–C), 6 (A–D) and 8 (D–C) form the sides; resistors 2 (B to centre), 7 (centre to D), 4 (A to centre) and 5 (centre to C) connect the vertices to the central node.

a. What is the equivalent resistance between points A and C? [10 т.]

b. What is the equivalent resistance between points A and B? [15 т.]

Отговори

Решение

Оригинал в Архива: Semifinal 2003.pdf · официални решения: Semifinal 2003 Solutions.pdf