USAPhO 2007, Полуфинал — Задача 2. A2

AAPT AIP United States Physics Team 2007 — 2007 Semi-Final Exam · 25 т.

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. 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.

  • Do not discuss the contents of this exam with anyone until after March 27th.

  • Good luck!

Possibly Useful Information - (Use for both part A and for part B)
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=1/4πεo=8.99×109 N⋅m2/C2k = 1/4\pi\varepsilon_o = 8.99 \times 10^{9}\ \mathrm{N\cdot m^2/C^2}
Biot-Savart constantkm=μo/4π=10−7 T⋅m/Ak_m = \mu_o/4\pi = 10^{-7}\ \mathrm{T\cdot m/A}
Speed of light in a vacuumc=3.00×108 m/sc = 3.00 \times 10^{8}\ \mathrm{m/s}
Boltzmann’s constantkB=1.38×10−23 J/Kk_B = 1.38 \times 10^{-23}\ \mathrm{J/K}
Avogadro’s numberNA=6.02×1023 (mol)−1N_A = 6.02 \times 10^{23}\ \mathrm{(mol)^{-1}}
Ideal gas constantR=NAkB=8.31 J/(mol⋅K)R = N_A k_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=1.602×10−19 Ce = 1.602 \times 10^{-19}\ \mathrm{C}
1 electron volt1 eV=1.602×10−19 J1\ \mathrm{eV} = 1.602 \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=9.109×10−31 kg=0.511 MeV/c2m = 9.109 \times 10^{-31}\ \mathrm{kg} = 0.511\ \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 - \frac{1}{2}\theta^2

Условие

A simple gun can be made from a uniform cylinder of length L0L_0 and inside radius rcr_c. One end of the cylinder is sealed with a moveable plunger and the other end is plugged with a cylindrical cork bullet. The bullet is held in place by friction with the walls of the cylinder. The pressure outside the cylinder is atmospheric pressure, P0P_0. The bullet will just start to slide out of the cylinder if the pressure inside the cylinder exceeds PcrP_{cr}.

a. (16 т.)

There are two ways to launch the bullet: either by heating the gas inside the cylinder and keeping the plunger fixed, or by suddenly pushing the plunger into the cylinder. In either case, assume that an ideal monatomic gas is inside the cylinder, and that originally the gas is at temperature T0T_0, the pressure inside the cylinder is P0P_0, and the length of the cylinder is L0L_0.

i. Assume that we launch the bullet by heating the gas without moving the plunger. Find the minimum temperature of the gas necessary to launch the bullet. Express your answer in terms of any or all of the variables: rc,T0,L0,P0r_c, T_0, L_0, P_0, and PcrP_{cr}. [8 т.]

ii. Assume, instead that we launch the bullet by pushing in the plunger, and that we do so quickly enough so that no heat is transferred into or out of the gas. Find the length of the gas column inside the cylinder when the bullet just starts to move. Express your answer in terms of any or all of the variables: rc,T0,L0,P0r_c, T_0, L_0, P_0, and PcrP_{cr}. [8 т.]

b. (9 т.)

It is necessary to squeeze the bullet to get it into the cylinder in the first place. The bullet normally has a radius rbr_b that is slightly larger than the inside radius of the cylinder; rb−rc=Δrr_b - r_c = \Delta r, is small compared to rcr_c. The bullet has a length h≪L0h \ll L_0. The walls of the cylinder apply a pressure to the cork bullet. When a pressure PP is applied to the bullet along a given direction, the bullet’s dimensions in that direction change by

Δxx=−PE\frac{\Delta x}{x} = \frac{-P}{E}

for a constant EE known as Young’s modulus. You may assume that compression along one direction does not cause expansion in any other direction. (This is true if the so-called Poisson ratio is close to zero, which is the case for cork.)

If the coefficient of static friction between the cork and the cylinder is μ\mu, find an expression for PcrP_{cr}. Express your answer in terms of any or all of the variables: P0,μ,h,E,ΔrP_0, \mu, h, E, \Delta r, and rcr_c.

Отговори

Решение

Оригинал в Архива: olympiad_2007_semi-final.pdf · официални решения: olympiad_2007_semi-final_soln.pdf