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USAPhO 2018, Полуфинал — Задача 3. Question A3

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Всички материали от USAPhO 2018

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

Внимание: Бележка към темата

The supervisor and student security dates differ in the original packet. The printed diode key retains its positive negative-voltage exponential and positive discharge derivative despite surrounding text/graphs. The radiation-pressure key retains inconsistent intensity/energy-density terminology and its classical electromagnetic definitions. Exact original sources and detailed defect evidence accompany the transcription.

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

This examination consists of two parts: Part A has three questions and is allowed 90 minutes; Part B also has three 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 3 parts: the cover sheet (page 2), Part A (pages 3-5), Part B (pages 7-9), and several answer sheets for one of the questions in Part A (pages 11-13). 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 21, 2018.
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 three 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 three problems. Each question is worth 25 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 proctor’s AAPT ID number, the question number and the page number/total pages for this problem, in the upper right hand corner of each page. For example,

Student AAPT ID # Proctor AAPT ID # 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 13, 2018.

Possibly Useful Information. You may use this sheet for both parts of the exam.

$g$	$9.8\ \mathrm{N/kg}$
$k=1/(4\pi\varepsilon_0)$	$8.99\times10^9\ \mathrm{N\cdot m^2/C^2}$
$c$	$3.00\times10^8\ \mathrm{m/s}$
$N_A$	$6.02\times10^{23}\ (\mathrm{mol})^{-1}$
$\sigma$	$5.67\times10^{-8}\ \mathrm{J/(s\cdot m^2\cdot K^4)}$
$1\ \mathrm{eV}$	$1.602\times10^{-19}\ \mathrm J$
$m_e$	$9.109\times10^{-31}\ \mathrm{kg}=0.511\ \mathrm{MeV}/c^2$
$\sin\theta$	$\approx\theta-\frac16\theta^3$ for $\vert \theta\vert \ll1$
$G$	$6.67\times10^{-11}\ \mathrm{N\cdot m^2/kg^2}$
$k_m=\mu_0/(4\pi)$	$10^{-7}\ \mathrm{T\cdot m/A}$
$k_B$	$1.38\times10^{-23}\ \mathrm{J/K}$
$R=N_Ak_B$	$8.31\ \mathrm{J/(mol\cdot K)}$
$e$	$1.602\times10^{-19}\ \mathrm C$
$h$	$6.63\times10^{-34}\ \mathrm{J\cdot s}=4.14\times10^{-15}\ \mathrm{eV\cdot s}$
$(1+x)^n$	$\approx1+nx$ for $\vert x\vert \ll1$
$\cos\theta$	$\approx1-\frac12\theta^2$ for $\vert \theta\vert \ll1$

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

Ideally the test supervisor will divide the question paper into 3 parts: the cover sheet (page 2), Part A (pages 3-10), Part B (pages 12-18), and several answer sheets for one of the questions in Part A (pages 20-22). Examinees should be provided parts A and B individually, although they may keep the cover sheet. The answer sheets should be printed single sided!

Условие

A vacuum system consists of a chamber of volume $V$ connected to a vacuum pump that is a cylinder with a piston that moves left and right. The minimum volume in the pump cylinder is $V_0$ , and the maximum volume in the cylinder is $V_0+\Delta V$ . You should assume that $\Delta V\ll V$ .

Full native vacuum pump illustration, motor/crank, cylinder/piston and inlet/outlet valves, chamber V and every label.

The cylinder has two valves. The inlet valve opens when the pressure inside the cylinder is lower than the pressure in the chamber, but closes when the piston moves to the right. The outlet valve opens when the pressure inside the cylinder is greater than atmospheric pressure $P_a$ , and closes when the piston moves to the left. A motor drives the piston to move back and forth. The piston moves at such a rate that heat is not conducted in or out of the gas contained in the cylinder during the pumping cycle. One complete cycle takes a time $\Delta t$ . You should assume that $\Delta t$ is a very small quantity, but $\Delta V/\Delta t=R$ is finite. The gas in the chamber is ideal monatomic and remains at a fixed temperature of $T_a$ .

Start with assumption that $V_0=0$ and there are no leaks in the system.

a. At $t=0$ the pressure inside the chamber is $P_a$ . Find an equation for the pressure at a later time $t$ .

b. Find an expression for the temperature of the gas as it is emitted from the pump cylinder into the atmosphere. Your answer may depend on time.

For the remainder of this problem $0<V_0<\Delta V\ll V$ .

c. Find an expression for the minimum possible pressure in the chamber, $P_{\min}$ .

STOP: Do Not Continue to Part B

If there is still time remaining for Part A, you should review your work for Part A, but do not continue to Part B until instructed by your exam supervisor.

Решение

a.

During each cycle, the system sucks gas out of the chamber and pushes it into the atmosphere. Since $V_0=0$ , the inlet valve opens the moment the piston starts moving to the left. When the piston is all the way to the left, a fraction $\Delta V/(V+\Delta V)$ of the gas is in the cylinder. As the piston moves to the right, all of this gas is pushed out, so after a single cycle,

P_f=P_i\left(\frac V{V+\Delta V}\right)

and in general,

P(t)=P_a\left(\frac V{V+\Delta V}\right)^{t/\Delta t}.

While this is technically correct, it can be simplified significantly. Write

P(t)=P_a\left(1+\frac{\Delta V}V\right)^{-t/\Delta t}=P_a\left((1+x)^{1/x}\right)^{-Rt/V}

where $x=\Delta V/V\ll1$ . Then using the definition of $e$ ,

e=\lim_{x\to0}(1+x)^{1/x}

we have

P(t)=P_ae^{-Rt/V}.

b.

When the piston is all the way to the left, the pressure is $P(t)$ and the temperature is $T_a$ . As the piston moves to the right, the gas is adiabatically compressed until its pressure reaches $P_a$ and the outlet valve opens. Since $PV^\gamma$ is constant during adiabatic compression and $PV/T$ is constant by the ideal gas law,

T_{\mathrm{out}}(t)=T_a\left(\frac{P_a}{P(t)}\right)^{1-1/\gamma}=T_a\left(\frac{P_a}{P(t)}\right)^{2/5}=T_ae^{2Rt/5V}

where we used $\gamma=5/3$ for a monatomic ideal gas.

c.

Since $V_0>0$ , the inlet valve will not open immediately when the piston begins moving to the left; instead it will open once the pressure in the cylinder equals the pressure in the chamber. Since the expansion of the cylinder is adiabatic, $PV^\gamma$ is constant, so

P_{\min}=P_a\left(\frac{V_0}{V_0+\Delta V}\right)^\gamma=P_a\left(1+\frac{\Delta V}{V_0}\right)^{-\gamma}.

Оригинал в Архива: USAPhO-2018.pdf · официални решения: USAPhO-2018-Solutions.pdf