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USAPhO 2011, Полуфинал — A2

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Всички материали от USAPhO 2011

AAPT UNITED STATES PHYSICS TEAM AIP 2011 — Semifinal Exam · 25 т.

Внимание: Бележка към темата

Printed source errors are preserved: A1’s solution labels both quoted volumes V₀ and says “we used the fat that”; A2’s solution reverses its graph-variable description; A4 prints ω = f/(2π). The solutions repeat the question statements and common instructions; they are shown once here. No individual subpart point values are printed.

Important Instructions for the Exam Supervisor

DO NOT DISTRIBUTE THIS PAGE

Important Instructions for the Exam Supervisor

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 3 parts: the cover sheet (page 2), Part A (pages 3-4), and Part B (pages 6-7). Examinees should be provided parts A and B individually, although they may keep the cover sheet.
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 March 31, 2011.
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.
Please provide the examinees with graph paper for Part A.

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. 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 1, 2011.

Possibly Useful Information

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

$g = 9.8\ \mathrm{N/kg}$	$G = 6.67 \times 10^{-11}\ \mathrm{N \cdot m^2/kg^2}$
$k = 1/4\pi\epsilon_0 = 8.99 \times 10^{9}\ \mathrm{N \cdot m^2/C^2}$	$k_{\mathrm{m}} = \mu_0/4\pi = 10^{-7}\ \mathrm{T \cdot m/A}$
$c = 3.00 \times 10^{8}\ \mathrm{m/s}$	$k_{\mathrm{B}} = 1.38 \times 10^{-23}\ \mathrm{J/K}$
$N_{\mathrm{A}} = 6.02 \times 10^{23}\ (\mathrm{mol})^{-1}$	$R = N_{\mathrm{A}} k_{\mathrm{B}} = 8.31\ \mathrm{J/(mol \cdot K)}$
$\sigma = 5.67 \times 10^{-8}\ \mathrm{J/(s \cdot m^2 \cdot K^4)}$	$e = 1.602 \times 10^{-19}\ \mathrm{C}$
$1\mathrm{eV} = 1.602 \times 10^{-19}\ \mathrm{J}$	$h = 6.63 \times 10^{-34}\ \mathrm{J \cdot s} = 4.14 \times 10^{-15}\ \mathrm{eV \cdot s}$
$m_e = 9.109 \times 10^{-31}\ \mathrm{kg} = 0.511\ \mathrm{MeV/c^2}$	$(1+x)^n \approx 1 + nx$ for $\lvert x\rvert \ll 1$
$\sin\theta \approx \theta - \frac{1}{6}\theta^3$ for $\lvert\theta\rvert \ll 1$	$\cos\theta \approx 1 - \frac{1}{2}\theta^2$ for $\lvert\theta\rvert \ll 1$

Условие

A thin, uniform rod of length $L$ and mass $M = 0.258\ \mathrm{kg}$ is suspended from a point a distance $R$ away from its center of mass. When the end of the rod is displaced slightly and released it executes simple harmonic oscillation. The period, $T$ , of the oscillation is timed using an electronic timer. The following data is recorded for the period as a function of $R$ . What is the local value of $g$ ? Do not assume it is the canonical value of $9.8\ \mathrm{m/s^2}$ . What is the length, $L$ , of the rod? No estimation of error in either value is required. The moment of inertia of a rod about its center of mass is $(1/12)ML^2$ .

$R$ (m)	$T$ (s)
0.050	3.842
0.075	3.164
0.102	2.747
0.156	2.301
0.198	2.115

$R$ (m)	$T$ (s)
0.211	2.074
0.302	1.905
0.387	1.855
0.451	1.853
0.588	1.900

You must show your work to obtain full credit. If you use graphical techniques then you must plot the graph; if you use linear regression techniques then you must show all of the formulae and associated workings used to obtain your result.

Отговори

Решение

Using the parallel axis theorem, the period of such a physical pendulum is

T = 2\pi\sqrt{\frac{I}{mgR}} = 2\pi\sqrt{\frac{R^2 + L^2/12}{gR}}.

We can rearrange this into the linear form

y = mx + b, \quad y = R^2, \quad x = \frac{T^2 R}{4\pi^2}, \quad m = g, \quad b = -\frac{L^2}{12}.

Filling out a data table, we get

$R$	$T$	$T^2R/4\pi^2$	$R^2$
0.050	3.842	0.0187	0.0025
0.075	3.164	0.0190	0.0056
0.102	2.747	0.0195	0.0104
0.156	2.301	0.0209	0.0243
0.198	2.115	0.0224	0.0392
0.211	2.074	0.0230	0.0445
0.302	1.905	0.0278	0.0912
0.387	1.855	0.0337	0.1498
0.451	1.853	0.0392	0.2034
0.588	1.900	0.0538	0.3457

The graph of $T^2R/4\pi^2$ versus $R^2$ is a line with slope $g$ and intercept $-L^2/12$ , which should be plotted on graph paper.

Spreadsheet plot of the data: vertical axis from 0.0000 to 0.4000 in steps of 0.0500, horizontal axis from 0.0150 to 0.0600 in steps of 0.0050; ten square data points lie on a straight fitted line; the label reads f(x) = 9.786x - 0.180, R² = 1.000.

Looking at the graph, we read off the results

g = 9.79\ \mathrm{m/s^2}, \quad L = 1.47\ \mathrm{m}.

Also note that most of the first few data points are not useful. To get full credit, at least half of the data points should be used; it is not necessary to use all of them.

Оригинал в Архива: USAPhO-2011.pdf · официални решения: USAPhO-2011-Solutions.pdf