OPhO 2020, Теория — Задача 1. A Party Trick

2020 Online Physics Olympiad: Invitational Contest v1.4 · 20 т.

Внимание: Бележка към темата

The v1.4 question paper, native Partial Solutions and separately attributed organizer clarifications are retained. The key is incomplete: Problem 1 is only marked TBD, Problem 8 has no key for parts f–h or an explicit critical power in e, and Problem 10 has no key for Part B f–h. Problems 2 and 9 also have source gaps noted below. Printed mathematical defects and differences between repeated questions in the key and the question original are not silently repaired. Problem2(e) gives a time-dependent spin result but no explicit answer in orbital angle; key2(c) refers to an absent diagram. Problem9(a) gives its ODE and initial conditions without an explicit speed, and 9(b) does not state whether the pressure increases or decreases. Native source typography is retained, including the Planck-constant value, missing electron-charge unit, missing kappa in key3(a), mismatched labels in key4, inconsistent theta indices and ring factors in key6, the negative probability/energy signs and r_B/K_B/k variants in key7, radiation expressions and fusion-rate/proton-rate difference in key8, and image/flux signs and field factors in key10. All these are printed-source issues, not repaired answers. Question2 omits the word define while the repeated key question includes it; the repeated key question7 adds For simplicity, ignore spin. The 4-page organizer clarification contradicts question1(d)’s frictionless table and also contains both only one final submission and multiple submissions instructions. The native original and the separately attributed clarifications remain visible.

Original credits

2020 Online Physics Olympiad (OPhO): Invitational Contest v1.4

OPhO Lead Organizers and Writers: Adithya Balachandran, Eddie Chen, Ashmit Dutta, Ethan Hu, Viraj Jayam, Rishab Parthasarathy, Kushal Thaman, Max Wang, Qilin Xue

OPhO Website Developer: Ben Oommen

Advisors: Dr. Branislav Kisačanin, Kye Shi, Kevin Zhou

July 3 - 6, 2020

Указания

If you wish to request a clarification, please use this form. To see all clarifications, see this document.

  • This test contains 10 long answer questions.
  • The total base score for the exam is 300 points; a factor incorporating the number of teams who solved the question will be added in the marking scheme. The final scores of all the teams will be available a few days after the contest ends.
  • The team leader should submit their final solution document in this here. We don’t anticipate a tie, but in the rare circumstance that there is one, the time you submit will be used to break it.
  • You can write on paper, type it up online, or a mix of both. If you wish to use a pre-made LaTeX template, we made one which you can choose to use here.
  • Since this is a long answer response, you will be judged on the quality of your work. To receive full points, you need to show your work, including deriving equations. As a general rule of thumb, any common equations (such as the ones in this formula sheet) can be cited without proof. Remember to state any approximations made, which system of equations were solved after every step. Explicitly showing every step of algebra is not necessary.
  • You may leave all final answers in symbolic form (in terms of variables) unless otherwise specified.2
  • Be sure to state all assumptions.
Sponsors

WOLFRAM · AwesomeMath (making x,y,z as easy as a,b,c) · Physics Olympiad · AoPS.com Art of Problem Solving

List of Constants
  • Proton mass, mp=1.67⋅10−27 kgm_p=1.67\cdot10^{-27}\ \mathrm{kg}
  • Neutron mass, mn=1.67⋅10−27 kgm_n=1.67\cdot10^{-27}\ \mathrm{kg}
  • Electron mass, me=9.11⋅10−31 kgm_e=9.11\cdot10^{-31}\ \mathrm{kg}
  • Avogadro’s constant, N0=6.02⋅1023 mol−1N_0=6.02\cdot10^{23}\ \mathrm{mol^{-1}}
  • Universal gas constant, R=8.31 J/(mol⋅K)R=8.31\ \mathrm{J/(mol\cdot K)}
  • Boltzmann’s constant, kB=1.38⋅10−23 J/Kk_B=1.38\cdot10^{-23}\ \mathrm{J/K}
  • Electron charge magnitude, e=1.60⋅10−19e=1.60\cdot10^{-19}
  • 1 electron volt, 1 eV=1.60⋅10−19 J1\ \mathrm{eV}=1.60\cdot10^{-19}\ \mathrm J
  • Speed of light, c=3.00⋅108 m/sc=3.00\cdot10^8\ \mathrm{m/s}
  • Universal Gravitational constant, G=6.67⋅10−11 (N⋅m2)/kg2G=6.67\cdot10^{-11}\ \mathrm{(N\cdot m^2)/kg^2}
  • Acceleration due to gravity, g=9.81 m/s2g=9.81\ \mathrm{m/s^2}
  • 1 unified atomic mass unit, 1 u=1.66⋅10−27 kg=931 MeV/c21\ \mathrm u=1.66\cdot10^{-27}\ \mathrm{kg}=931\ \mathrm{MeV}/c^2
  • Planck’s constant, h=6.63⋅10−34 J⋅s=4.41⋅10−15 eV⋅sh=6.63\cdot10^{-34}\ \mathrm{J\cdot s}=4.41\cdot10^{-15}\ \mathrm{eV\cdot s}
  • Permittivity of free space, ϵ0=8.85⋅10−12 C2/(N⋅m2)\epsilon_0=8.85\cdot10^{-12}\ \mathrm{C^2/(N\cdot m^2)}
  • Coulomb’s law constant, k=14πϵ0=8.99⋅109 (N⋅m2)/C2k=\frac1{4\pi\epsilon_0}=8.99\cdot10^9\ \mathrm{(N\cdot m^2)/C^2}
  • Permeability of free space, μ0=4π⋅10−7 T⋅m/A\mu_0=4\pi\cdot10^{-7}\ \mathrm{T\cdot m/A}
  • Magnetic constant, μ04π=1⋅10−7 (T⋅m)/A\frac{\mu_0}{4\pi}=1\cdot10^{-7}\ \mathrm{(T\cdot m)/A}
  • 1 atmospheric pressure, 1 atm=1.01⋅105 N/m2=1.01⋅105 Pa1\ \mathrm{atm}=1.01\cdot10^5\ \mathrm{N/m^2}=1.01\cdot10^5\ \mathrm{Pa}
  • Wien’s displacement constant, b=2.9⋅10−3 m⋅Kb=2.9\cdot10^{-3}\ \mathrm{m\cdot K}
  • Stefan-Boltzmann constant, σ=5.67⋅10−8 W/m2/K4\sigma=5.67\cdot10^{-8}\ \mathrm{W/m^2/K^4}
Native key title

2020 Online Physics Olympiad (OPhO): Invitational Contest Partial Solutions

The same organizer, writer, developer and advisor credits appear on this key cover.

Organizer clarification document — major notes

Separately quoted from the organizer clarification document, titled 2020 OPhO Invitational Clarifications:

MAJOR NOTES

  • Hi everyone! All team members are welcome to fill out this (optional) feedback form. Everyone who submits (and consents) will be entered in a draw to receive a $25 Amazon gift card.
  • Use this link to submit clarification requests. The problem writers may or may not deem it necessary to answer all clarification requests.
  • Link to the overleaf template (the link has been fixed) for answer submission.
  • Reminder: There is only one final submission at the end of the competition via the submission form only. Do NOT email us a copy of your submission. In case of internet connectivity issues, any team member may email us the file.
  • Make sure to read over the instructions. You are expected to give symbolic answers unless explicitly otherwise stated. If the numerical values for a symbol are stated, you do not need to solve the problem for a general case.
  • Uploading a .jpeg on the overleaf for a graph is allowed.
  • Please check the newest version v1.4 for revisions.
  • Please provide clear reasoning or derivations of any obscure results. Directly citing them would not necessarily fetch full points.
  • We will be awarding a special prize for whichever team produces the most creative and/or elegant solution to certain problems. We will not be disclosing which problems until after the competition has concluded.

Note on submission of solutions:

It is highly recommended that the solutions to the invitational round be submitted in LaTeXed form. We recommend using Overleaf. It is preferred that the solutions are quite detailed; this means that while you do not need to explicitly write all your algebra, all the relevant equations must be clearly written. A brief wording on how you solve a given system of equations is also optimal. Clearly state any approximations you make. Solutions typed in .doc/.docx, and ones neatly written by hand will also be accepted. Some students have a handwriting that’s hard to read; if that’s you, please read this. You can submit solutions multiple times. Only the latest submission will count.

Условие

It is a well known party trick that by pulling the tablecloth very quickly and suddenly, the plates on top of the table can stay nearly in place.

a) (2 pt) Start with a circular table D=1 mD=1\ \mathrm m in diameter with a very flat and small (and dimensionless for now) plate in the centre. How fast must you pull the tablecloth so the plate remains on the table? Assume the static friction coefficient is μs=0.5\mu_s=0.5 and kinetic friction is μk=0.3\mu_k=0.3 (for any contacts with the tablecloth) and that the tablecloth accelerates instantaneously. The tablecloth does not overhang the table.

b) (4 pts) Mythbusters famously attempted to replicate the same trick with a giant tablecloth and a motorbike. We will simplify the experiment by assuming that the table was only set with plates placed d=0.5 md=0.5\ \mathrm m apart from each other and from both ends. Assuming a ℓ=7 m\ell=7\ \mathrm m long table, how fast must the bike be travelling to successfully carry out this experiment? The mass of the cloth and plates are negligible compared to the mass of the motorbike. Assume small plates and that the tablecloth does not overhang the table.

c) (6 pts) A young boy places his toy car in the centre on the table from part (b). He believes that since it has wheels, it will stay on the table easier. Confirm or deny this statement and find the significant speeds that the tablecloth is pulled at that would cause the car to stay on the table. Assume the car (including wheels) has mass 2m2m, and that each wheel is a uniform disk of mass m/4m/4. The car is small in comparison to the table.

d) (8 pts) We run the experiment one last time with the glass, with same mass as a plate, placed on top of each plate on the tablecloth, which in turn, is on a frictionless table. The static and kinetic coefficient between the glass and the plate are μs′=0.30\mu_s'=0.30 and μk′=0.15\mu_k'=0.15 respectively. How fast must the tablecloth be pulled so that the glasses stay completely on the plate and the plates stay completely on the table? Do not assume that the either plate or glass is dimensionless this time. A diagram is provided below:

Two glasses on plates and a tablecloth pulled right; all 10cm,15cm,30cm,50cm dimensions and cloth arrow retained.

Published organizer clarifications (separate source)

Separately quoted from the organizer clarification document:

Problem 1: The coefficient of friction (0.5 static, 0.3 kinetic) applies to the interaction of all surfaces contacting the tablecloth and table unless otherwise stated. Note that there was previously an incorrect clarification here, This applies to part b as well, and carries over to d. Since the speed of the table cloth is constant, find the speed of the tablecloth for parts a and b and d.

  • This means the table + object(s) surface has friction.
  • Assume a 4-wheeled car.
  • For the car, we are interested if the car remains on the table, no matter in which direction it falls off from

Решение

Внимание: Непълно решение

The original available Partial Solutions contains only TBD for this problem, with no worked solution. No answer has been invented.

Organizer clarification document — other notes

Separately quoted from the organizer clarification document:

Other Notes

  • Uploading images: View this link to find how to upload images on overleaf.
  • You are allowed to mix LaTeX and handwritten PDFs for your answer sheet. Note that you need to write your solution to the problem alongside your final answers.
  • You can use basic formulae related to topics on the IPhO syllabus without derivation but all other formulae which go beyond that must be derived.
  • If you find that space is running out on your LaTeX file, try to use \newpage\verb|\newpage| and see if that fixes it.

Оригинал в Архива: invitational.pdf · официални решения: invitational_sol.pdf · допълнителни данни: Clarifications.pdf