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OPhO 2020, Теория — Задача 5. Mathematical Physics

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Всички материали от OPhO 2020

2020 Online Physics Olympiad: Invitational Contest v1.4 · 18 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, $m_p=1.67\cdot10^{-27}\ \mathrm{kg}$
Neutron mass, $m_n=1.67\cdot10^{-27}\ \mathrm{kg}$
Electron mass, $m_e=9.11\cdot10^{-31}\ \mathrm{kg}$
Avogadro’s constant, $N_0=6.02\cdot10^{23}\ \mathrm{mol^{-1}}$
Universal gas constant, $R=8.31\ \mathrm{J/(mol\cdot K)}$
Boltzmann’s constant, $k_B=1.38\cdot10^{-23}\ \mathrm{J/K}$
Electron charge magnitude, $e=1.60\cdot10^{-19}$
1 electron volt, $1\ \mathrm{eV}=1.60\cdot10^{-19}\ \mathrm J$
Speed of light, $c=3.00\cdot10^8\ \mathrm{m/s}$
Universal Gravitational constant, $G=6.67\cdot10^{-11}\ \mathrm{(N\cdot m^2)/kg^2}$
Acceleration due to gravity, $g=9.81\ \mathrm{m/s^2}$
1 unified atomic mass unit, $1\ \mathrm u=1.66\cdot10^{-27}\ \mathrm{kg}=931\ \mathrm{MeV}/c^2$
Planck’s constant, $h=6.63\cdot10^{-34}\ \mathrm{J\cdot s}=4.41\cdot10^{-15}\ \mathrm{eV\cdot s}$
Permittivity of free space, $\epsilon_0=8.85\cdot10^{-12}\ \mathrm{C^2/(N\cdot m^2)}$
Coulomb’s law constant, $k=\frac1{4\pi\epsilon_0}=8.99\cdot10^9\ \mathrm{(N\cdot m^2)/C^2}$
Permeability of free space, $\mu_0=4\pi\cdot10^{-7}\ \mathrm{T\cdot m/A}$
Magnetic constant, $\frac{\mu_0}{4\pi}=1\cdot10^{-7}\ \mathrm{(T\cdot m)/A}$
1 atmospheric pressure, $1\ \mathrm{atm}=1.01\cdot10^5\ \mathrm{N/m^2}=1.01\cdot10^5\ \mathrm{Pa}$
Wien’s displacement constant, $b=2.9\cdot10^{-3}\ \mathrm{m\cdot K}$
Stefan-Boltzmann constant, $\sigma=5.67\cdot10^{-8}\ \mathrm{W/m^2/K^4}$

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

The study of mathematics has almost always paved the way for the development of new ideas in physics. Newtonian mechanics could not be possible without first inventing calculus, and general relativity could not have existed without heavy development in tensors. However, there are numerous cases where physical insight have paved the way for mathematics.

Perhaps the most notable would be the Brachistochrone problem, which asks for the path that leads to the fastest descent influenced by gravity between two given points. While it is solvable through the calculus of variations, Newton proposed an easier solution by modelling the path of light through a medium with a variable index of refraction. You may read about this problem and the fascinating history behind it here.

We will not be dealing with this specific problem, but rather multiple short mathematics problems that can be represented with a physical analog. To receive points, you must use the suggested physical set-up.

a) (4 pts) Show that for small values of $x$ , we have

\cos(x)=1-\frac{x^2}{2}.

Physical Setup: Consider a small object moving in a circle.

b) (8 pts) A ladder of length $\ell$ with a thickness of $0.3\ \mathrm m$ is transported around a right angled corner where the two hallways leading up to it have a width of $3\ \mathrm m$ and $5\ \mathrm m$ . What is the maximum length of the ladder such that it can be successfully transported across?

Physical Setup: Consider the ladder as a compressed spring that can freely expand in only its longitudinal direction. Do not explicitly take derivatives. Instead, consider a force/torque balance. You may or may not need to solve an equation numerically.

c) (6 pts) Prove the AM-QM inequality:

\frac{a_1+a_2+\cdots+a_n}{n}\le\sqrt{\frac{a_1^2+a_2^2+\cdots+a_n^2}{n}}

Physical Setup: Design circuit(s) and compare their measurable quantities with each other qualitatively (or otherwise).

Published organizer clarifications (separate source)

Separately quoted from the organizer clarification document:

Problem 5: Please use the setups given.

Problem 5 a)

Do not assume sin x ~= x for small x.
To be more flexible, we decided that we will not take off any marks if the object does not start from rest, as there may be multiple solutions.

Problem 5 b)

For “force balance”, you may choose to take into account forces and/or torques.
This is a diagram of the setup (not to scale), from the top down.

Rectangular ladder across a right-angle corridor; complete outer and inner wall lines and ladder corners retained.

The solution can be numerical, though not necessarily. You may or may not need to use a graphing calculator for this question (but cannot for other questions).

Problem 5 c)

It is not necessary to use two circuits. You may use more or less, but the solution has to be based on circuitry.
You can upload an image of the circuit if needed.

Решение

Solution 5:

(a) Consider a small particle starting from rest on a circular path. It quickly and uniformly accelerates to a speed $v$ in a very small time interval $\Delta t$ such that $\frac{v\Delta t}{R}\ll1$ where $R$ is the radius of the circle.

Since this time interval is so small, the acceleration is directed inwards. Without loss of generality, let us define the inwards direction as the positive $+y$ direction. The displacement $\Delta y$ is given by $R(1-\cos\theta)$ which gives:

R(1-\cos\theta)=\frac12at^2

At time $t$ , since the particle hasn’t moved very far, the inwards acceleration is still pointed in the $+y$ direction and using equations for circular motion, it has a magnitude of:

a=\frac{v^2}{R}

Making the substituting, we can simplify:

1-\cos\theta=\frac12\left(\frac{vt}{R}\right)^2

Since $vt=R\theta$ is the distance traveled, this simplifies to:

1-\cos\theta=\frac12\theta^2\implies\boxed{\cos\theta=1-\frac12\theta^2}

(b) Treating the ladder as a spring, we see that its length is directly related to its potential energy. Determining the point at which the ladder’s length is at a minimum is equivalent to determining the point at which the rigid spring’s potential energy is at a minimum. Using the fact that:

F=-\frac{dU}{dx}

implies that when the potential energy is at a minimum, the net force must be equal to zero. Therefore, we can set up a force and torque balance in order to solve for the shortest length. There are three normal forces to take into account. Balancing forces gives two equations, and a torque balance gives the third. Solving a system of three equations will allow us to solve the problem.

Perhaps a slicker way is to realize that these three forces must be concurrent. At least two of the normal force vectors will intersect, and performing a torque balance where the pivot is selected to be that intersection point, these two normal force vectors will not contribute to a torque. In order to satisfy $\sum\tau=0$ , the third normal force vector must also intersect at this same place. This turns it into a geometry problem.

Solving the problem geometrically or analytically, you end up solving the equation:

\frac{a\cos\theta-\ell}{\sin^2\theta}=\frac{b\sin\theta-\ell}{\cos^2\theta}

which is possible to be solved numerically (or solved through an approximation). This wasn’t specified on the exam, so most teams who arrived at this equation would have received the marks.

Solving, we get $L\approx10.6\ \mathrm m$ .

(c) We connect several batteries with an electromotive force $\varepsilon_i$ and an identical internal resistance $r$ in parallel with one another, and the entire system is in parallel with a wire with zero resistance. Using Kirchoff’s Loop rule, the voltage drops across each internal resistor is going to be the same. Therefore, the power dissipation is:

P_1=\frac{\varepsilon_1^2+\varepsilon_2^2+\cdots+\varepsilon_N^2}{r}

if there are $N$ such resistors. We then create a second circuit where all of these resistors are in series. The current is now the same and the total power dissipation is:

P_2=\frac{(\varepsilon_1+\varepsilon_2+\cdots+\varepsilon_n)^2}{Nr}

If we make the claim that $P_1\ge P_2$ , the AM-QM inequality will naturally follow. To do this, we can look at the amp-hour capacity of batteries in parallel vs in series. In series, the amp-hour capacity of each resistor must be the same (and be equal to the capacity of the weakest battery). If this was not the case, then some batteries will be depleted before others. This is not the case for parallel and will only be the same if $\varepsilon_1=\varepsilon_2=\cdots=\varepsilon_N$ , in which case we have $P_1=P_2$ . Due to this, the power dissipation in the first circuit must be equal or greater than the one in the second circuit, proving the AM-QM inequality.

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 $\verb|\newpage|$ and see if that fixes it.

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