INPhO 2019, Теория — Задача 4. A compressible liquid

Indian National Physics Olympiad – 2019 · 3 февруари 2019 г. · 7 т.

Внимание: Бележка към темата

The complete retained question booklet includes summary answer boxes, eighteen blank detailed-answer pages and three blank graph sheets. All matching archived HBCSE key content is retained, including both full handwritten graph sheets and accepted alternatives/ranges. Printed exponent, sign, factor, notation and unit inconsistencies remain literal with labelled source notes.

Native cover and instructions

Indian National Physics Olympiad – 2019

Date: 03 February 2019. Time: 09:00–12:00 (3 hours). Maximum Marks: 75.

Roll Number: 1 9 □ □−□ □ □ □−□ □ □ □1\,9\,\square\,\square-\square\,\square\,\square\,\square-\square\,\square\,\square\,\square. Extra sheets attached: [blank box]. Centre (e.g. Kota): [blank line].

(Do not write below this line)

  1. This booklet consists of 29 pages (excluding this page) and total of 7 questions.
  2. This booklet is divided in two parts: Questions with Summary Answer Sheet and Detailed Answer Sheet. Write roll number at the top wherever asked.
  3. The final answer to each sub-question should be neatly written in the box provided below each sub-question in the Questions & Summary Answer Sheet.
  4. You are also required to show your detailed work for each question in a reasonably neat and coherent way in the Detailed Answer Sheet. You must write the relevant Question Number(s) on each of these pages.
  5. Marks will be awarded on the basis of what you write on both the Summary Answer Sheet and the Detailed Answer Sheet. Simple short answers and plots may be directly entered in the Summary Answer Sheet. Marks may be deducted for absence of detailed work in questions involving longer calculations. Strike out any rough work that you do not want to be considered for evaluation.
  6. Adequate space has been provided in the answersheet for you to write/calculate your answers. In case you need extra space to write, you may request additional blank sheets from the invigilator. Write your roll number on the extra sheets and get them attached to your answersheet and indicate number of extra sheets attached at the top of this page.
  7. Non-programmable scientific calculators are allowed. Mobile phones cannot be used as calculators.
  8. Use blue or black pen to write answers. Pencil may be used for diagrams/graphs/sketches.
  9. This entire booklet must be returned at the end of the examination.

Table of Constants

ConstantSymbolValue
Speed of light in vacuumcc3.00×108 m⋅s−13.00\times10^8\ \mathrm{m\cdot s^{-1}}
Planck’s constanthh6.63×10−34 J⋅s6.63\times10^{-34}\ \mathrm{J\cdot s}
ℏ\hbarh/2πh/2\pi
Universal constant of GravitationGG6.67×10−11 N⋅m2⋅kg−26.67\times10^{-11}\ \mathrm{N\cdot m^2\cdot kg^{-2}}
Magnitude of electron chargeee1.60×10−19 C1.60\times10^{-19}\ \mathrm C
Rest mass of electronmem_e9.11×10−31 kg9.11\times10^{-31}\ \mathrm{kg}
Value of 1/4πϵ01/4\pi\epsilon_09.00×109 N⋅m2⋅C−29.00\times10^9\ \mathrm{N\cdot m^2\cdot C^{-2}}
Avogadro’s numberNAN_A6.022×1023 mol−16.022\times10^{23}\ \mathrm{mol^{-1}}
Acceleration due to gravitygg9.81 m⋅s−29.81\ \mathrm{m\cdot s^{-2}}
Universal Gas ConstantRR8.31 J⋅K−1⋅mol−18.31\ \mathrm{J\cdot K^{-1}\cdot mol^{-1}}
RR0.0821 l⋅atm⋅mol−1⋅K−10.0821\ \mathrm{l\cdot atm\cdot mol^{-1}\cdot K^{-1}}
Permeability constantμ0\mu_04π×10−7 H⋅m−14\pi\times10^{-7}\ \mathrm{H\cdot m^{-1}}
QuestionMarksScore
19
211
312
47
59
614
713
Total75

HOMI BHABHA CENTRE FOR SCIENCE EDUCATION

Tata Institute of Fundamental Research

V. N. Purav Marg, Mankhurd, Mumbai, 400 088

Native key cover

Solutions of Indian National Physics Olympiad – 2019

The native key cover repeats the nine instructions, twelve constant rows, seven-question score table totalling 75 marks, roll-number form and HBCSE address from the question cover. It prints “20 pages (excluding this page)”; the retained key contains twenty-one PDF pages, including two appended handwritten graph sheets.

Условие

Consider a long narrow cylinder of cross section AA filled with a compressible liquid up to height hh whose density ρ\rho is a function of the pressure P(z)P(z) as ρ(z)=ρ02(1+P(z)P0)\rho(z)=\frac{\rho_0}{2}\left(1+\frac{P(z)}{P_0}\right) where P0P_0 and ρ0\rho_0 are constants. The depth zz is measured from the free surface of the liquid where the pressure is equal to the atmospheric pressure (PatmP_{\mathrm{atm}}).

Complete original cylinder, liquid free surface, h/z dimensions, atmospheric-pressure label and downward g arrow.

a) Obtain the pressure (P(z)P(z)) as a function of zz. Obtain the mass (MM) of liquid in the tube.

Summary answer boxes: P(z)=P(z)= [blank]; M=M= [blank]. [5 т.]

b) Let Pi(z)P_i(z) be the pressure at zz, if the liquid were incompressible with density ρ0/2\rho_0/2. Assuming that P0≫ρ0gzP_0\gg\rho_0gz obtain an approximated expression for ΔP=P(z)−Pi(z)\Delta P=P(z)-P_i(z).

Summary answer box: ΔP≈\Delta P\approx [blank].

Detailed answers can be found on page numbers: [blank]. [2 т.]

Решение

Оригинал в Архива: INPhO2019-Question.pdf · решение от архива: INPhO2019-Solution-20190219.pdf