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INPhO 2019, Теория — Задача 7. A magnet on a metallic inclined plane

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Всички материали от INPhO 2019

Indian National Physics Olympiad – 2019 · 3 февруари 2019 г. · 13 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: $1\,9\,\square\,\square-\square\,\square\,\square\,\square-\square\,\square\,\square\,\square$ . Extra sheets attached: [blank box]. Centre (e.g. Kota): [blank line].

(Do not write below this line)

This booklet consists of 29 pages (excluding this page) and total of 7 questions.
This booklet is divided in two parts: Questions with Summary Answer Sheet and Detailed Answer Sheet. Write roll number at the top wherever asked.
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.
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.
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.
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.
Non-programmable scientific calculators are allowed. Mobile phones cannot be used as calculators.
Use blue or black pen to write answers. Pencil may be used for diagrams/graphs/sketches.
This entire booklet must be returned at the end of the examination.

Table of Constants

Constant	Symbol	Value
Speed of light in vacuum	$c$	$3.00\times10^8\ \mathrm{m\cdot s^{-1}}$
Planck’s constant	$h$	$6.63\times10^{-34}\ \mathrm{J\cdot s}$
	$\hbar$	$h/2\pi$
Universal constant of Gravitation	$G$	$6.67\times10^{-11}\ \mathrm{N\cdot m^2\cdot kg^{-2}}$
Magnitude of electron charge	$e$	$1.60\times10^{-19}\ \mathrm C$
Rest mass of electron	$m_e$	$9.11\times10^{-31}\ \mathrm{kg}$
Value of $1/4\pi\epsilon_0$		$9.00\times10^9\ \mathrm{N\cdot m^2\cdot C^{-2}}$
Avogadro’s number	$N_A$	$6.022\times10^{23}\ \mathrm{mol^{-1}}$
Acceleration due to gravity	$g$	$9.81\ \mathrm{m\cdot s^{-2}}$
Universal Gas Constant	$R$	$8.31\ \mathrm{J\cdot K^{-1}\cdot mol^{-1}}$
	$R$	$0.0821\ \mathrm{l\cdot atm\cdot mol^{-1}\cdot K^{-1}}$
Permeability constant	$\mu_0$	$4\pi\times10^{-7}\ \mathrm{H\cdot m^{-1}}$

Question	Marks	Score
1	9
2	11
3	12
4	7
5	9
6	14
7	13
Total	75

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

As shown, a magnet of mass $m=19.00\ \mathrm g$ slides on a rough non-magnetic metallic inclined plane which makes an angle $\theta$ with the horizontal. One can change the angle of inclination of the plane. Due to relative motion between the magnet and the plane, eddy currents are generated inside the metal which retard motion of the magnet.

Complete native inclined plane, magnet, motion arrow, metallic-plane label and theta/horizontal geometry.

Assume that the magnitude of the resistive force on the magnet is $bv$ where $b$ is a positive constant and $v$ is the instantaneous speed of the magnet. Let $\mu$ be the coefficient of kinetic friction between the magnet and the plane.

a) If the magnet starts moving from rest at time $t=0$ , obtain the expression of its terminal velocity $V_T$ . Also obtain the displacement $S(t)$ along the inclined plane as a function of time. Take $S(0)=0$ .

Summary answer boxes: $V_T=$ [blank]; $S(t)=$ [blank]. [2 т.]

b) For a fixed $\theta$ a student records $S$ for all values of $t$ as shown in the table below. Draw a suitable graph and obtain the terminal velocity $V_T$ of the magnet from this graph. For this and the next part, three graph papers are provided with this booklet. No extra graph papers will be provided.

$t$ (s)	$S$ (m)
0.016	0.001
0.049	0.003
0.070	0.006
0.090	0.010
0.120	0.017
0.174	0.029
0.230	0.046
0.270	0.058
0.320	0.074
0.370	0.091

Summary answer box: $V_T=$ [blank].

Graph is plotted on page no.: [blank]. [4 т.]

c) The above process is repeated for various values of $\theta$ . The obtained values of terminal velocities for the different $\theta$ are given below.

$\theta$ (degree)	$V_T$ (m/s)
19	0.15
24	0.23
28	0.29
35	0.40
40	0.46
45	0.53
48	0.58
52	0.62

Plot a suitable graph and obtain the values of $b$ and $\mu$ from this graph.

Summary answer boxes: $b=$ [blank]; $\mu=$ [blank].

Graph is plotted on page no.: [blank].

Full original orange blank graph sheet, printed page26, native header, blank question-number field, all grid boundaries and roll field where printed.

Full original orange blank graph sheet, printed page27, native header, blank question-number field, all grid boundaries and roll field where printed.

Full original orange blank graph sheet, printed page28, native header, blank question-number field, all grid boundaries and roll field where printed.

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

END OF THE QUESTION PAPER [7 т.]

Решение

(a)

V_T=\frac{mg(\sin\theta-\mu\cos\theta)}b\qquad (7.1)

S(t)=V_T\left[t-\frac mb\left(1-e^{-tb/m}\right)\right]\qquad (7.2)

(b)

$V_T=$ slope of the graph $=0.315\ \mathrm{m/s}$ .

Accepted range $0.301\ \mathrm{m/s}\le V_T\le0.329\ \mathrm{m/s}$ .

Full original scanned handwritten7b graph sheet, all10points/scales/fit/slope/labels/page26 header and entire graph grid.

The native handwritten sheet is labelled7b, printed page26, with $S$ (m) versus $t$ (s), a fitted terminal straight line, the full plotted points and scale annotations: on x axis1cm=0.025sec; on y axis1cm=0.005m. It marks two points used for the fitted line and writes slope0.315m/s. The exact handwritten point annotations are preserved in the complete original image; the first coordinate label is not unambiguously legible as text.

(c)

\frac{V_T}{\cos\theta}=\frac{mg}b\tan\theta-\frac{\mu mg}b\qquad (7.3)

Graph is plotted for $V_T/\cos\theta$ vs $\tan\theta$ .

$\tan\theta$	$V_T/\cos\theta$ (m/s)
0.34	0.16
0.45	0.25
0.53	0.33
0.70	0.49
0.84	0.60
1.00	0.75
1.11	0.87
1.28	1.01

Slope of the graph $=0.91\ \mathrm{m/s}=\frac{mg}b\Rightarrow b=0.21\ \mathrm{N\cdot s/m}$ .

Intercept $=0.16=\frac{\mu mg}b\Rightarrow\mu=0.18$ .

Accepted range of slope $=0.86$ – $0.96\ \mathrm{m/s}$ .

Accepted range of intercept $=0.12$ – $0.19\ \mathrm{m/s}$ .

Respective ranges of $b$ and $\mu$ are

0.19\le b\le0.22\ \mathrm{m/s}\quad\text{and}\quad0.14\le\mu\le0.20

Determination of $b$ and $\mu$ from a graph of $V_T$ vs $\theta$ using an appropriate method also gains credit.

Full original scanned handwritten7c graph sheet, all8points/scales/negative-intercept/fit/slope/labels/page27 header and entire graph grid.

The native handwritten sheet is labelled7c, printed page27, with $V_T/\cos\theta$ (m/s) versus $\tan\theta$ , eight plotted points, the full extended line and scales. It writes scale on x axis1cm=0.1; on y axis1cm=0.05m/s; slope0.91m/s; intercept−0.155m/s. The fitted line crosses the horizontal axis near $\tan\theta=0.17$ .

Source note: The printed key calls the positive magnitude0.16 the “Intercept”, while the fitted line in the original handwritten graph has a negative vertical intercept and labels−0.155m/s. The printed accepted range for $b$ uses m/s although its stated result0.21 uses N·s/m. The literal values, units, signs, alternate-method credit and full native handwritten sheets are retained.

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

END OF THE QUESTION PAPER

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