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    HBCSE

    Indian National Physics Olympiad 2013

    Roll Number:P 1 3Date: 3rd February 2013

    Time : 09:00-12:00 (3 hours) Maximum Marks: 60

    Please fill in all the data below correctly. The contact details provided herewould be used for all further correspondence.

    Full Name (BLOCK letters) Ms. / Mr.:

    Male / Female T-shirt Size: S/M/L/XL/XXL Date of Birth (dd/mm/yyyy):

    Name of the school / junior college:

    Class: XI/ XII Board: ICSE / CBSE / State Board / Other

    Address for correspondence (include city and PIN code):

    PIN Code:

    Telephone (with area code): Mobile:

    Email address:

    Ipermit / do not permit (strike out one) HBCSE to make my academic performanceand personal details public unless I communicate my consent in writing.

    Besides the International Physics Olympiad, do you also want to be considered for theAsian Physics Olympiad? Orientation-cum-Selection Camp (OCSC) for physics will beorganised from April 11-20 in Mumbai. For APhO - 2013 pre-departure training, yourpresence will be required in this camp from April 11-17 and in Delhi-cum-Indonesia fromMay 01-13. In principle you can participate in both olympiads.

    Yes/No.

    I have read the procedural rules for INPhO and agree to abide by them. Signature(Do not write below this line)

    ==================================================

    MARKS

    Question: 1 2 3 4 5 6 7 Total

    Marks: 5 5 8 8 8 13 13 60

    Score:

    HOMI BHABHA CENTRE FOR SCIENCE EDUCATIONTata Institute of Fundamental Research

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

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    Instructions:

    1. Write your Roll Number on every page of this booklet.2. Fill out the attached performance card. Do not detach it from this booklet.3. Booklet consists of 26 pages (excluding this sheet) and 7 questions.4. Questions consist of sub-questions. Write your detailed answer in the blank

    space provided below the sub-question and final answer to the sub-question inthe smaller box which follows the blank space.

    5. Extra sheets are also attached at the end in case you need more space. You mayalso use these extra sheets for rough work.

    6. Computational tools such as calculators, mobiles, pagers, smart watches, slide rules,log tables etc. are notallowed.

    7. This entire booklet must be returned.

    Table of Information

    Speed of light in vacuum c = 3.00 108 ms1

    Plancks constant h = 6.63 1034 Js

    Universal constant of Gravitation G = 6.67 1011 Nm2kg2

    Magnitude of the electron charge e = 1.60 1019 C

    Mass of the electron me = 9.11 1031 kg = 0.51MeVc2

    Stefan-Boltzmann constant = 5.67 108 Wm2 K4

    Permittivity constant 0 = 8.85 1012 Fm1

    Value of 1/40 = 9.00 109 Nm2C2

    Permeability constant 0 = 4 107 Hm1

    Universal Gas Constant R = 8.31 J K1

    mole1

    Molar mass of air = 29.0 kgkmol1

    HOMI BHABHA CENTRE FOR SCIENCE EDUCATIONTata Institute of Fundamental Research

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

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    INPhO - 2013 Roll Number:P 1 3

    1. A parallel plate capacitor with plate areaAand separationd(d lateral dimensions)is filled with a dielectric of dielectric constant k = ex where is a constant ( >0)and plate is at x= 0. The capacitor is charged to a potential V. [Marks: 5]

    (a) Obtain the capacitance C. [2]

    C=

    (b) Sketch Cvs d. [1]

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    INPhO - 2013 Roll Number:P 1 3

    (c) Obtain the charge on the plates. [1]

    Charge =

    (d) Obtain the electric field E(x). [1]

    E(x) =

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    INPhO - 2013 Roll Number:P 1 3

    2. The figure below depicts the reflection of normally incident monochromatic wave bytwo neighbouring surface atoms of crystal. Angle of reflection and nearest neighbourdistance are and d respectively. [Marks: 5]

    d

    Two reflected waves will interfere. The interference is due to deBroglie wave of theelectron accelerated from rest by less then 1 keV.

    (a) Obtain a relationship between , d and the kinetic energy (K) of the electronwith rest mass m0 for the case of constructive interference. [3]

    =

    (b) Calculate dif first maxima occurs at = 30 deg for K=100 eV electron. [2]

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    INPhO - 2013 Roll Number:P 1 3

    d =

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    INPhO - 2013 Roll Number:P 1 3

    3. Consider a bicycle in vertical position accelerating forward without slipping on astraight horizontal road. The combined mass of the bicycle and the rider is M andthe magnitude of the accelerating torque applied to the rear wheel by the pedal andgear system is . The radius and the moment of inertia of each wheel is R and I(with respect to axis) respectively. The acceleration due to gravity is g. [Marks: 8]

    (a) Draw the free body diagram of the system (bicycle and rider). [2]

    RR

    Wheel 2 Wheel 1

    (b) Obtain the acceleration ain terms of the above mentioned quantities. [2]

    a=

    (c) For simplicity assume that the centre of mass of the system is at heightR fromthe ground and equidistant at 2Rfrom the center of each of the two wheels. Letbe the coefficient of friction (both static and dynamic) between the wheels andthe ground. Consider M I/R2 and no slipping. Obtain the conditions for themaximum acceleration am of the bike. [3]

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    am=

    (d) For= 1.0 calculate am. [1]

    am=

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    INPhO - 2013 Roll Number:P 1 3

    4. The figure below depicts a concave mirror with center of curvature C focusF, and ahorizontally drawn OF Cas the optic axis. The radius of curvature is R (OC= R)andOF =R/2). A ray of lightQP, parallel to the optical axis and at a perpendiculardistance w (w R/2) from it, is incident on the mirror at P. It is reflected to thepoint B on the optical axis, such that BF = k. Here k is a measure of lateralaberration. [Marks: 8]

    O A B F C

    P Q

    k

    (a) Expressk in terms of{w, R}. [3]

    k=

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    INPhO - 2013 Roll Number:P 1 3

    (b) Sketchk vs w for w [0, R/2]. [2]

    (c) Consider pointsP1, P2, ....Pnon the concave mirror which are increasingly furtheraway from the optic centerO and approximately equidistant from each other(seefigure below). Rays parallel to the optic axis are incident at P1, P2, ....Pn andreflected to points on the optic axis. Consider the points where these rays re-flected from Pn, Pn1, .....P2 intersect the rays reflected from Pn1, Pn2, ....P1respectively. Qualitatively sketch the locus of these points in figure below for amirror (shown with solid line) with radius of curvature 2 cm. [3]

    P1

    Pn

    P2

    O

    +1 cm

    1 cm

    F C

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    5. Two long parallel wires in theyzplane at a distance 2aapart carry a steady currentIin opposite directions. Midway between the wires is a rectangular loop of wire 2b dcarrying current Ias shown in the figure (b < a < 2b). The loop is free to rotateabout thezaxis and the currents remain fixed toIirrespective of the relative motionbetween the loop and the wire. [Marks: 8]

    I I

    I

    z z

    y

    2a

    2b

    d

    (a) Obtain the torque tending to rotate the rectangular loop about its axis as afunction of. Here is the angle that plane of loop makes with the plane ofwires. [5]

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    Torque is

    (b) Obtain the value offor which the torque is maximum? [3]

    =

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    INPhO - 2013 Roll Number:P 1 3

    6. A planet of mass m is orbiting a star of mass M (M m) under the influence ofa central force. A central force (F(r)r) is a force that is directed from the particletowards (or away from) a fixed point in space, the center, and whose magnitude onlydepends on the distance from the planet to the center. Thus gravitational force iscentral force and it is generally known that the orbit is a conic section of a bodyunder the influence of gravitational force The planets position at some point P canbe shown in polar coordinates (r, ) as shown in figure below:

    r

    P

    x

    y

    The planets position (r), velocity (v) and acceleration (a) can be written as

    r= rr, v= rr+ r anda= (r r2)r+ (2r+ r)

    Consider central force to be gravitational force for parts (c-h). [Marks: 13]

    (a) One can write scalar form of newtons second law as

    d2r

    dt2 =g(r,,r, )

    Obtain g(r,,r, ). [1]

    g(r,,r, ) =

    (b) Show that the magnitude of the angular momentum (L) of planet is constant. [1/2]

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    (c) Obtain total energy (E) of the planet at distance r from the center of the force.Take potential energy to be zero at large r. [1]

    E=

    (d) State the energy partV(r) =E mr2/2 in terms ofL and other quantities. [2]

    V(r) =

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    (e) Sketch V(r). [11/2]

    (f) Comment on the allowed regions and shape of the orbit if planets energyE >0,and E

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    INPhO - 2013 Roll Number:P 1 3

    r0 =

    (h) Hence forward we take that orbit is circular of radius r0 and planet is slightlydisturbed from its position such that its position is r =r0+where /r0 1.Show that planet will oscillate simple harmonically around mean position r0.Obtain time period (Tr) for these radial oscillations. [2]

    T(r) =

    (i) For this part we assume that planet moves under the force expressible as a powerofr as F = crn where c (c > 0) is a constant. For what values ofn a stableorbit is possible? [2]

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    n=

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    INPhO - 2013 Roll Number:P 1 3

    7. Brunt-Vaisala Oscillations: A balloon containing inert gas Helium is used tostudy the weather. It rises up and rests at an equilibrium height in the atmosphere,where its weight is exactly balanced by the upward buoyant force. We shall assumeideal gas law to hold for all processes and neglect the mass of the balloon. If theballoon is displaced vertically, it is often found to oscillate about the equilibriumposition. The frequency of this oscillation is called Brunt-Vaisala frequency. Weshall investigate this phenomenon. [Marks: 13]

    (a) Suppose the balloon is displaced upwards adiabatically from the equilibriumposition. In the displaced position, let Tb and mb be the temperature and molarmass of the gas in the balloon. LetTa and ma be the temperature and molarmass of the outside atmosphere at the same level. Let dTb/dz and dTa/dz bethe rate of change of temperature with height for the gas in the balloon and theoutside air respectively where the positive zaxis is vertically upwards and z isthe height from the ground.

    Derive and Calculate the lapse rate (a = dTa/dz) for atmosphere given that themolar mass of air is 29 kgkmol1 and the gas constant is R= 8.31 J Kmol1.Assume that acceleration due to gravity (g) remains constant and all air processesare adiabatic. [1]

    a =

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    (b) Assuming that the pressure inside the balloon is the same as that outside, derivean expression for b = dTb/dz in terms of the molar specific heat at constantpressure (Cb) of the gas in the balloon. [2]

    b =

    (c) Obtain an expression for the vertical acceleration (z) of the balloon in terms oftemperatures and molar masses. [2]

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    z=

    (d) Find the equilibrium height (z0) in terms of molar masses, Cband relevant quan-tities? Assume that for both ballon and air, T =T0 at z= 0. [2]

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    z0 =

    (e) State the condition for balloon to oscillate simple harmonically. Also obtain thefrequency of oscillation () for balloon. [4]

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    Condition:

    =

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    (f) Calculate approximately the time period () of the oscillating balloon ifT0 =300 K in a balloon consisting of Ar-He mixture in the mass ratio 1:1 (AtomicWeights of Ar = 36 amu and He = 4 amu). [2]

    =

    **** END OF THE QUESTION PAPER ****

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    Extra Sheet

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    Extra Sheet

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    Extra Sheet

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    Extra Sheet

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    Extra Sheet

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    HOMI BHABHA CENTRE FOR SCIENCE EDUCATIONTata Institute of Fundamental Research

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

    Indian National Physics Olympiad - 2013

    Solutions

    PLEASE NOTE THAT ALTERNATE/EQUIVALENT SOLUTIONS MAY EXIST. Brief so-lutions are given below.

    1. (a) C= 0A

    1 ed(b)

    0A

    d

    C

    (c) Charge = 0AV1 ed

    (d) E(x) = V ex

    1 ed x

    2. (a) = sin1

    nh

    d

    2m0K

    (b) d 2.4A

    3. (a) Heref1, f2 are frictional forces and N1, N2 are normal reactions.

    N 2

    N 1

    f1

    f2

    RR

    mg

    Wheel 1Wheel 2

    (b) a=

    M R2 + 2IR

    (c) a g/2

    1

    4

    (d) am= 2g/3

    4. (a) k =R

    2

    R

    (R2 2)1/2 1

    (b)

    w

    k

    R/2

    1

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    (c)

    P1

    Pn

    P2

    O

    1 cm

    C

    +1 cm

    0 5 cm

    0 5 cm

    F

    5. (a) Torque = 20I

    2(a2 +b2)abd sin

    (a2 b2)2 + 4a2b2 sin2 z(b) = ;; ; +

    where= sin1 [(a2 b2)/2ab]

    6. (a) g(r,,r, ) =

    F(r)

    m +r2

    (b) From null azimuthal component

    2r+r = 0d

    dt(r2) = 0

    d

    dt(mr2) = 0

    d

    dtL = 0

    L= constant

    (c) E=m(r2 +r22)

    2 GMm

    r

    (d) V(r) = L2

    2mr2 GMm

    r(e)

    rc r0

    V0

    V(r)

    r

    wherer0= L2

    GMm2, rc=

    L2

    2GM m2 and V0 = G

    2M2m3

    2L2

    (f) For E >0 orbit is hyperbola. For E

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    (h) Tr = 2

    r30GM

    (i) n > 3

    7. (a) a=(1 a)mag

    aR ormag

    Cawherea = 7/5 (ratio of specific heats at constant pressure and volume) and Ca ismolar specific heat at constant pressure for air.

    |a| 0.01

    K m1

    (b) b= magTbCbTa

    (c) z=g

    maTbTamb

    1

    (d) z0= T0a

    1

    mbma

    1/(1)

    where= Ca/Cb.

    (e) Condition: Ca > Cb

    = g

    ma( 1)

    CaT0

    mamb

    1/(1)(f) 95 s