Indian National Physics Olympiad 2017

Transcription

1 Indian National Phsics Olmpiad 2017 Date: 29 th Januar 2017 Solutions Roll Number: Time : 09:00-12:00 (3 hours) Maimum Marks: 75 I permit/do not permit (strike out one) HBCSE to reveal m academic performance and personal details to a third part. Besides the International Phsics Olmpiad (IPhO) 2017, do ou also want to be considered for the sian Phsics Olmpiad (PhO) 2017? For PhO 2017 and its pre-departure training, our presence will be required in Delhi and Russia from pril 26 to Ma 10, In principle, ou can participate in both olmpiads. Yes/No. Full Name (BLOCK letters) Ms./Mr.: Etra sheets attached : 0 Date Centre(e.g.Jaipur) Signature (Do not write below this line) ================================================== Instructions 1. This booklet consists of 6 pages (ecluding this sheet) and total of 6 questions. 2. This booklet is divided in two parts: Questions with Summar nswer Sheet and Detailed nswer Sheet. Write roll number at the top wherever asked. 3. The final answer to each sub-question should be neatl written in the bo provided below each sub-question in the Questions & Summar nswer Sheet. 4. You are also required to show our detailed work for each question in a reasonabl neat and coherent wa in the Detailed nswer Sheet. You must write the relevant Question Number on each of these pages. 5. Marks will be awarded on the basis of what ou write on both the Summar nswer Sheet and the Detailed nswer Sheet. Simple short answers and plots ma be directl entered in the Summar nswer Sheet. Marks ma be deducted for absence of detailed work in questions involving loner calculations. Strike out an rough work that ou do not want to be evaluated. 6. dequate space has been provided in the answersheet for ou to write/calculate our answers. In case ou need etra space to write, ou ma request for additional blank sheets from the invigilator. Write our roll number on the etra sheets and get them attached to our answersheet and indicate number of etra 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 ma be used for diagrams/graphs/sketches. 9. This entire booklet must be returned at the end of the eamination. Table of Constants Speed of light in vacuum c m s 1 Planck s constant h J s Universal constant of Gravitation G N m2 kg Magnitude of electron charge e C Mass of electron m e kg Value of 1/4πɛ N m2 C 2 Universal Gas Constant R 8.31 J K 1 mole 1 Question Marks Score Total 75 HOMI BHBH CENTRE FOR SCIENCE EDUCTION Tata Institute of Fundamental Research V. N. Purav Marg, Mankhurd, Mumbai,

2 INPhO 2017 Page 1 Questions & Summar nswers 1. massive star of mass M is in uniform circular orbit around a supermassive black hole of mass M b. Initiall, the radius and angular frequenc of the orbit are R and ω respectivel. ccording to Einstein s theor of general relativit the space around the two objects is distorted and gravitational waves are radiated. Energ is lost through this radiation and as a result the orbit of the star shrinks graduall. One ma assume, however, that the orbit remains circular throughout and Newtonian mechanics holds. The power radiated through gravitational wave b this star is given b [1] L G = Kc G M 2 R 4 ω 6 where c is the speed of light, G is the universal gravitational constant, and K is a dimensionless constant. Obtain and b dimensional analsis. = 5, = 1 Obtain the total mechanical energ (E) of the star in terms of M, M b, and R. E = GM bm 2R Derive an epression for the rate of decrease in the orbital period (dt/dt) in terms of the masses, period T and constants. From Kepler s law Using previous part lso de/dt = L G. This ields T 2 = 4π2 R 3 GM b E = (GM b2π) 2/3 M 2T 2/3 dt 3KG5/3 dt = M 2/3 b M(2π) 8/3 c 5 T 5/3 2. The free surface of mercur (Hg) is a good reflecting surface. tall clinder partl filled with Hg and possessing total moment of inertia I is rotated about its ais with the constant angular velocit ω 0 as shown in figure. The Hg surface attains a paraboloidal profile. The radius of curvature ρ of a general profile is given b [1 + (d/d) 2 ] 3/2 ρ = d 2 /d 2 where the smbols have their usual meaning. Free surface (0, 0 ) ω 0 (0,0) Obtain the epression for ρ of the Hg surface in terms of ω 0, the distance from the clinder ais, and g. Hg [1] [3] [3]

3 INPhO 2017 Page 2 Questions & Summar nswers Roll Number: ρ = ( ) ω g ω 2 0 /g 3/2 Calculate the value of ρ at the lowest point of the Hg surface, that is (0,0), when ω 0 = 78 rpm (revolutions per minute). [1] ρ =0 = g ω 2 0 = 14.7 cm Consider a point object at (0, 0 ) as shown in the figure. Obtain an epression for the image position i in terms of given quantities. State conditions on 0 for the formation of real and virtual images. Using mirror equation where f = ρ =0 /2. This gives 1 f = 1 1 i 0 g 0 i = g 2ω0 2 0 For real images 0 > g/2ω 2 0. For virtual images 0 < g/2ω Two identical blocks and B each of mass M are placed on a long inclined plane (angle of inclination = θ) with higher up than B. The coefficients of friction between the plane and the blocks and B are respectivel µ and µ B with tan θ > µ B > µ. The two blocks are initiall held fied at a distance d apart. t t = 0 the two blocks are released from rest. t what time t 1 will the two blocks collide? 2d t 1 = (µ B µ )g cos θ [3] [2] Consider each collision to be elastic. t what time t 2 and t 3 will the blocks collide a second and third time respectivel? [4] t 2 = 3t 1, t 3 = 5t 1 Draw a schematic velocit-time diagram for the two blocks from t = 0 till t = t 3. Draw below them on a single diagram and use solid line ( ) to depict block and dashed line ( ) to depict block B. [5]

4 INPhO 2017 Page 3 Questions & Summar nswers velocit B t 1 2t 1 3t 1 4t 1 5t 1 time 4. One mole of an ideal gas (c p /c v = γ where smbols have their usual meanings) is subjected to an Otto ccle (-B-C-D) as shown in the following P - V diagram. Path -B and C-D are adiabats. The temperature at B is T B = T 0. Diagram is not to scale. P 2P 0 P 0 B C D V 0 2V 0 Find the temperatures at,c, and D in terms of T 0 and pressures at and D in terms of P 0. T = T 0 2 γ 1 ; T C = 2T 0 ; T D = T 0 2 γ 2 ; P = P 0 2 γ ; P D = P 0 2 γ 1 Find total heat absorbed ( Q) b the sstem, the total work done ( W ) and efficienc (η) of the Otto ccle in terms of γ and related quantities. ( Q = c v (T C T B ) + c v (T T D ) = c v T W = Q η = 1 Q BC Q D = γ 1 2 γ 1 ) V [4] [3 1 / 2 ] Draw below corresponding P -T and T -S(entrop) diagrams for the ccle. [6 1 / 2 ]

5 INPhO 2017 Page 4 Questions & Summar nswers Roll Number: P C T C B B D D T S 5. metallic rod of mass m and length L (thick line in the figure below) can slide without friction on two perpendicular wires (thin lines in the figures). Entire arrangement is located in the horizontal plane. constant magnetic field of magnitude B eists perpendicular to this plane in the downward direction. The wires have negligible resistance compared to the rod whose resistance is R. Initiall, the rod is along one of the wires so that one end of it is at the junction of the two wires (see Fig. ). L B B L The rod is given an initial angular speed Ω such that it slides with its two ends alwas in contact with the two wires (see Fig. ), and just comes to rest in an aligned position with the other wire (see Fig. ). Determine Ω. Neglect the self-inductance of the sstem. B L B [15] φ L n intermediate position of the rod is shown in figure. The coordinates of the centre of mass of the rod are given b Thus the kinetic energ T of the rod at an instant is cm = L cos φ (1) 2 cm = L sin φ (2) 2 T = 1 2 mv2 cm I cm φ 2 = ml2 ω 2 6 (3)

6 INPhO 2017 Page 5 Questions & Summar nswers Magnitude of the induced emf E, is given b E = 1 2 BL2 cos(2φ) φ (4) The power dissipated due to the current is equal to the loss of kinetic energ of the rod. We have dt = E2 dt R 0 φ = 3B2 L 2 Ω 4mR = Ω = 3πB2 L 2 16mR π 2 0 (5) cos 2 (2φ)dφ (6) (7) 6. White light is incident at an angle i on a prism of angle placed in air as shown. i Let D be the angular deviation (not necessaril a minimum) suffered b an emer- white light dispersed light gent ra of a particular wavelength. Obtain an epression for sin(d + i) in terms of the refractive inde n and trigonometric [3] functions of i and onl. ( ) sin(d + i) = µ sin sin 1 sin i µ Let = and i = Obtain the refractive inde (n λ ) for a ra of wavelength λ which has suffered deviation D = n λ = detailed microscopic theor ields the relation between the refractive inde, n, of the material of the prism and the angular frequenc ω = 2πc/λ of the incident light as n 2 ( ) 1 n 2 = Ne ɛ 0 m e ω0 2 ω2 Here N is the electron densit and ω 0 = 2πc/λ 0 the natural frequenc of oscillation of the electron of the material. The other smbols have their usual meaning. The table below lists the refractive indices at si wavelengths. [2] [10] λ (nm) n Re-epress the above equation to get a linear relationship in terms of β = (n 2 + 2)/(n 2 1) and a suitable power of λ. Tabulate and plot so that ou ma obtain N and ω 0. (Two graph papers are provided with this booklet in case ou make a mistake). Linear relation: 1 λ 2 = 1 Ne 2 λ 2 β 0 3ɛ 0 m e (2πc) 2. Graph is plotted between β vs 1/λ2 at the end of this booklet.

7 INPhO 2017 Page 6 Questions & Summar nswers Roll Number: 1/λ 2 ( 10 6 nm 2 ) β (d) Calculate the values of N, ω 0 from the graph ou plotted. Which part of the electromagnetic spectrum does λ 0 belong to? [4] From the drawn graph, N = m 3 ; ω 0 = Hz. λ 0 belongs to the ultraviolet part of the electromagnetic spectrum. ccepted values: N m 3 and ω Hz. (e) n X-ra of energ kev is incident on the prism. If we write n = 1 + δ then obtain the numerical value of δ for this ra. [3] (f) δ = For the X-ra of the previous part let i c be the critical angle and θ c = 90 i c be the corresponding grazing angle. Obtain θ c. θ c = 0.68 **** END OF THE QUESTION PPER **** [1]

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