Data Provided: A formula sheet and table of physical constants is attached to this paper. DEPARTMENT OF PHYSICS & Autumn Semester ASTRONOMY

Size: px

Start display at page:

Download "Data Provided: A formula sheet and table of physical constants is attached to this paper. DEPARTMENT OF PHYSICS & Autumn Semester ASTRONOMY"

Garey McCoy
6 years ago
Views:

1 PHY221 PHY472 Dt Provided: Formul sheet nd physicl constnts Dt Provided: A formul sheet nd tble of physicl constnts is ttched to this pper. DEPARTMENT OF PHYSICS & Autumn Semester ASTRONOMY DEPARTMENT OF PHYSICS AND ASTRONOMY ADVANCED QUANTUM MECHANICS 2 hours Autumn 2014 Clssicl Physics Answer question ONE (Compulsory) nd TWO other questions, one ech from section A nd section B. Instructions: All questions Answer question re mrked ONE (Compulsory) out of ten. nd The 2brekdown other questions. on the right-hnd side of the pper is ment s guide to the mrks tht cn be obtined from ech prt. 2 hours All questions re mrked out of 20. The brekdown on the right-hnd side of the pper is ment s guide to the mrks tht cn be obtined from ech prt. Plese clerly indicte the question numbers on which you would like to be exmined on the front cover of your nswer book. Cross through ny work tht you do not wish to be exmined. PHY221 TURN OVER 1

2 PHY221 SECTION A COMPULSORY 1. () Imgine one-dimensionl sound wves trvelling in ir in the x direction in time t. i. Write down the pproprite wve eqution in terms of the ir pressure, P, nd the phse velocity of sound, c. [2] ii. Show tht P = P 0 cos (kx + ωt) is solution to this wve eqution, where k is the wve number nd ω is the ngulr frequency. [4] (b) Consider driven dmped hrmonic oscilltor with ngulr frequency 1.5 rd s 1 nd dmping fctor γ = s 1. i. Clculte the qulity fctor. [2] ii. Give one physicl mening of the qulity fctor. [2] (c) Consider the following Lgrngin for n object of mss m t position (x, y, z) in grvittionl field: L = 1 2 m(ẋ 2 + ẏ 2 + ż 2 ) mgz i. Identify the cyclic coordintes nd find the constnts of motion. [4] ii. Wht is the physicl mening of the constnts of motion? [2] (d) Wht is the direction of the Coriolis force cting on n object moving long the x xis in frme of reference rotting with ngulr frequency ω prllel to the positive z xis? [4] PHY221 CONTINUED 2

3 PHY221 SECTION B Answer 2 questions from this section 2. An injured person on mountin hers the sound of rescue helicopter. () Given tht c = 342 m s 1 nd the density of ir is ρ = 1.24 kg m 3 clculte the impednce of sound in ir Z ir. [2] (b) The sound bounces off rocky cliffs cusing n echo. Given tht the reflection coefficient r = (Z 1 Z 2 ) 2 (Z 1 +Z 2 ) 2 nd the impednce of the rock is Z rock = kg m 2 s 1, wht is the frction of sound intensity fter two reflections? [4] (c) Explin wht is ment by the group velocity. [4] (d) Clculte the group velocity given the dispersion reltion: ω = Ak 1 b 2 k 2, where ω is the ngulr frequency, k is the wve number nd A nd b re constnt. [4] (e) Find n expression for the group velocity t low frequencies bk 1 by expnding your nswer to prt (d) up to second order in bk. [4] (f) By considering the expression given in prt (d), does the injured person her lower or higher frequencies first? Explin your resoning. [2] 3. A mountin rescue officer is lowered from helicopter by rope nd swings with smll ngle oscilltions, θ. The system cts like pendulum performing simple hrmonic motion. () Write down the eqution of motion ssuming the rope hs length l nd the person cn be treted s pendulum bob of mss m. Include ir resistnce s dmping fctor γ. (b) By solving this eqution, show tht the generl solution cn be expressed s: [2] θ = (A 1 cos ω d t + A 2 sin ω d t)e γt, where A 1 nd A 2 re rbitrry constnts nd ω d is constnt to be found. [6] (c) Given the initil conditions tht t time t = 0 the rope is verticl nd the mss hs velocity v 0, find the rbitrry constnts (A 1 nd A 2 ) nd give the solution to the eqution of motion. (d) Given tht γ = s 1 nd l = 4.2 m, clculte the dmped ngulr frequency. [2] (e) How long does the person hve to wit for the oscilltions to die wy to e 1 of the strting mplitude? [8] [2] PHY221 TURN OVER 3

4 PHY A mountin rescue officer of mss m hngs from helicopter on the end of rope of length l. She connects nother rope of length l from herself to n injured person, lso of mss m. The two people oscillte with smll ngles (θ 1 nd θ 2 respectively) nd the system cn be treted s double pendulum. Dmping (ir resistnce) my be ignored. l θ 1 l m θ 2 m () Wht is the totl kinetic energy of the system? [4] (b) Find the totl potentil energy of the system. Simplify your nswer by tking up to qudrtic order in smll ngles. [5] (c) Using your nswer to prt () nd (b), write down the Lgrngin of the system nd use it to derive Lgrnge s equtions of motion for the system. [5] (d) Solve these equtions to find the ngulr frequencies. [6] 5. A beetle of mss m t position x i is crwling with constnt velocity v i rdilly outwrds long helicopter blde tht is rotting with constnt ngulr frequency ωk. () Wht re the two fictitious forces cting on the beetle? [2] (b) Find expressions for these fictitious forces in terms of the Crtesin unit bsis vectors in the rotting frme of reference i, j or k. [8] (c) Write down Newton s eqution of motion for the beetle in the rotting frme of reference, denoting the totl rel force s F rel. [4] (d) Give the direction nd physicl origin of the horizontl nd verticl rel forces. [6] END OF EXAMINATION PAPER 4

5 PHYSICAL CONSTANTS & MATHEMATICAL FORMULAE Physicl Constnts electron chrge e = C electron mss m e = kg = MeV c 2 proton mss m p = kg = MeV c 2 neutron mss m n = kg = MeV c 2 Plnck s constnt h = J s Dirc s constnt ( = h/2π) = J s Boltzmnn s constnt k B = J K 1 = ev K 1 speed of light in free spce c = m s m s 1 permittivity of free spce ε 0 = F m 1 permebility of free spce µ 0 = 4π 10 7 H m 1 Avogdro s constnt N A = mol 1 gs constnt R = J mol 1 K 1 idel gs volume (STP) V 0 = 22.4 l mol 1 grvittionl constnt G = N m 2 kg 2 Rydberg constnt R = m 1 Rydberg energy of hydrogen R H = 13.6 ev Bohr rdius 0 = m Bohr mgneton µ B = J T 1 fine structure constnt α 1/137 Wien displcement lw constnt b = m K Stefn s constnt σ = W m 2 K 4 rdition density constnt = J m 3 K 4 mss of the Sun M = kg rdius of the Sun R = m luminosity of the Sun L = W mss of the Erth M = kg rdius of the Erth R = m Conversion Fctors 1 u (tomic mss unit) = kg = MeV c 2 1 Å (ngstrom) = m 1 stronomicl unit = m 1 g (grvity) = 9.81 m s 2 1 ev = J 1 prsec = m 1 tmosphere = P 1 yer = s

6 Polr Coordintes x = r cos θ y = r sin θ da = r dr dθ 2 = 1 ( r ) + 1r 2 r r r 2 θ 2 Sphericl Coordintes Clculus x = r sin θ cos φ y = r sin θ sin φ z = r cos θ dv = r 2 sin θ dr dθ dφ 2 = 1 ( r 2 ) + 1 r 2 r r r 2 sin θ ( sin θ ) + θ θ 1 r 2 sin 2 θ 2 φ 2 f(x) f (x) f(x) f (x) x n nx n 1 tn x sec 2 x e x e x sin ( ) 1 x ln x = log e x 1 x cos 1 ( x sin x cos x tn ( 1 x cos x sin x sinh ( ) 1 x cosh x sinh x cosh ( ) 1 x sinh x cosh x tnh ( ) 1 x ) ) 1 2 x x 2 2 +x 2 1 x x x 2 cosec x cosec x cot x uv u v + uv sec x sec x tn x u/v u v uv v 2 Definite Integrls x n e x dx = n! (n 0 nd > 0) n+1 π e x2 dx = π x 2 e x2 dx = 1 2 Integrtion by Prts: 3 b u(x) dv(x) dx dx = u(x)v(x) b b du(x) v(x) dx dx

7 Series Expnsions (x ) Tylor series: f(x) = f() + f () + 1! n Binomil expnsion: (x + y) n = (1 + x) n = 1 + nx + k=0 ( ) n x n k y k k n(n 1) x 2 + ( x < 1) 2! (x )2 f () + 2! nd (x )3 f () + 3! ( ) n n! = k (n k)!k! e x = 1+x+ x2 2! + x3 x3 +, sin x = x 3! 3! + x5 x2 nd cos x = 1 5! 2! + x4 4! ln(1 + x) = log e (1 + x) = x x2 2 + x3 3 n Geometric series: r k = 1 rn+1 1 r k=0 ( x < 1) Stirling s formul: log e N! = N log e N N or ln N! = N ln N N Trigonometry sin( ± b) = sin cos b ± cos sin b cos( ± b) = cos cos b sin sin b tn ± tn b tn( ± b) = 1 tn tn b sin 2 = 2 sin cos cos 2 = cos 2 sin 2 = 2 cos 2 1 = 1 2 sin 2 sin + sin b = 2 sin 1( + b) cos 1 ( b) 2 2 sin sin b = 2 cos 1( + b) sin 1 ( b) 2 2 cos + cos b = 2 cos 1( + b) cos 1 ( b) 2 2 cos cos b = 2 sin 1( + b) sin 1 ( b) 2 2 e iθ = cos θ + i sin θ cos θ = 1 ( e iθ + e iθ) 2 nd sin θ = 1 ( e iθ e iθ) 2i cosh θ = 1 ( e θ + e θ) 2 nd sinh θ = 1 ( e θ e θ) 2 Sphericl geometry: sin sin A = sin b sin B = sin c sin C nd cos = cos b cos c+sin b sin c cos A

8 Vector Clculus A B = A x B x + A y B y + A z B z = A j B j A B = (A y B z A z B y ) î + (A zb x A x B z ) ĵ + (A xb y A y B x ) ˆk = ɛ ijk A j B k A (B C) = (A C)B (A B)C A (B C) = B (C A) = C (A B) grd φ = φ = j φ = φ x î + φ y ĵ + φ z ˆk div A = A = j A j = A x x + A y y + A z z ) curl A = A = ɛ ijk j A k = ( Az y A y z φ = 2 φ = 2 φ x + 2 φ 2 y + 2 φ 2 z 2 ( φ) = 0 nd ( A) = 0 ( A) = ( A) 2 A ( Ax î + z A ) ( z Ay ĵ + x x A ) x y ˆk

Data Provided: A formula sheet and table of physical constants is attached to this paper.

Data Provided: A formula sheet and table of physical constants is attached to this paper. PHY106 PHY47 Dt Provided: Formul sheet nd physicl constnts Dt Provided: A formul sheet nd tble of physicl constnts is ttched to this pper. DEPARTMENT OF PHYSICS & Autumn Semester 009-010 ASTRONOMY DEPARTMENT