Mechanics, Oscillations and Waves

Transcription

1 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 2015 Mechnics, Oscilltions nd Wves 2 hours Answer question ONE (Compulsory) nd TWO other questions, one ech from section A nd section B. Instructions: All questions Answer the re compulsory mrked out question of ten. of ech The section brekdown (Q1 ndon Q4), the nd right-hnd ONE dditionl side question of the of your pper choice is ment froms EACH guide section to Athe ndmrks B. tht cn be obtined from ech prt. All questions re mrked out of twenty. The brekdown on the right-hnd side of the pper is ment s guide to the mrks tht cn be obtined from ech prt. Answers to different sections must be written in seprte books, the books tied together nd hnded in s one. 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. TURN OVER 1

2 1. COMPULSORY SECTION A Mechnics () A circus performer spins hoop of rdius R nd mss M round their wrist. Show tht the moment of inerti of this hoop is given by I = 2MR 2 You should derive ny moment of inerti bout the centre of mss reltionships you use. (b) A womn weighing 600 N steps on bthroom scle tht contins stiff spring. In equilibrium, the spring is compressed 1.0 cm under her weight. Find the force constnt of the spring nd the totl work done on it during the compression. [6] (c) The steroid Plls hs n orbitl period of 4.62 yers, mss of kg nd n orbitl eccentricity of Find the semi-mjor xis of its orbit. [6] [8] 2. () Show tht the rnge of projectile is given by R = (v2 sin 2θ) 0, g where v 0 nd θ re the initil velocity nd firing ngle respectively. [6] (b) Romeo wnts to drop love letter t the feet of Juliet by firing it over the grden wll using slingshot. If the slingshot fires the letter (in smll rigid continer) t n ngle of 25 to the horizontl, wht rnge of velocities could Romeo fire the letter t so tht it psses through the 1.0 metre tll window, the bse of which is locted 5.0 metres from the ground nd Romeo is 18 metres wy? [10] (c) After getting her ttention, Juliet gives Romeo chllenge. If he cn hit n pple dropped from her window using pebble in his slingshot, then Juliet will run wy with him. If the slingshot is fired t the sme time the pple is dropped should Romeo im bove, below or t the initil position of the pple? Explin your resoning nd stte ny ssumptions you hve mde. [4] CONTINUED 2

3 3. () A block is suspended from light, inextensible cble, nd bullet is fired t nd becomes embedded in the block. Using bllistic pendulum setup, show tht the initil velocity of bullet v i is given by v i = 1 + M 2gh, m where m nd M re the msses of the bullet nd pendulum block respectively, nd h is the mximum height the pendulum block reches. [8] (b) A tired scientist believes tht initil velocity of the bullet is ctully given by v i = 1 + M v f, m where v f is the velocity of the bullet-block system fter impct. Wht ssumptions hve they mde to derive this eqution? Explin whether these ssumptions re vlid or not. [6] (c) Insted of using bllistic pendulum, bullet with velocity v = 100 m/s is fired t sttionry solid bll resting on surfce. If the bullet deflects t n ngle of 30 to its originl pth nd the bll is nine times more mssive thn the bullet, wht is the velocity of the bll fter the impct? You should ssume tht the bll only moves horizontlly nd does not bounce or lift from the surfce. [6] TURN OVER 3

4 SECTION B Oscilltions nd Wves 4. COMPULSORY Consider 2 kg mss suspended on spring where the displcement (in cm) s function of time (in seconds) is given by x(t) = Acos(ωt φ), where A = 8.5 cm, ω = 4.2 rd s 1 nd φ = 2.5 rd. () Clculte the time for one complete vibrtion. [2] (b) Find the force constnt of the spring. [4] (c) Find the mximum speed of the mss. [2] (d) Clculte the mximum force on the mss nd the mximum kinetic energy. [4] (e) Find the position, speed nd ccelertion of the mss t t = 1.0 nd the force on the mss t tht time. (f) Stte one ssumption mde in order for your clcultions to be correct. [2] [6] 5. Consider single violin string, comprising thin tut wire fixed t both ends. When oscillting in the third hrmonic its shpe is described by the stnding wve eqution y(x, t) = Asin(k x) sin(ω t), where A = 8.0 cm, k = rd cm 1 nd ω = 50 rd s 1. () Drw sketch to illustrte the form of the stnding wve pttern. [2] (b) Wht is the mplitude of the two trvelling wves tht together form the stnding wve? (c) Clculte the length of the string. [4] (d) Find the period, frequency, wvelength nd speed of the trvelling wve. [4] (e) Write n eqution for the trnsverse speed of point on the string nd hence clculte the mximum trnsverse speed. [4] (f) Consider the possibility tht the string is vibrting in its eighth hrmonic. Wht would be the eqution for y(x, t) in this cse? [4] [2] CONTINUED 4

5 6. Consider person listening to source of sound comprising single frequency wve, trvelling through ir t norml tmospheric pressure nd density. The speed of sound v = 344 m s 1, the bulk modulus B = P, the density ρ = 1.20 kg m 3 nd the mximum pressure vritions p mx = P. () Write n eqution for the pressure of the wve p(x, t) in terms of the wvenumber k, the displcement mplitude A, the bulk modulus B nd ω. Hence show tht the eqution for the mximum pressure fluctution is p mx = BkA. [4] (b) Clculte the mximum displcement if the frequency of the sound wve is 500 Hz. (c) The intensity I of sound wve is the time verged vlue of p(x, t)v y (x, t) where p(x, t) is the force per unit re nd v y (x, t) is the prticle velocity. By using the reltionship v 2 = B/ρ, show tht [2] I = p2 mx 2ρv, nd hence clculte the intensity of our sound wve in W m 2. [4] (d) Show tht the pressure mplitude cn be written s constnt tht does not depend on frequency. (e) Now clculte the pressure nd displcement mplitudes for 50 Hz sound wve with the sme intensity s the 500 Hz sound wve. [4] (f) If the person moves fctor three further wy from the source of the sound, by how mny decibels does the sound intensity level drop, ssuming the intensity obeys the inverse squre lw? [2] [4] [4] END OF EXAMINATION PAPER 5

6 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

7 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

8 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

9 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