DEPARTMENT OF PHYSICS AND ASTRONOMY Dt Provided: A formul sheet nd tble of physicl constnts is ttched to this pper. Liner grph pper is vilble. Autumn Semester 2014-2015 PHYSICS : Elements of Physics 2 HOURS Answer questions ONE nd FOUR, plus ONE other from ech section, FOUR questions in ll. Answers to different sections must be written in seprte books, the books tied together nd hnded in s one. 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, nd scles with difficulty. 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. 1 TURN OVER
SECTION A 1. COMPULSORY () For question 1, ssume the bsence of drg due to ir resistnce, ssume the speed of light is infinitely fst, nd the speed of sound is c = 340 ms -1. (i) You drop stone into deep well. The well is lit, you see the stone splsh into the wter t the bottom of the well fter 4 seconds. Clculte the depth of this well. [2] (ii) You drop nother stone into nother well. This well is drk, but you her the splsh of the stone hitting the wter t the bottom 4 seconds fter dropping the stone. Clculte the depth of the second well. [7] (b) A fireworks rocket hs mss of 290 g. When ignited, it produces thrust of 6 N. (i) Clculte the initil ccelertion of the rocket when it is lunched verticlly from the Erth s surfce. [2] (ii) Assuming the rocket s ccelertion remins t its initil mgnitude, how long will the rocket tke to rise to 50 m bove ground? [1] (c) The vectors s 1 nd s 2 represent two displcements, in m: s 1 = 5 ˆ x + 6 ˆ y nd s 2 = 4 ˆ x 2 ˆ y Wherein ˆ x nd ˆ y stnd for unit vectors in the x nd y directions. Clculte (tht mens clculte, not drw nd mesure): (i) the mgnitudes of both vectors. [2] (ii) the ngle (in degrees) between s 1 nd s 2. [2] (d) A prticle moves in circulr pth, rdius r, with constnt velocity v. Write down expressions, in terms of r nd v, for: (i) (ii) the ngulr velocity, ω, of the prticle, the centripetl ccelertion,, of the prticle. [1] [1] (e) In the gme of Rugby, plyers try to get the better of ech other when their bodies collide t high speed. (i) (ii) Nme the physicl conservtion lw tht controls the behviour of colliding bodies. At the pek of his creer, Jonh Lomu s body mss ws 120 kg (1.6 times tht of n verge helthy mle dult), yet he sprinted 100 m in 10.89 seconds (only bout 10% slower thn the world record). Explin his success s rugby plyer despite limited technicl bility. [1] [1] 2 CONTINUED
2. () (b) Give the work done by constnt force cting for given distnce, nd stte its SI unit. In your nswer, ccount for the vector chrcter of both force, nd distnce. [2] A firground ride releses from rest crt with pssengers t height, H, bove ground. The crt first runs downhill on frictionless trck to pick up speed. As it reches ground level, it then enters into verticl circulr loop of rdius, R, with 2R < H. Drg nd friction cn be neglected. (i) Drw n nnotted sketch of the described ride, including coordinte system with sensibly chosen origin. Put the most importnt vlues of height bove ground on the y xis. [3] (ii) Clculte the centrifugl ccelertion on pssenger t the lowest, nd the highest, point of the loop. Your expression shll be in terms of R, H, g, nd the pssenger s mss M, if required. Also give the direction of the ccelertion in terms of the unit vectors of your coordinte system from (i). [6] (iii) For given R, find minimum mgnitude of H so tht pssengers cnnot fll out of the crt t the highest point of the loop, even in the bsence of sfety fetures like belts. [2] (c) (d) Collisions between bodies my be either elstic or inelstic. Which type(s) of collision(s) conserve(s) kinetic energy? Which type(s) of collision(s) conserve(s) momentum? [2] Two gliders re plced on frictionless, horizontl ir trck. Glider A hs mss of 0.40 kg nd initilly is sttionry ner the middle of the trck. Glider B hs mss of 0.25 kg nd velocity 0.25 m s -1 when it collides with A. The gliders stick together fter the collision. Clculte: (i) the velocities of the gliders fter the collision, [2] (ii) the chnge in kinetic energy s result of the collision. [3] 3 TURN OVER
3. () (i) Define ccelertion. [1] (ii) A cr, trvelling long stright rod, ccelertes with constnt ccelertion to increse its velocity from 20 m s 1 to 36 m s 1 in 4.0 s. Clculte the mgnitude of ccelertion, nd the displcement of the cr during the 4 second period. [2] (iii) If the cr s mss is 1000 kg, by how much hs its kinetic energy incresed? [2] (b) Explin the difference between elstic nd inelstic collisions. [2] (c) A crne lifts pllet of bricks, mss 500 kg, through verticl height of 20 m in 35 s. Clculte: (i) the work done by the crne, [2] (ii) the power of the crne. [1] (d) A bllistic pendulum cn be used to mesure the speed of bullet. It consists of soft wooden block (mss M) suspended by wire of negligible mss. A bullet (mss m) fired into the block becomes embedded into it following n inelstic collision. The block (nd bullet) rise through verticl height h. Clculte: (i) the increse in grvittionl potentil energy when the block nd bullet hve risen to their gretest height, [2] (ii) the velocity of the block nd bullet t its initil position, immeditely fter the bullet hs embedded itself. [2] (iii) the velocity of the bullet s it enters the block. [4] (iv) lower limit for the impulse (known s kick bck ) tht the gun firing the bullet hs imprted on the gunner. [2] mss of block, M = 2.485 kg mss of bullet, m = 0.015 kg verticl rise, h = 29.36 cm 4 CONTINUED
4. COMPULSORY SECTION B () (i) Define electric field strength nd stte its SI unit. [2] (ii) Clculte the electric field strength t distnce r = 30 cm from point chrge Q = +15 µc. [2] (b) A rel bttery is modelled by n idel electrochemicl cell of unknown electromotive force (emf) E (in Volts, not to be confused with electric field), in series with n internl resistnce, r. The bttery is to be chrcterised with the help of vrible mesurement resistor by circuit s shown: E r I A The mesurement resistor is set to different settings, nd vlues of potentil difference (V) cross the mesurement resistor, nd corresponding vlues of current in the circuit (I), re mesured nd tbulted: V [V] 0.9 1.3 2.3 3 3.6 4 I [A] 9 7.9 5.6 4.2 2.8 2 V V Use the dt in the tble to drw plot, using the supplied grph pper. From tht plot find the emf E, nd the highest current the bttery cn deliver. Wht is the internl resistnce r of the bttery? [10] (c) Define cpcitnce of cpcitor, nd give its SI units. [2] (d) A solenoid is 75 cm long nd hs 300 turns. Use the formul B = µ 0nI to clculte the mgnetic flux density B t the centre of the solenoid when current of 1.4 A flows in it. [2] (e) A sinusoidl lternting current wveform hs pek vlue of 5.0 A nd period of 20 ms. Clculte the frequency, nd the root-men-squre (r.m.s.) vlue, of the current. [2] 5 TURN OVER
5. () Two cpcitors (2 µf nd 3 µf) re connected in series. One electrode of the 2 µf cpcitor is connected to 0V (ground), one electrode of the 3µF cpcitor is connected to 12 V. The respective other electrodes re connected to ech other. Clculte: (i) the overll cpcitnce, [1] (ii) the chrge on ech cpcitor, [2] (iii) the potentil difference cross the 2 µf cpcitor, [2] (iv) the energy stored in the 3 µf cpcitor. [2] (b) Two resistors (4 kω nd 6 kω) re connected in prllel to 12V bttery with negligible internl resistnce. Clculte: (i) the totl current flowing in the circuit. [2] (ii) the power dissipted in ech resistor. [2] (c) (d) Show in digrm how three 8 kω resistors cn be connected to give totl resistnce of 12 kω. [3] The heting element of cr windscreen consists of ten identicl conductors connected in prllel. Ech conductor is 1.8 m long nd mde from mteril with resistivity 1.0 10-5 Ω m, nd hs rectngulr cross-section of 1.0 mm by 0.30 mm. Clculte: (i) the resistnce of one of the conductors in the heting element, [2] (ii) the totl resistnce of the heting element, [2] (iii) the power delivered to the heting element when it is connected to 12 V bttery with negligible internl resistnce. [2] 6 CONTINUED
6. () A 4.7 mf cpcitor is first chrged to 9.4 V, nd then dischrged through 1 kω resistor. Clculte: (i) the energy tht is stored in the cpcitor initilly. [1] (ii) the time it tkes for the voltge cross the cpcitor to decy to 3.458 V. [1] (iii) the totl mount of energy tht will be dissipted in the resistor until the cpcitor is dischrged completely. [1] (b) A solenoid is 10 cm long, hs rdius of 0.5 cm, nd is mde of 1000 turns of copper (Cu) wire. (i) Clculte the totl length of the copper wire. [1] (ii) The copper wire is cylindricl with dimeter of 0.1 mm. The resistivity of copper is ρ = 1.68 x 10-8 Ωm. Clculte the resistnce of the solenoid. [2] (iii) The solenoid is connected to 12V bttery with negligible internl resistnce. Clculte the current tht will flow in the solenoid, nd the power dissipted in the solenoid. Also, clculte the resulting mgnetic field B in the centre of the solenoid, using B = µ ni [3] o (iv) Explin why solenoids for high B fields (in the order 1T) require wter cooling. [1] (c) (d) Explin why mgnetic field lines re closed loops, without beginning or end. Compre nd contrst to electric field lines. [2] A bem of protons moving t 1.6 10 6 m s -1 enters region of uniform mgnetic field (flux density 0.3T) perpendiculr to the field lines. Clculte: (i) the mgnetic force cting on ech proton in the bem. [2] (ii) the rdius of the circulr pth followed by the protons in the mgnetic field. [2] (iii) the frequency with which protons will complete this circulr pth. [3] (e) A trnsformer hs primry coil of 100 turns, nd secondry coil of 1000 turns. When it is driven with n AC current of frequency f = 50 Hz flowing in the primry coil, wht frequency will the current flowing in the secondry coil hve? [1] END OF EXAMINATION PAPER 7 TURN OVER
PHYSICAL CONSTANTS & MATHEMATICAL FORMULAE Physicl Constnts electron chrge e = 1.60 10 19 C electron mss m e = 9.11 10 31 kg = 0.511 MeV c 2 proton mss m p = 1.673 10 27 kg = 938.3 MeV c 2 neutron mss m n = 1.675 10 27 kg = 939.6 MeV c 2 Plnck s constnt h = 6.63 10 34 J s Dirc s constnt ( = h/2π) = 1.05 10 34 J s Boltzmnn s constnt k B = 1.38 10 23 J K 1 = 8.62 10 5 ev K 1 speed of light in free spce c = 299 792 458 m s 1 3.00 10 8 m s 1 permittivity of free spce ε 0 = 8.85 10 12 F m 1 permebility of free spce µ 0 = 4π 10 7 H m 1 Avogdro s constnt N A = 6.02 10 23 mol 1 gs constnt R = 8.314 J mol 1 K 1 idel gs volume (STP) V 0 = 22.4 l mol 1 grvittionl constnt G = 6.67 10 11 N m 2 kg 2 Rydberg constnt R = 1.10 10 7 m 1 Rydberg energy of hydrogen R H = 13.6 ev Bohr rdius 0 = 0.529 10 10 m Bohr mgneton µ B = 9.27 10 24 J T 1 fine structure constnt α 1/137 Wien displcement lw constnt b = 2.898 10 3 m K Stefn s constnt σ = 5.67 10 8 W m 2 K 4 rdition density constnt = 7.55 10 16 J m 3 K 4 mss of the Sun M = 1.99 10 30 kg rdius of the Sun R = 6.96 10 8 m luminosity of the Sun L = 3.85 10 26 W mss of the Erth M = 6.0 10 24 kg rdius of the Erth R = 6.4 10 6 m Conversion Fctors 1 u (tomic mss unit) = 1.66 10 27 kg = 931.5 MeV c 2 1 Å (ngstrom) = 10 10 m 1 stronomicl unit = 1.50 10 11 m 1 g (grvity) = 9.81 m s 2 1 ev = 1.60 10 19 J 1 prsec = 3.08 10 16 m 1 tmosphere = 1.01 10 5 P 1 yer = 3.16 10 7 s
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 2 1 2 x 2 2 +x 2 1 x 2 + 2 1 x 2 2 2 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 0 + + 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
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
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