Data Provided: A formula sheet and table of physical constants is attached to this paper. MEDICAL PHYSICS: Physics of Living Systems 2 2 HOURS

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1 Dt Provided: A formul sheet nd tble of physicl constnts is ttched to this pper. Ancillry Mteril: None DEPARTMENT OF PHYSICS AND ASTRONOMY Spring Semester (2015/2016) MEDICAL PHYSICS: Physics of Living Systems 2 2 HOURS The pper is divided into two Sections: A nd B The student should nswer ll questions in Section A. One sentence nswers re sufficient for ll questions in this section. The student should nswer two questions from Section B. 1 TURN OVER

2 SECTION A - COMPULSORY (Answer ll questions in this Section: 2 mrks ech) 1. A coin is thrown 5 times. Wht is the probbility of getting 5 heds? 2. How is strin within mteril best chrcterised; is it sclr, vector or tensor quntity? 3. An elstic wire of relxed length L 0 is stretched to new length L 1. Write down n expression describing its strin. 4. Write down the mthemticl eqution tht defines the bulk modulus of mteril. 5. Poisson's rtio is often quoted in reltion to deformtion of mterils. Wht is ment by Poisson's rtio? 6. Provide n pproximte vlue for the stiffness of corticl bone. 7. Wht is the nme given to the idelised viscoelstic model represented by dshpot nd spring in series? 8. Curvture refers to ngulr chnge per unit length of curve. Wht is the curvture of circle of dimeter d? 9. A liner elstic spring of relxed length L 0 nd spring constnt k is extended so tht it is now of length L 1. Write n expression for the energy stored in the spring. 10. Write down the ffine trnsform responsible for uniformly scling n object in 2D spce. 11. In which vessel is blood flow most pulstile? 12. Write down the Bernoulli eqution. 13. Wht is the conversion fctor between the SI unit of pressure (P) nd the clinicl unit of blood pressure, mmhg? 14. By wht proportion is the pek velocity higher thn the men velocity for fully developed lminr flow in stright tube? 15. Write down the expression for the Womersley prmeter which defines the nture of pulstile flow in tube. 16. Wht effect does the elsticity of the wll of n rtery hve on its fluid dynmic chrcteristics? 17. By how much does the pressure in the vessels in the foot increse from the supine (lying down) to the stnding position? 2 CONTINUED

3 18. Wht is most often tken s the Reynolds number for the onset of turbulence in tube? 19. In which vessels is it resonble to ssume tht blood behves s Newtonin fluid? 20. At bifurction of the rteril system if the vessel dimeter of the dughter vessel is 80% of the prent vessel, wht is the percentge increse in totl cross-sectionl re of the vsculr bed? 3 TURN OVER

4 SECTION B (Answer two questions in this Section: 30 mrks ech) B1. A syringe with mssless, zero friction plunger hs short outlet nozzle with orifice dimeter 2mm. The plunger trvels 10cm to displce volume of 50ml. Sketch the dimensions of the syringe. [5] b. An experiment is undertken in stndrd temperture nd pressure conditions (ie. 15 deg.c, MP) in which the plunger is fully depressed nd then withdrwn (in ir) by 50ml. Next, the nozzle is completely blocked so tht the ssembly now cts like spring. With the syringe held still nd 2N force pplied to the plunger clculte: (i) the equilibrium pressure of the compressed gs to 4 significnt figures [6] (ii) the expected displcement of the syringe plunger. (Assume tht Boyle's Lw pplies {ie. Pressure x Volume = constnt} nd tht temperture effects cn be ignored). [6] (iii) the effective spring constnt of this system. Declre ny ssumptions you hve mde. [3] c. Consider tht the blockge in the nozzle is now removed. Clculte the mximum velocity of the plunger if the force of 2N is mintined. In this instnce, ssume tht Poiseuille flow through the outlet is the flow limiting fctor, given tht the nozzle is ttched to 2mm dimeter tube of 10cm length. (Note: Coefficient of viscosity of ir = 1.81x10-5 Ps). [8] d. The coefficient of viscosity of ir is criticl fctor in the behviour of this system. How is viscosity defined? [2] 4 CONTINUED

5 B2. Consider n object tht is described by the following coordintes x 1,y 1 = 1,1 x 2,y 2 = 3,1 x 3,y 3 = 2,2.73 Wht kind of object is this? [2] b. Write down the ffine trnsform tht represents displcement of this object 3 units long the x-xis nd 5 units long the y-xis. [2] c. Write down the ffine trnsform tht represents rottion of this object 30 degrees clockwise bout the origin. [2] d. Write down the trnsform tht represents rottion of this object 30 degrees bout its centroid. [10] e. A sphygmomnometer is used to mesure blood pressure by compressing the rtery in the upper rm nd monitoring the sound of blood squirting through the vessel s the pressure is decresed. Assuming the dimeter of the brchil rtery is 4 mm nd the pek blood flow through the vessel is 300ml/min compute the Reynolds number under the following conditions: i) When the cuff pressure equls systolic pressure, ssuming the effective dimeter of the vessel is one tenth the norml vlue, due to vessel compression ii) When the cuff pressure equls distolic pressure nd the vessel hs returned to the norml vlue [8] f. Discuss the difference in the flow regime under these two conditions nd describe how this is used to mesure systolic nd distolic blood pressure [6] 5 TURN OVER

6 B3. The simplest model we might use for the temporl vrition of pressure nd flow out of the ort due to the systemic circultion is shown in the figure below Wht is the nme given to this type of model? [1] b. Given P out = 0, derive the eqution which reltes P in to Q in in this model. You cn ssume peripherl resistnce = R nd the elstic tube hs rdius, r, thickness, h, nd elstic modulus, E. [6] c. Sketch the electricl nlogue of this model [3] d. Describe the behviour represented by this model during systole nd distole [6] e. Sketch the form of the vrition in pressure with time predicted by the model during distole [3] f. Comment on how you might expect the model prmeters to differ between child nd n elderly individul, wht process is responsible for this chnge? [3] g. Write down the eqution for the speed of the pressure pulse nd discuss how you expect the wvespeed to vry with incresing ge [4] h. Given typicl wvespeed in the ort of 5ms -1 estimte the elstic modulus of the ort, stting ny ssumptions you mke [4] END OF EXAMINATION PAPER 6

7 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

8 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

9 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

10 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 are attached to this paper. DEPARTMENT OF PHYSICS & ASTRONOMY Spring Semester

Data Provided: A formula sheet and table of physical constants are attached to this paper. DEPARTMENT OF PHYSICS & ASTRONOMY Spring Semester Dt Provided: A formul sheet nd tble of physicl constnts re ttched to this pper. Ancillry Mteril: None DEPARTMENT OF PHYSICS & ASTRONOMY Spring Semester 2016-2017 MEDICAL PHYSICS: Physics of Living Systems