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

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1 Dt Provided: A formul sheet nd tble of physicl constnts re ttched to this pper. Ancillry Mteril: None DEPARTMENT OF PHYSICS & ASTRONOMY Spring Semester 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. Clerly indicte the question numbers on which you would like to be exmined on the front cover of your nswer books. Cross through ny work tht you do not wish to be exmined. 1 TURN OVER

2 SECTION A COMPULSORY (Answer ll questions in this Section: 2 mrks ech) 1. Wht is the SI unit of stress? 2. A vector field is described by the function f(x,y)=xi + yj. Sketch the field. 3. In sttistics, wht is the nme given to the distribution tht is chrcterised by bell shped curve? 4. Clculte the men nd stndrd devition of the following 5 numbers: 1,2,3,4,5 5. Sketch second clss lever. 6. Wht is the nme given to the idelised viscoelstic model represented by dshpot nd spring in prllel? 7. Consider the x,y coordinte (2,0) on 2D plne, where force vector of 5N is pplied in the direction of the vector 3i+4j. Clculte the moment cting bout the origin. 8. Wht is the pproximte stiffness of steel? How does this compre to bone? 9. Wht clss of objects re susceptible to buckling, nd under wht conditions? 10. Sketch n I-bem. 11. In wht clinicl context might form of the Bernoulli eqution be pplied? 12. Wht is typicl pressure wvespeed in the humn ort? 13. Write down n expression for Reynolds number for flow in cylindricl tube. 14. Does Reynolds number typiclly increse or decrese t n rteril bifurction? 15. In systems in which blood cn be regrded s Newtonin, by wht fctor is its viscosity higher thn tht of wter? 16. By how much does the pressure in the vessels in the foot increse from the supine (lying down) to the stnding position? 17. Wht is the definition of Newtonin fluid? 18. Write down the Poiseuille eqution. 19. Wht is the dimeter of cpillry in the humn crdiovsculr system? 20. Where in the vessel is sher stress t its mximum for fully developed lminr flow in stright tube? 2 CONTINUED

3 SECTION B (Answer two questions in this Section: 30 mrks ech) 1. () Describe the conditions under which the Poiseuille eqution will give n ccurte prediction of the reltion between pressure grdient nd flow rte in tube [4] (b) Consider the forces cting on n nnulus of fluid within stright tube s shown in the figure below Define the mening of the following terms ( r, R, P, z, τ - s shown in the figure bove) nd provide their SI units. [5] (c) The forces cting on ech fce of the nnulus re shown in the figure below Write down n expression which describes the blnce of forces on the nnulus due to pressure nd sher stress. [8] 3 TURN OVER

4 (d) From this expression show tht the sher stress cting on the nnulus is given by the following formul: [10] τ = r. P 2 z (e) Bsed on this formultion, where do you expect the sher stress to rech its mximum vlue? Why is this importnt when considering flow within blood vessels? [3] 4 CONTINUED

5 2. () Produce sketch illustrting the vrition in pressure throughout the rteril system, strting t the hert nd moving out through the rteries nd the cpillry bed, nd bck through the veins to the hert. Indicte on the sketch the mgnitude of the pressure, distinguishing between systole nd distole. [7] (b) Discuss the effects of the elsticity of the vessel wlls on crdiovsculr flow, mking prticulr reference to the speed of the pressure pulse. Describe the Windkessel effect nd discuss how this phenomenon ffects the flow in the ort. Wht re the elements of the simple one dimensionl model tht might be used to describe this system? [8] (c) The Engineers Theory of Bending reltes curvture (1/R) to bending lod through n pplied moment M. It lso tkes ccount of the Youngs modulus of the mteril nd the second moment of re of the bem. Write down this eqution nd clrify wht ech of the terms men. [6] (d) Show tht the second moment of re of curved solid bem of squre cross-section is proportionl to its bredth (b), rised to the 4 th power. [6] (e) How does the bove expression compre with the second moment of re of cylindricl bem of dimeter b nd solid cross-section, s shown below? How much energy is required to bend such bem (curvture 1/R) if it is of length L? [3] 5 TURN OVER

6 3. () With the help of digrm, describe Hookes Lw, in the context of liner elstic spring. [4] (b) Derive n effective spring constnt (k) of system in which two springs re connected in series, one of which is the sme relxed length but hlf the stiffness of the other. Relte k to the spring constnt of the less stiff spring. [7] (c) Contrst the lod/extension behviour described in () bove, with description bsed on stress/strin. If single spring system is replced by two identicl springs in series, how does this modify the overll spring constnt of the combined system nd the Youngs modulus? [7] (d) Consider the double spring system of (b) in the context of energy. Clculte the reltive mount of work done by ech spring in the combined system (ie. unequl springs in series). [6] (e) Write brief description of ech of the following quntities i) Potentil energy ii) Kinetic energy iii) Strin energy [6] 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