Homework_5_Solutions

Homework_5_Solutions - 14:125:208: Introduction to...

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14:125:208: Introduction to Biomechanics Spring 2007 Homework Assignment #5 Due 3/08/07 Problem 1: Ozkaya and Nordin Problem 7.6 In order to complete this problem we first write out the equations of Static Equilibrium (a) = + = = + + = 0 2 0 0 0 2 2 2 1 1 1 2 2 1 1 lW T d W l T d M W T W T R F A A y Now we have 2 equations and 3 unknowns (T 1 , T 2 , and R A ). So we need another equation which relates T 1 to T 2 due to geometric constraints. 2 1 2 8 1 8 2 2 1 1 2 8 6 9 2 2 2 2 2 2 2 2 2 1 1 8 6 9 1 1 1 1 1 1 1 1 1 1 2 2 2 2 1 1 1 1 1 2 2 1 1 2 2 1 1 0417 . 1 3 10 25 . 6 1 10 2 10 25 . 6 10 400 10 80 2 10 2 10 500 10 200 2 tan T T T T d d T T A E h T A E T h T T A E h T A E T h A T E A T E h h d d = × = × = × = × × = = = × = × × = = = = = = = = = = = δ ε σ α Now we plug back into our equations of static equilibrium to solve for T 1 , T 2 and R A . N W T W T R N T N T T T T lW T d W l T d A 409 400 445 100 464 464 445 1800 0417 . 4 0 400 4 3 100 2 0417 . 1 1 0 2 2 2 1 1 1 2 2 2 2 2 2 2 1 1 1 = + = + = = = = = + = + (b) Now to calculate stress we use GPa A T GPa A T 113 . 1 10 400 445 928 . 0 10 500 464 6 2 2 2 6 1 1 1 = × = = = × = =
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14:125:208: Introduction to Biomechanics Spring 2007 Homework Assignment #5 Due 3/08/07 Problem 2: (a) Write an equation which relates the pressure of a spherical shell to the wall stress. From the notes: () σ π φ 2 2 0 2 0 2 2 i r r r r r r wall r r r dr r drd r F o i o i o i = = = = ∫∫ 2 2 2 0 2 2 2 0 2 2 2 0 2 0 2 0 2 2 0 0 2 2 00 2 2 2 0 2 sin 4 1 2 2 sin 2 cos sin sin sin sin sin sin sin sin i i i i i i i i i i i i i i i i i i i i i i p r r r p p r r r p r r r p r p r d p r d p r d d p r d d r p dA p F θ ππ = = = = = = = = = = Now what happens as the wall thickness (r o -r i )=h becomes vanishingly small? ( ) ( ) ( ) i i i i i i i o i o i i i r h r h r r h h r r r r r r r r r p 2 2 2 2 2 2 2 2 2 0 = + = + = =
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14:125:208: Introduction to Biomechanics Spring 2007 Homework Assignment #5 Due 3/08/07 Now as h 0 it is more convenient to define the wall as a surface tension T c (F/m) rather than a stress r r h p i i γ σ 2 2 = = (b) In cell mechanics studies, a micropipette aspiration technique is often used to relate membrane tension (T c ) (not shown) to the pressure applied to the membrane by the micropipette (P p ). Given a cell with a radius, R c and a pipette of radius, R p , derive an expression which relates P p to T C . In order to relate P p to T c we need to perform a force balance on the section of membrane which has been pulled into the pipette. P p R c R p P c
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14:125:208: Introduction to Biomechanics Spring 2007 Homework Assignment #5 Due 3/08/07 Now, P c is unkown so we need to relate P c to T c . This can be done by considering the rest of the cell So Now we find = = + = + = c P c p p p c p P c p p c c p p p c p c P R R T P P R R R R T P R R T R P R P R T R 1 1 2 2 2 2 2 2 2 2 2 2 2 π T c R c P c T c T c R p P p P c T c c c c c c c c R T P P R T R 2 2 2 = = p p c p c P P R P R T R 2 2 2 + =
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14:125:208: Introduction to Biomechanics Spring 2007
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Homework_5_Solutions - 14:125:208: Introduction to...

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