(Pr. 1) In an isotropic medium, Poisson's ratio v relates Young's modulus to either the shear modulus G or thebulk modulus K (inverse of the compressibility) as: Y=2(1 + v)G=3(1-2v)K (see your theory slides).(a) A certain material has K = 8 -10³ Pa and Y = 10° Pa. What is its Poisson's ratio? What is its shear modulus?A vulcanized natural rubber (cis-polyisoprene) has K = 2. 10° Pa and Y = 1.3 106 Pa. What is Poisson's ratiofor this vulcanized natural rubber? What is its shear modulus?(b) Show that in the limit where K is much larger than G, Poisson's ratio is equal to 1/2. This limit is valid formost liquids (including polymer melts and solutions), which is why they are termed "incompressible".(c) Show that the bulk modulus of an ideal gas is equal to its pressure. Assuming that a water molecule is asphere of diameter 0.3 nm, estimate the bulk modulus of water assuming that the ideal gas result is valid evenfor liquid water. Compare your value with the actual bulk modulus of water (2 GPa).(Pr. 2) A simple viscometer consists of two square plates, of side 10 cm, separated by a distance of 1 mm. Thelower plate is fixed while the upper one can slide.(a) If the space between the plates is filled with an oil whose viscosity is 1 Pa s, find the force that must beapplied to the upper plate to keep it moving at a speed of 0.5 cm s ¹.(b) If the gap between the plates is filled with a rubber with a shear modulus of 5 105 Pa, and the upper plate isshifted from the rest position of the rubber yielding an angular deformation of 0.01 radians, what force must beapplied to keep the upper plate in this position?(c) A parallel-plate rheometer consists of a motor able to rotate a disk of radius R parallel to a flat horizontalsurface, and a heighth above it. The space between the disk and the surface is filled with a rather viscous liquidof viscosity 7. Show that, for the disk to rotate at a constant angular velocity about its centre, the torqueapplied to the disk must be proportional to R¹, and find the constant of proportionality. Why can the moment ofinertia of the disk be neglected under steady conditions (constant ) ? Hint: integrate the definition of viscosity.(Pr. 3) Consider a phase made of molecules with permanent dipole moment p, in an applied electric fieldEexternal. Each dipole has energy W=-p. E= μ E cose, where E is the local electric field at a molecule'sposition, 0 is the angle between E and p, and is the modulus of the vector p. Assume that E is parallel to thez axis.(a) Explain why the average molecular dipole moment is given by: (p)Spexp: (p₂)2 =Sexp ΦΩL-Eμ(cose) 2, where do do sinode = do d(cose) is the differential of the solid-angle. Why is the solid angle usedhere? How is the elementary probability dw that the dipole moment points in a given direction, related to thedifferential d?(b) By integrating by parts the numerator of the above expression for (p), show that its modulus is equal to:[exp(a)+exp(-a)(P₂) = μ(cose):=H*exp(x)dx[exp(a) exp(-a)], where a = µ E /(kB7)aThe function L(x) ==μlexp(x)dxex+e-xex-e-x-= coth(x) is known as Langevin function. (coth = hyperbolic cotangent)(c) Use the approximate expression ex1 + x (Taylor expansion to linear order), valid for small x, inside bothintegrals in the expression for expansion (p) given in (a), to show that (p)show that (p) = 37. Call K the effective3kgT3kgTpolarizability, deff.(d) Explain the meaning of local electric field, and why is it different, in general, from the externally appliedfield Eexternal. Show that, if the molecules are in the gas phase, the local field is simply equal to E = Eexternal. Showthat, for a gas of polar molecules, the polarization of the sample, defined as P = n (p), where n (=N/V) is thenumber density of molecules, can be written as P = (-1) Eo E, where to is the vacuum permittivity, and givethe expression for & for this gas of polar molecules, as a function of the effective polarizability. Substitute thevalue you found in (c) to obtain the expected temperature dependence of &(e) If instead the phase is a polar liquid, combine the result of point (c) with either the Clausius-Mossotti cavityfield, or the "reaction" field (see theory slides), to obtain two different expressions for &. Compare the twoexpressions with one another, and with the Kirkwood formula in the theory slides.

Question

(Pr. 1) In an isotropic medium, Poisson's ratio v relates Young's modulus to either the shear modulus G or thebulk modulus K (inverse of the compressibility) as: Y=2(1 + v)G=3(1-2v)K (see your theory slides).(a) A certain material has K = 8 -10³ Pa and Y = 10° Pa. What is its Poisson's ratio? What is its shear modulus?A vulcanized natural rubber (cis-polyisoprene) has K = 2. 10° Pa and Y = 1.3 106 Pa. What is Poisson's ratiofor this vulcanized natural rubber? What is its shear modulus?(b) Show that in the limit where K is much larger than G, Poisson's ratio is equal to 1/2. This limit is valid formost liquids (including polymer melts and solutions), which is why they are termed &#34;incompressible&#34;.(c) Show that the bulk modulus of an ideal gas is equal to its pressure. Assuming that a water molecule is asphere of diameter 0.3 nm, estimate the bulk modulus of water assuming that the ideal gas result is valid evenfor liquid water. Compare your value with the actual bulk modulus of water (2 GPa).(Pr. 2) A simple viscometer consists of two square plates, of side 10 cm, separated by a distance of 1 mm. Thelower plate is fixed while the upper one can slide.(a) If the space between the plates is filled with an oil whose viscosity is 1 Pa s, find the force that must beapplied to the upper plate to keep it moving at a speed of 0.5 cm s ¹.(b) If the gap between the plates is filled with a rubber with a shear modulus of 5 105 Pa, and the upper plate isshifted from the rest position of the rubber yielding an angular deformation of 0.01 radians, what force must beapplied to keep the upper plate in this position?(c) A parallel-plate rheometer consists of a motor able to rotate a disk of radius R parallel to a flat horizontalsurface, and a heighth above it. The space between the disk and the surface is filled with a rather viscous liquidof viscosity 7. Show that, for the disk to rotate at a constant angular velocity about its centre, the torqueapplied to the disk must be proportional to R¹, and find the constant of proportionality. Why can the moment ofinertia of the disk be neglected under steady conditions (constant ) ? Hint: integrate the definition of viscosity.(Pr. 3) Consider a phase made of molecules with permanent dipole moment p, in an applied electric fieldEexternal. Each dipole has energy W=-p. E= μ E cose, where E is the local electric field at a molecule'sposition, 0 is the angle between E and p, and is the modulus of the vector p. Assume that E is parallel to thez axis.(a) Explain why the average molecular dipole moment is given by: (p)Spexp: (p₂)2 =Sexp ΦΩL-Eμ(cose) 2, where do do sinode = do d(cose) is the differential of the solid-angle. Why is the solid angle usedhere? How is the elementary probability dw that the dipole moment points in a given direction, related to thedifferential d?(b) By integrating by parts the numerator of the above expression for (p), show that its modulus is equal to:[exp(a)+exp(-a)(P₂) = μ(cose):=H*exp(x)dx[exp(a) exp(-a)], where a = µ E /(kB7)aThe function L(x) ==μlexp(x)dxex+e-xex-e-x-= coth(x) is known as Langevin function. (coth = hyperbolic cotangent)(c) Use the approximate expression ex1 + x (Taylor expansion to linear order), valid for small x, inside bothintegrals in the expression for expansion (p) given in (a), to show that (p)show that (p) = 37. Call K the effective3kgT3kgTpolarizability, deff.(d) Explain the meaning of local electric field, and why is it different, in general, from the externally appliedfield Eexternal. Show that, if the molecules are in the gas phase, the local field is simply equal to E = Eexternal. Showthat, for a gas of polar molecules, the polarization of the sample, defined as P = n (p), where n (=N/V) is thenumber density of molecules, can be written as P = (-1) Eo E, where to is the vacuum permittivity, and givethe expression for & for this gas of polar molecules, as a function of the effective polarizability. Substitute thevalue you found in (c) to obtain the expected temperature dependence of &(e) If instead the phase is a polar liquid, combine the result of point (c) with either the Clausius-Mossotti cavityfield, or the &#34;reaction&#34; field (see theory slides), to obtain two different expressions for &. Compare the twoexpressions with one another, and with the Kirkwood formula in the theory slides.

Accepted Answer

Hi,   As per the guidelines, we can solve only first question for you since the exact one is nor…