March 30, 2017

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A hanging wire made out of titanium with diameter 0.080 cm is initially 2.3 m long. When a 78 kg mass is hung from it, the wire stretches an amount 2.89 cm. A mole of titanium has a mass of 47.9 grams, and its density is 4.51 g/cm3. Based on these experimental measurements, what is the Young's modulus for titanium?

Find the effective spring stiffness of one interatomic bond in titanium.

  • Physics - ,

    Young's modulus = (tensile stress)/(strain)

    In this case the strain is
    (delta L)/L = 2.89cm/230 cm
    = 1.257*10^-2

    and the stress is

    sigma = (78*9.8 N)/[(pi/4)*(8^10^-4m)^2] = 764.4 N/5.027*10^-7 m^2)
    = 1.52*10^9 N/m^2

    So E = 1.21*10^11 N/m^2
    = 121 GPa

    A spring stiffness for an individual Ti-Ti interatomic pair can be estimated by dividing
    (Tensile force)/(area occupied by one molecule)
    by (stretch per intermolecular molecule pair).

    You will need a characteristic intermolecular distance or diameter for Ti atoms in the solid. Call it d. Get that from the number density of Ti atoms, n.
    d = n^(-1/3)

    n = [4.51 g/cm^3/(47.9g/mole)]*6.02*10^23 atom/mole = 5.56*10^22 atom/cm^3
    n^-1/3 = d = 2.6*10^-8 cm
    = 2.6*10^-10 m
    spring stiffness = k
    = E/d = 1.21*10^11/2.6*10^-10
    = 4.6*10^20 N/m

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