7. 0/4 points I Previous Answers M134. 12.P084. My Notes One mole of tungsten (6

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7. 0/4 points I Previous Answers M134. 12.P084. My Notes One mole of tungsten (6 x 1023 atoms) has a mass of 184 grams, and its density is 19.3 grams per cubic centimeter. You have a long thin bar of tungsten, 2.5 m long, with a square cross section, 0.08 cm on a side. You hang the rod vertically and attach a 118 kg mass to the bottom, and you observe that the bar becomes 1.25 cm longer. Calculate the effective stiffness of the interatomic bond, modeled as a "spring" Next you remove the 118 kg mass, place the rod horizontally, and strike one end with a hammer. How much time At will elapse before a microphone at the other end of the bar will detect a disturbance? Submit AnswerSave Progress

Explanation / Answer

We start by finding the bond length d, and also determine the Young's modulus Y, from which we can derive the bond stiffness k(b). See ref. 1 for a more detailed explanation of the method.
Bond length d is assumed equal to the atom's "diameter", which is actually just the cube root of its (cubic) volume.
d = (Volume/NAtoms)^(1/3)
= [184/19.3 (cm^3/mole) / 6.02E23 (atoms/mole)] ^ (1/3)
= 2.51E-8 cm/atom = 2.51E-10 m/atom
Young's modulus Y:
Y = F/(AL/L) = FL/(AL) = 9.8*118*2.5/(0.0008^2*0.0125) = 3.6138E11 Pa
Now we need to determine the number of atoms in the rod length N1 and cross section N2.
N1 = L/d
N2 = A/d^2
Elongation of each bond L(b) = L/N1 = dL/L
Force per bond F(b) = F/N2 = Fd^2/A
k(b) = F(b)/L(b) = (Fd^2/A)/(dL/L) = FL/(AL)*d^2/d = Yd
k(b) = 3.60138E11*2.51E-10 = 90.3438 N/m (answer)
Sound speed c = sqrt[Y(1-v)/(density(1+v)(1-2v))] where v is Poisson's ratio, ~0.3 for most metals. See ref. 2. (Note: input density in kg/m^3.)
t = L/c (answer)

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