One mole of tungsten (6 *10^23 atoms) has a mass of 184 grams, and its density i

Question

One mole of tungsten (6 *10^23 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, 3.0 m long, with a square cross section, 0.14 cm on a side.

You hang the rod vertically and attach a 364 kg mass to the bottom, and you observe that the bar becomes 1.52 cm longer. Calculate the effective stiffness of the interatomic bond, modeled as a "spring":
ks =( )N/m

Next you remove the 364 kg mass, place the rod horizontally, and strike one end with a hammer. How much time t will elapse before a microphone at the other end of the bar will detect a disturbance?
delta t =( )s

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/(A?L/L) = FL/(A?L) = 9.8*414*2.6/(0.0015^2*0.013) = 3.6064E11 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 = d?L/L Force per bond F(b) = F/N2 = Fd^2/A k(b) = F(b)/?L(b) = (Fd^2/A)/(d?L/L) = FL/(A?L)*d^2/d = Yd k(b) = 3.6064E11*2.51E-10 = 90.52064 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. (Note: input density in kg/m^3.) t = L/c (answer)

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