Question

One mole of nickel (6.02 ✕ 10²³ atoms) has a mass of 59 g, and its density is 8.9 g/cm³. You have a bar of nickel 2.48 m long, with a square cross section, 2.3 mm on a side. You hang the rod vertically and attach a 39 kg mass to the bottom, and you observe that the bar becomes 0.9 mm longer. Next you remove the 39 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? (Assume a simple cubic lattice for nickel.)

Answer 1

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).
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)
= [59/8.9 (cm^3/mole) / 6.02E23 (atoms/mole)] ^ (1/3)
= 2.22E-8 cm/atom = 2.22E-10 m/atom
Young's modulus Y:
Y = F/(AΔL/L) = FL/(AΔL) = 9.8*39*2.48/(0.0023^2*0.0009) = 1.99E11 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 ΔL(b) = ΔL/N1 = dΔL/L
Force per bond F(b) = F/N2 = Fd^2/A
k(b) = F(b)/ΔL(b) = (Fd^2/A)/(dΔL/L) = FL/(AΔL)*d^2/d = Yd
k(b) = 1.99E11*2.22E-10 = 44.18 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.
t = L/c = 2.48/0.3 = 8.27 s (answer)