One mole of magnesium (6 1023 atoms) has a mass of 24 grams, and its density is

Question

One mole of magnesium (6 1023 atoms) has a mass of 24 grams, and its density is 1.74 grams per cubic centimeter. You have a long thin bar of magnesium, 2.9 m long, with a square cross section, 0.12 cm on a side. 1. You hang the rod vertically and attach a 32 kg mass to the bottom, and you observe that the bar becomes 1.58 cm longer. Calculate the effective stiffness of the interatomic bond, modeled as a "spring": 2. Next you remove the 32 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?

Explanation / Answer

The force applied to the bar is 34*9,8 = 333.2 N. The extension of the wire ?L = 0.0145 m, so the bar stiffness is F/?L = 2.30*10^4 N/m The number of atoms in one layer of cross section is area of the bar divided by the area of one atom =0.0012²/(2.84*10^-10)² = 1.785*10^13. No no of bonds along the length is 2.5/(2.84*10^-10) = 8.80*10^9 The applied force is divided among all the bonds along the length, and then divided among all the atoms in a cross-section layer The force applied to each atom in a layer is then 333.2/(1.785*10^13*8.80*10^9) = 2.12*10^-21 N The strain (fractional length increase, ?L/L) on the bar is 0.0145/2.5 = 0.0058. This will also be the strain between the atom layers. The bond extension ?L is then 2.84*10^-10 * 0.0058 = 1.6472*10^-12 m. The bond stiffness F/?L = (2.12*10^-21)/(1.6472*10^-12) = 1.29*10^-9 N/m

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