Question

We have a rod made of a certain metal. The rod’s length is L = 1.2 m and has a circular cross section with radius r= 3.8mm. If we hang the rod vertically and hang a mass of m = 22 kg at its end we find that it is stretched by d = 2.5 mm. (a) Using the information given, estimate the Young’s modulus for this metal rod, in N/m.
(b) If we know that this metal has a density of 12 g/cm^3, what would be the speed of sound in this metal in m/s?

Answer 1

Answer:

         Y =  2.286 *$10^{9}$)  N/m

         V =  436.46 m/s

Explanation:

Length, L= 1.2 m

Radius, R=3.8 mm=3.8*$10^{-3}$

mass, m = 22 kg

Extension, ΔL = 2.5 mm

(a)

Forces  P= mg =22*9.8 =215.6 N

               Stress  = P/A

                            = 215.6/[($\pi$(r)²]

                Stress = ( 215.6 ) / [(3.14* (3.18*$10^{-3}$)²]

                Stress= 4.755 * $10^{6}$  N/m²

    Strain in rod  =  ΔL/L = (2.5 *$10^{-3}$)/(1.2)

                           =(2.08 *$10^{-3}$)

     Now, Toung midulus  =Y= Stress/ Strain

                                          =(4.755 * $10^{6}$ )/(2.08 *$10^{-3}$)

                                          =(2.286 *$10^{9}$)  N/m

(b)

                     Density of metal  = 12 g/$cm^{3}$

                                                   = 12*10³ kg/m

   Speed of sound in the metal=

v=$\sqrt{\frac{Y}{density} }  = \sqrt{\frac{2.286*10^{9} }{12*10^{3} } }$

                               V=436.46 m/s