According to a table in Kinsler and Frey, the maximum tensile strength of steel

Question

According to a table in Kinsler and Frey, the maximum tensile strength of steel is 10 9 Pa (for f per unit area, lPa=lN/m 2), and the volume density is 7700 kg/m 3. (NOTE: The maximum tensile strength is the force applied to the wire at the moment it breaks divided by its cross sectional area of the wire, so it has units of pressure.) A.) If we make strings from this material and tension them to a given fraction of their maximum tensile strength, you should be able to show that the phase speed of a disturbance on the string is independent of the radius of the wire. B.) Derive an expression for the length of a steel wire needed to produce a given frequency when tensioned to half of its maximum tensile strength, then |c.) Calculate the range of lengths needed f

Explanation / Answer

a)   Here, phase velocity = c/sqrt[ 1 - (wo/w)2]

=> phase speed of the disturbance on the string , is independent of the radius of wire .

b) Expression for length of steel wire to produce a given frequency

=>   f = sqrt(T/k)/2L

where,   k =   mass per unit length of wire .

             T = Tension in wire .

c) Here, smallest length =    sqrt(7700)/(2 * 4186)

                                        = 1.048 cm

       largest length =    sqrt(7700)/(2 * 27.5)

                                        = 159.54 cm

=> Range of length = [ 1.048 cm   to 159.54 cm ]

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