Question

A steel wire of length 31.0 m and a copper wire of length 17.0 m, both with 1.00-mm diameters, are connected end to end and stretched to a tension of 122 N. During what time interval will a transverse wave travel the entire length of the two wires

Answer 1

The time taken is
`t = 0.356 \ s`

Step-by-step explanation:

From the question we are told that

The length of steel the wire is
`l_1 = 31.0 \ m`

The length of the copper wire is
`l_2 = 17.0 \ m`

The diameter of the wire is
`d = 1.00 \ m = 1.0 *10^(-3) \ m`

The tension is
`T = 122 \ N`

The time taken by the transverse wave to travel the length of the two wire is mathematically represented as

`t = t_s + t_c`

Where
`t_s` is the time taken to transverse the steel wire which is mathematically represented as

`t_s = l_1 * [ \sqrt{ (\rho * \pi * d^2 )/( 4 * T) } ]`

here
`\rho_s` is the density of steel with a value
`\rho_s = 8920 \ kg/m^3`

So

`t_s = 31 * [ \sqrt{ (8920 * 3.142* (1*10^(-3))^2 )/( 4 * 122) } ]`

`t_s = 0.235 \ s`

And

`t_c` is the time taken to transverse the copper wire which is mathematically represented as

`t_c = l_2 * [ \sqrt{ (\rho_c * \pi * d^2 )/( 4 * T) } ]`

here
`\rho_c` is the density of steel with a value
`\rho_s = 7860 \ kg/m^3`

So

`t_c = 17 * [ \sqrt{ (7860 * 3.142* (1*10^(-3))^2 )/( 4 * 122) } ]`

`t_c =0.121`

So

`t = t_c + t_s`

`t = 0.121 + 0.235`

`t = 0.356 \ s`