Answer:
The time taken by the wave to travel along the combination of two wires is 458 ms.
Explanation:
Given that,
Length of steel wire= 27.0 m
Length of copper wire = 48.0 m
Tension = 145 N
Radius of both wires = 0.450 mm
Density of steel wire ![\rho_{s}= 7.86\times10^{3}\ kg/m^{3}](https://tex.z-dn.net/?f=%5Crho_%7Bs%7D%3D%207.86%5Ctimes10%5E%7B3%7D%5C%20kg%2Fm%5E%7B3%7D)
Density of copper wire ![\rho_{c}=8.92\times10^{3}\ kg/m^3](https://tex.z-dn.net/?f=%5Crho_%7Bc%7D%3D8.92%5Ctimes10%5E%7B3%7D%5C%20kg%2Fm%5E3)
We need to calculate the linear density of steel wire
Using formula of linear density
![\mu_{s}=\rho_{s}A](https://tex.z-dn.net/?f=%5Cmu_%7Bs%7D%3D%5Crho_%7Bs%7DA)
![\mu_{s}=\rho_{s}\times\pi r^2](https://tex.z-dn.net/?f=%5Cmu_%7Bs%7D%3D%5Crho_%7Bs%7D%5Ctimes%5Cpi%20r%5E2)
Put the value into the formula
![\mu_{s}=7.86\times10^{3}\times\pi\times(0.450\times10^{-3})^2](https://tex.z-dn.net/?f=%5Cmu_%7Bs%7D%3D7.86%5Ctimes10%5E%7B3%7D%5Ctimes%5Cpi%5Ctimes%280.450%5Ctimes10%5E%7B-3%7D%29%5E2)
![\mu_{s}=5.00\times10^{-3}\ kg/m](https://tex.z-dn.net/?f=%5Cmu_%7Bs%7D%3D5.00%5Ctimes10%5E%7B-3%7D%5C%20kg%2Fm)
We need to calculate the linear density of copper wire
Using formula of linear density
![\mu_{c}=\rho_{s}A](https://tex.z-dn.net/?f=%5Cmu_%7Bc%7D%3D%5Crho_%7Bs%7DA)
![\mu_{c}=\rho_{s}\times\pi r^2](https://tex.z-dn.net/?f=%5Cmu_%7Bc%7D%3D%5Crho_%7Bs%7D%5Ctimes%5Cpi%20r%5E2)
Put the value into the formula
![\mu_{c}=8.92\times10^{3}\times\pi\times(0.450\times10^{-3})^2](https://tex.z-dn.net/?f=%5Cmu_%7Bc%7D%3D8.92%5Ctimes10%5E%7B3%7D%5Ctimes%5Cpi%5Ctimes%280.450%5Ctimes10%5E%7B-3%7D%29%5E2)
![\mu_{c}=5.67\times10^{-3}\ kg/m](https://tex.z-dn.net/?f=%5Cmu_%7Bc%7D%3D5.67%5Ctimes10%5E%7B-3%7D%5C%20kg%2Fm)
We need to calculate the velocity of the wave along the steel wire
Using formula of velocity
![v_{s}=\sqrt{\dfrac{T}{\mu_{s}}}](https://tex.z-dn.net/?f=v_%7Bs%7D%3D%5Csqrt%7B%5Cdfrac%7BT%7D%7B%5Cmu_%7Bs%7D%7D%7D)
![v_{s}=\sqrt{\dfrac{145}{5.00\times10^{-3}}}](https://tex.z-dn.net/?f=v_%7Bs%7D%3D%5Csqrt%7B%5Cdfrac%7B145%7D%7B5.00%5Ctimes10%5E%7B-3%7D%7D%7D)
![v_{s}=170.3\ m/s](https://tex.z-dn.net/?f=v_%7Bs%7D%3D170.3%5C%20m%2Fs)
We need to calculate the velocity of the wave along the steel wire
Using formula of velocity
![v_{c}=\sqrt{\dfrac{T}{\mu_{c}}}](https://tex.z-dn.net/?f=v_%7Bc%7D%3D%5Csqrt%7B%5Cdfrac%7BT%7D%7B%5Cmu_%7Bc%7D%7D%7D)
![v_{c}=\sqrt{\dfrac{145}{5.67\times10^{-3}}}](https://tex.z-dn.net/?f=v_%7Bc%7D%3D%5Csqrt%7B%5Cdfrac%7B145%7D%7B5.67%5Ctimes10%5E%7B-3%7D%7D%7D)
![v_{c}=159.9\ m/s](https://tex.z-dn.net/?f=v_%7Bc%7D%3D159.9%5C%20m%2Fs)
We need to calculate the time taken by the wave to travel along the combination of two wires
![t=t_{s}+t_{c}](https://tex.z-dn.net/?f=t%3Dt_%7Bs%7D%2Bt_%7Bc%7D)
![t=\dfrac{l_{s}}{v_{s}}+\dfrac{l_{c}}{v_{c}}](https://tex.z-dn.net/?f=t%3D%5Cdfrac%7Bl_%7Bs%7D%7D%7Bv_%7Bs%7D%7D%2B%5Cdfrac%7Bl_%7Bc%7D%7D%7Bv_%7Bc%7D%7D)
Put the value into the formula
![t=\dfrac{27.0}{170.3}+\dfrac{48.0}{159.9}](https://tex.z-dn.net/?f=t%3D%5Cdfrac%7B27.0%7D%7B170.3%7D%2B%5Cdfrac%7B48.0%7D%7B159.9%7D)
![t=0.458\ sec](https://tex.z-dn.net/?f=t%3D0.458%5C%20sec)
![t=458\ ms](https://tex.z-dn.net/?f=t%3D458%5C%20ms)
Hence, The time taken by the wave to travel along the combination of two wires is 458 ms.