Traverse E wave travel Time on wires

[Husker70]

Homework Statement

A 30.0m steel wire and a 20.0m copper wire, both with 1.00mm diameter,
are connected end to end and stretched to a Tension of 150N. How
long does it take a transverse e wave to travel the entire length of
the two wires?

Homework Equations

Voulume of Rod = pie r^2(L)
v = sqrrt T/mass per unit length
v = dt/dx

The Attempt at a Solution

I found the volume of the steel rod to be pie(.0001m)^2(30.0m) = 9.42x10^-7 m^3
I found the volume of the copper rod to be pie(.0001m)^2(20.0m) = 6.42x10^-7 m^3

I looked up the density of steel to be 7850 kg/m^3
I looked up the density of copper to be 8230 kg/m^3

The weight of steel rod should be 9.42x10^-7 m^3/7850 kg/m^3 = 1.2 x 10^-10 kg
Then divide by 30.0m = 4.0x10^-10 kg/m

The weight of copper rod should be 6.28 x 10^-7 m^3/8230 kg/m^3 = 7.63 x 10^-11 kg
Then divide by 20.0m = 3.82 x 10^-12 kg/m

Using V = sqrrt T/mass unit of length

Steel wire is sqrrt 150N/4.0 x 10^-12 kg/m = velocity of steel = 6.12 x 10^6 m/s
Copper wire is sqrrt 150N/3.82 x 10^-12 kg/m = velocity of copper = 3.93 x 10^13 m/s

I added them together amd divided by 2 to get an average of 1.97 x 10^13 m/s
across the whole wire. Then divided that number by 50.0m to get 3.93 x 10^11s

I know that isn't right as the answer is .329s
I'm not sure what I'm doing wrong. Any help would be appreciated.
Thanks,
Kevin

 

[Redbelly98]

That "shortcut" of averaging the two velocities is a common misconception, but wrong. Even if the wires were the same length!

How long does it take the wave to travel down the steel part?
How long does it take the wave to travel down the copper part?

 

[thomate1]

Hello Kevin,

Thank you for sharing your attempt at solving this problem. It seems like you have the right equations and are on the right track, but there are a few errors in your calculations that are leading to an incorrect answer.

Firstly, when calculating the weight of the steel and copper rods, you should be using the volume of the wire (given by the formula you listed) and not the length of the wire. This will give you the correct mass for each wire, which is needed in the equation for velocity (v = sqrt(T/mass per unit length)).

Secondly, when calculating the velocity for each wire, you are using the incorrect values for the mass per unit length. Instead of dividing by the length of the wire, you should be dividing by the volume of the wire (which you have now corrected). This will give you the correct velocities for each wire.

Lastly, when finding the average velocity, you should be using the harmonic mean formula (1/v_avg = (1/v1 + 1/v2)/2) instead of simply averaging the velocities. This will give you the correct average velocity for the entire wire.

By making these corrections, you should be able to arrive at the correct answer of 0.329s. It is always helpful to double check your calculations and make sure you are using the correct values and equations. Keep up the good work!