Velocity, Period, and Transverse Wave

[Soaring Crane]

Homework Statement

Three pieces of string, each of length L, are joined together end-to-end, to make a combined string of length . The first piece of string has mass per unit length mu_1, the second piece has mass per unit length mu_1*4.00 , and the third piece has mass per unit length mu_1/4.00. If the combined string is under tension F, how much time does it take a transverse wave to travel the entire length 3L? Give your answer in terms of F, mu_1, and L.

Homework Equations

v = sqrt(F/mu)
T = position/velocity = 3L/v ??

The Attempt at a Solution

The answer is (7L/2)*sqrt(mu_1/F).

I understand the sqrt portion, but I am having difficulty with the multiplicative factor.

I thought the velocity would be:

v = sqrt(F/mu_total), where mu_total = mu_1 + 4*mu_1 + .25*mu_1 = 5.25*mu_1

T = 3L/sqrt(F/5.25*mu_1) = 3L*sqrt(5.25)*sqrt(mu_1/F)

How do I get the correct factor?

Thanks.

 

[OlderDan]

The densities do not add. You need to find an expression for the time it takes for the wave to traverse each part of the string and add those to find the total time.

 

[m2003]

I would first commend the student for their attempt at solving the problem and for asking for assistance when they encountered difficulty. I would also point out that their equations and approach are generally correct.

To get the correct factor, the student needs to consider the mass per unit length of each piece of string. The first piece has a mass per unit length of mu_1, the second has a mass per unit length of 4*mu_1, and the third has a mass per unit length of mu_1/4. When these three pieces are joined together, the total mass per unit length of the combined string is:

mu_total = mu_1 + 4*mu_1 + mu_1/4 = 5.25*mu_1

This means that the velocity of the wave can be calculated as:

v = sqrt(F/mu_total) = sqrt(F/5.25*mu_1)

To find the time it takes for the wave to travel the entire length 3L, we can use the equation T = position/velocity. In this case, the position is 3L and the velocity is sqrt(F/5.25*mu_1), so the time is:

T = 3L/sqrt(F/5.25*mu_1) = (3L*sqrt(5.25))/sqrt(F/mu_1) = (3L*sqrt(5.25))*sqrt(mu_1/F)

Thus, the correct factor is (3L*sqrt(5.25)), which gives us the final answer of:

T = (3L*sqrt(5.25))*sqrt(mu_1/F) = (7L/2)*sqrt(mu_1/F)

In summary, the correct factor takes into account the mass per unit length of each piece of string, which affects the velocity of the wave and ultimately the time it takes for the wave to travel the entire length 3L.