How Does Wire Diameter Affect Wave Speed?

[freeofwork]

Homework Statement

Two steel wires are stretched with the same tension. The first wire has a diameter of .00059 meters and the second wire has a diameter of .00089m. If the speed of waves traveling along the first wire is 54.0 m/s, what is the speed of the waves along the second wire?

Homework Equations

v=√Ft/μ
μ=m/L

The Attempt at a Solution

I have no idea how to get the mass or length of the string...

 

[LawrenceC]

Hint: You know the diameters so you know the mass per unit length of each.

 

[freeofwork]

Hint: You know the diameters so you know the mass per unit length of each.

what would be the equation for that... i only know μ=m/L...

 

[LawrenceC]

You know the formula for velocity. The tension is the same for both wires. Mu is mass per unit length which you can determine by realizing that a wire is a cylinder. You know the velocity of the wave on one of the wires. Therefore, you can solve this without knowing the respective wire lengths.

 

[freeofwork]

You know the formula for velocity. The tension is the same for both wires. Mu is mass per unit length which you can determine by realizing that a wire is a cylinder. You know the velocity of the wave on one of the wires. Therefore, you can solve this without knowing the respective wire lengths.

imdont have e height though... how can you convert the diameter into something usable...

 

[LawrenceC]

m/l is mass per unit length.

 

[LawrenceC]

What's the formula for the volume of a cylinder?

 

[freeofwork]

What's the formula for the volume of a cylinder?

pi r^2h

 

[LawrenceC]

That's correct or you could also say (pi d^2/4)h. So what would be the mass per unit length?

 

[freeofwork]

radical x over 54^2?
x being the tension?..

 

[LawrenceC]

Do you know what mass per unit length means?

 

[freeofwork]

linear density

 

[LawrenceC]

In the case of a wire which is a cylinder the mass per unit length is:

m/L = rho * Volume/L = rho * pi * d^2/4 * L/L = rho * pi * d^2/4

where rho is the density in kg/meter^3.

You have the formula for the velocity of one wave at a specific m/L. You want the velocity of the wave in the other wire. The tensions are the same.

V1^2 = F/mu1 and V2^2 = F/mu2. Think about (V1/V2)^2.

 

[LawrenceC]

Got to hurry here. Giants-Falcons game starts in 11 minutes!

 

[freeofwork]

how do you find the density or linear density? this is crazy

 

[LawrenceC]

You do not the actual number; you only need the ratio of the two.

V1^2 = F/mu1 and V2^2 = F/mu2. Think about (V1/V2)^2.

 

[freeofwork]

idk...

 

[LawrenceC]

idk...

D
What is idk?

 

[freeofwork]

i don't know.

 

[LawrenceC]

Ok, I shall provide more help. I cannot do it for you -forum rules.

Form a ratio for v1/v2. You know one of the v's. This ratio equals square root of mass per unit length ratios. You can compute this ratio. This leaves one unknown.

 

[freeofwork]

what do u mean by mass per unit ratios?

 

[LawrenceC]

(M1/Ll)/(M2/L)

M's are dependent on diameters. L's drop out.

 

[freeofwork]

o how would u get e mass?
so the equation who look something like this?
v₁/v₂=√m₁m₂

 

[LawrenceC]

You almost have it. Check your algebra. The radical should be m2/m1. Relate the m's to their respective diameters...

 

[freeofwork]

i don't know how the mass are related to the diameters...

 

[LawrenceC]

Volume of a cylinder...

 

[freeofwork]

i don't really get what to plug in there sir.

 

[LawrenceC]

Mass = rho * volume /g

rho is density and g is acceleration of gravity. If you are forming a ratio of mass per unit length of the same materials, everything cancels except the volumes.

The volume per unit length is simply (pi * d^2)/4 which is the cross sectional area of a cylinder. So the ratios of masses per unit length is the ratio of diameters squared.

 

[freeofwork]

what is volume per unit length?

 

[LawrenceC]

I do not think you are serious about this problem.