Standing Wave Period for Third Harmonic Oscillation

[i_hate_math]

Homework Statement

A rope, under a tension of 209 N and fixed at both ends, oscillates in a second-harmonic standing wave pattern. The displacement of the rope is given by
y=0.49sin(πx/3)sin(12πt)
What are (a) the length of the rope, (b) the speed of the waves on the rope, and (c)the mass of the rope? (d) If the rope oscillates in a third-harmonic standing wave pattern, what will be the period of oscillation?

Homework Equations

T=2π/w

The Attempt at a Solution

I have solved part a,b,c but I got part d wrong. I simply used T=2π/w
==> T=2π/(12π)=0.1666666...=0.167s
And this is not the correct answer.
Please point out where I did wrong

 

[haruspex]

Homework Statement

A rope, under a tension of 209 N and fixed at both ends, oscillates in a second-harmonic standing wave pattern. The displacement of the rope is given by
y=0.49sin(πx/3)sin(12πt)
What are (a) the length of the rope, (b) the speed of the waves on the rope, and (c)the mass of the rope? (d) If the rope oscillates in a third-harmonic standing wave pattern, what will be the period of oscillation?

Homework Equations

T=2π/w

The Attempt at a Solution

I have solved part a,b,c but I got part d wrong. I simply used T=2π/w
==> T=2π/(12π)=0.1666666...=0.167s
And this is not the correct answer.
Please point out where I did wrong

The given 12π is for the second harmonic. Part d asks about the third harmonic.

 

[i_hate_math]

The given 12π is for the second harmonic. Part d asks about the third harmonic.

I see, but is it okay to assume that the length of the rope remains the same?

 

[haruspex]

I see, but is it okay to assume that the length of the rope remains the same?

Yes.

 

[i_hate_math]

Yes.

With the same L, I got speed v=sqrt(T•L/m) where T is tension, so v remains the same too. the wavelength is changed since its now in third harmonic pattern, L=6=(3/2)λ.
==> λ=4
==> λ•f=v and from part b, v=36 is the correct answer
==> v=1/9=0.111...

 

[haruspex]

With the same L, I got speed v=sqrt(T•L/m) where T is tension, so v remains the same too. the wavelength is changed since its now in third harmonic pattern, L=6=(3/2)λ.
==> λ=4
==> λ•f=v and from part b, v=36 is the correct answer
==> v=1/9=0.111...

Did you mean T=0.111?

 

[i_hate_math]

Did you mean T=0.111?

Yes. Sorry about the typo

 

[haruspex]

Yes. Sorry about the typo

Looks right.

 

[i_hate_math]

Looks right.

I checked with the answers, it is!