What is the Frequency of the Third Harmonic in a Heated, Stretched Yarn?

[utkarshakash]

Homework Statement

A yarn of material that cannot dilate, length L, mass m and elastic constant K is trapped and stretched with negligible tension between the two supports A and B attached to the ends of the metal bar, CD, whose coefficient of expansion varies linearly from to , increasingly with temperature in the range of interest of the question. Determine the frequency of the third harmonic that is established in the rope when heated ΔT.

The Attempt at a Solution

$\alpha _{eq} = \dfrac{\alpha 1 + \alpha 2}{2}$

Since the metal bar expands, separation between A and B increases. This creates a tension in the string. The change in length is given by LαΔT.
F = KLαΔT
Frequency of third harmonic = 4v/2L
where $v=\sqrt{\dfrac{FL}{m}}$

If I substitute the value of F, the answer comes out to be wrong.

 

[utkarshakash]

Anyone?

 

[Saitama]

...linearly from to , increasingly ...

Something seems to be missing here.

The Attempt at a Solution

$\alpha _{eq} = \dfrac{\alpha 1 + \alpha 2}{2}$

[STRIKE]I am not sure but I think this is incorrect. The question says that coefficient of linear expansion varies linearly with temperature so I think you should find it as a function of temperature and then obtain the change in length through integration.[/STRIKE]

EDIT: Sorry, that is correct, integration yields the same result. So the only possible error is in your formula for frequency of third harmonic.

 

[utkarshakash]

Something seems to be missing here.


[STRIKE]I am not sure but I think this is incorrect. The question says that coefficient of linear expansion varies linearly with temperature so I think you should find it as a function of temperature and then obtain the change in length through integration.[/STRIKE]

EDIT: Sorry, that is correct, integration yields the same result. So the only possible error is in your formula for frequency of third harmonic.

Ah! That was a silly mistake. I confused "harmonics" with "overtones". Thanks for pointing out.

 

[utkarshakash]

Here's the correct answer

$\dfrac{3}{2} \sqrt{\dfrac{K Δ T (\alpha_1 + \alpha_2)}{2m}}$