# Frequency of third harmonic

1. Apr 29, 2014

### utkarshakash

1. The problem statement, all variables and given/known data
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.

3. 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.

2. Apr 30, 2014

### utkarshakash

Anyone?

3. Apr 30, 2014

### Pranav-Arora

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.

Last edited: Apr 30, 2014
4. Apr 30, 2014

### utkarshakash

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

5. Apr 30, 2014

### utkarshakash

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