Musical Frequencies Overtones, ratios

[tigerwoods99]

Homework Statement

Two strings on a musical instrument are tuned to play at 262 Hz (C) and 294 Hz (D).

Questions:
1. If the two strings have the same length and are under the same tension, what must be the ratio of their masses (MC/MD)?
2. If the strings, instead, have the same mass per unit length and are under the same tension, what is the ratio of their lengths (LC/LD)?
3. If their masses and lengths are the same, what must be the ratio of the tensions in the two strings? (TC/TD)

 

[Stonebridge]

These questions all require one formula, the one that expresses the frequency of vibration of the string in terms of the tension, length and mass per unit length.
Do you have this formula in your book or lecture notes?

 

[tigerwoods99]

V = Square root (Ft * L)/m

 

[Gigasoft]

Well, then you write down the equation for case 1 (262 Hz) and case 2 (294Hz). Then you solve the equations. Hopefully, you know how to solve first degree equations.

It should be noted that the frequencies given are musically inaccurate. 262 Hz differs from C by about 2.48 cents, and 294 Hz differs from D by about 1.98 cents.

 

[Stonebridge]

V = Square root (Ft * L)/m

This gives the speed of the wave on the string, and is usually written

v = √(T/μ) where T is tension in Newton, and μ is mass per unit length of string

You now need to say how this speed, v, is related to the frequency of the wave in the string.
You should also have a formula relating speed, frequency and wavelength for a wave.

Just remember that for the string wave, the wavelength is two times the length of the string. Substitute for v in the other equation and arrange it so that you have f= (instead of v=)

 

[tigerwoods99]

so : (Frequency)(2L) = square root (FT * L)/(m)

{(frequency)(2L)} ^2 = (Ft *L)/(m)
and then what do i do? for the first one and the rest of them?

thanks!

 

[Stonebridge]

so : (Frequency)(2L) = square root (FT * L)/(m)

{(frequency)(2L)} ^2 = (Ft *L)/(m)
and then what do i do? for the first one and the rest of them?

thanks!

Write the equation with just f on the left and put the frequency values in, then you will have two equations

262=
294=

To find the ratios the question wants, divide the one equation by the other
262/294 =
noting which quantities are the same in the two equations and cancelling them out when possible.