On a string instrument of the violin family, the effective length of a string is the distance between the bridge and the nut. For a violin, this distance is 29.3 cm, while for a cello it is 37.8 cm.
The string of a violin is placed in a cello with the intention of producing a sound of the same fundamental frequency.
To accomplish this, the string on the cello will be under a larger tension than on the violin.
By how much should the tension in the cello be increased with respect to the tension in the violin?
Express the result as a percentage, and to two significant figures. Only answer in numerical values, without the % sign.
For example, an increase of 11% corresponds to Tcello = (1.11) Tviolin, and should be entered as 11 in the answer box.
You want the answer in a box? Without the % sign?
Do you want fries with that?
If you want to learn the subject and not just fill in the blanks to get some meaningless degree, use the fact that the frequency is proportional to
(wave speed)/(string length)
To keep the frequency the same, the wave speed must increase by a factor 37.8/29.3 = 1.2901
The string lineal density remains the same, since it is the same string. Take a look at the formula for wave speed in a string under tension. If you don't know it, look it up.
It says that you have to increase Tension so than sqrt(tension) is increased by a factor 1.2901
Take it from there
To find the increase in tension in the cello compared to the tension in the violin, we need to understand the relationship between tension, frequency, and effective length of a string.
The fundamental frequency of a vibrating string is inversely proportional to the effective length of the string and directly proportional to the square root of the tension.
Mathematically, this relationship can be expressed as:
f ∝ 1 / √L
Where f is the frequency and L is the effective length.
Since we want to produce the same fundamental frequency on both instruments, we can set up the following equation:
fviolin = fcello
Using the relationship above, we can rewrite this equation by substituting the variables:
(1 / √Lviolin)√Tviolin = (1 / √Lcello)√Tcello
We are given the effective lengths of the strings, Lviolin = 29.3 cm and Lcello = 37.8 cm. Let's substitute these values:
(1 / √29.3)√Tviolin = (1 / √37.8)√Tcello
To find the increase in tension in the cello compared to the violin, we need to solve for Tcello/Tviolin:
(√Tcello / √29.3) = (√Tviolin / √37.8)
Simplifying further:
Tcello / Tviolin = (√37.8 / √29.3)
Tcello / Tviolin ≈ 1.102 (rounded to three decimal places)
To express this as a percentage, we subtract 1 and multiply by 100:
(Tcello / Tviolin - 1) * 100 = (1.102 - 1) * 100 ≈ 10.2
Therefore, the tension in the cello should be increased with respect to the tension in the violin by approximately 10.2%.