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 30 cm, while for a cello it is 36.6 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.
ANY! help would be appreciated.
Check your 2-3-11,8:58pm post.
To determine the increase in tension required in the cello string compared to the tension in the violin string, we need to understand the relationship between string tension, string length, and fundamental frequency.
The frequency of a vibrating string is inversely proportional to its effective length and directly proportional to the square root of its tension. Mathematically, we can represent this relationship as:
F ∝ 1 / L (1),
F ∝ √T (2),
where F represents the fundamental frequency, L represents the effective length of the string, and T represents the tension in the string.
Since we want to produce the same fundamental frequency on the cello as the violin, we can set the two formulas equal to each other:
1 / Lviolin = √Tviolin / Lcello (3).
Now we can solve equation (3) for Tcello, the tension in the cello string. Rearranging the equation, we get:
Tcello = (√Tviolin) * (Lcello / Lviolin) (4).
We are given that Lviolin = 30 cm and Lcello = 36.6 cm.
To calculate the tension increase as a percentage, we need to find the ratio of Tcello to Tviolin, and then express it as a percentage increase.
Let's substitute the known values into equation (4):
Tcello = (√Tviolin) * (36.6 cm / 30 cm)
Now, we can calculate the tension increase by dividing Tcello by Tviolin:
Increase = (Tcello / Tviolin) - 1
Expressing this as a percentage, we multiply the increase by 100:
Percentage increase = Increase * 100.
Finally, we can plug in the known values and calculate the result.
Let's assume the tension in the violin string is Tviolin. We'll call the tension increase as X, so the tension in the cello string would be (Tviolin + X).
Now, we can plug in the values:
(Tviolin + X) / Tviolin = (√Tviolin) * (36.6 cm / 30 cm)
(Tviolin + X) / Tviolin = (√Tviolin) * (1.22)
Simplifying further:
1 + (X / Tviolin) = (√Tviolin) * (1.22)
Rewrite the equation for X:
(X / Tviolin) = (√Tviolin) * (1.22) - 1
(X / Tviolin) = (√Tviolin) * (0.22)
(X / Tviolin) ≈ 0.468√Tviolin
To get the tension increase as a percentage, we multiply both sides by 100:
(100 X / Tviolin) ≈ 46.8√Tviolin
Since the value of Tviolin is not given, we can't determine the exact percentage increase without additional information. However, using this formula, you can calculate the percentage increase once you know the tension in the violin string (Tviolin).