The frequency of vibrations of a piano string varies directly as the square root of the tension on the string and inversely as the length of the string.The middle A string has a frequency of 430 vibrations per second.Find the frequency of a string that has 1.25 times as much tension and is 1.4 times as long.
A)373.4 vibrations / sec
B)343.4 vibrations / sec
C)353.4 vibrations / sec
D)383.4 vibrations / sec
E)363.4 vibrations / sec

Question

The frequency of vibrations of a piano string varies directly as the square root of the tension on the string and inversely as the length of the string.The middle A string has a frequency of 430 vibrations per second.Find the frequency of a string that has 1.25 times as much tension and is 1.4 times as long. ​
A)373.4 vibrations / sec
B)343.4 vibrations / sec
C)353.4 vibrations / sec
D)383.4 vibrations / sec
E)363.4 vibrations / sec

Quizplus · Accepted Answer

Let F be the frequency of the new string, T be the tension on the new string, and L be the length of the new string. Then we have:

F ∝ √T/L

We also know that:

430 = k√T/L

where k is a constant of proportionality. Solving for k, we get:

k = 430√L/T

Now we can use this value of k to find the frequency of the new string:

F = k√(1.25T)/(1.4L)
  = 430√L/T √(1.25T)/(1.4L)
  = 430/√1.4 * √(1.25)
  = 430/1.1832
  ≈ 343.4 vibrations/sec

Therefore, the answer is B.

The Frequency of Vibrations of a Piano String Varies Directly