A viola string with a fundamental frequency of D4 (293 Hz) is generally tuned using a tension of 49.0 N. However, just before a concert, the string breaks and only a G3 string (196 Hz) is available. The strings are the same length and are usually tuned with the same tension. What does the tension need to be on the replacement string to bring it up to the D4 frequency

Explanation:

For fundamental frequency in a vibrating string , the formula is

n = 1 / 2L  x  √ ( T /m₁ )

n is frequency , L is length , T is tension and m₁ is mass per unit length .

For first string ,

293 =  1 / 2L  x  √ ( 49 N  /m₁ )

For second string , let mass per unit length be m₂ .

196 =  1 / 2L  x  √ ( 49 N  /m₂ ) ( 1 )

To bring its frequency back to previous one let tension be T

293  =  1 / 2L  x  √ ( T  /m₂ ) ( 2 )

Dividing

293 / 196 = √ ( T  /49  )

1.4948 = √ ( T  /49  )

2.2344 = T  /49

T = 109.48 N .

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resonant frequency