CBSE Class 11-science Answered
frequency of vibration (n) of a stretched string depends on its length (l) , tension (t) and linear mass density taking proportionality constant k=½ find a relation between them
Asked by ss4172380 | 14 Nov, 2023, 08:05: AM
Let Frequency n
la , where l is length of string and a is number .
![begin mathsize 14px style proportional to end style](https://images.topperlearning.com/topper/tinymce/cache/a1d297e56ce9d5553b36ea9cc6dd4809.png)
Let Frequency n
Tb , where T is tension force and b is number .
![begin mathsize 14px style proportional to end style](https://images.topperlearning.com/topper/tinymce/cache/a1d297e56ce9d5553b36ea9cc6dd4809.png)
Let Frequency n
mc , where m is linear mass density and c is number .
![begin mathsize 14px style proportional to end style](https://images.topperlearning.com/topper/tinymce/cache/a1d297e56ce9d5553b36ea9cc6dd4809.png)
Hence we can write Frequency n as
n = k × l a × T b × m c .................... (1)
where k = 1/2 is proportionality constant .
Let M , L , T be the dimension symbol for mass, length and time respectively
Dimension of frequency = T-1
Dimension of length = M
Dimension of tension force = M L T-1/2
Dimension of linear mass density = mass / length = M L-1
Hence , if we apply dimensions to eqn.(1) , we get
![begin mathsize 14px style T to the power of negative 1 end exponent space equals space left square bracket space L space right square bracket to the power of a space space left square bracket space M space L space T to the power of negative 2 end exponent right square bracket to the power of b space left square bracket space M space L to the power of negative 1 end exponent space right square bracket to the power of c end style](https://images.topperlearning.com/topper/tinymce/cache/29afd0cbdec0da7eb675c1b299a014b8.png)
By equating the respective exponents on both sides, we get
a + b - c = 0
b + c = 0
-2b = -1
By solving above equations , we get , a = -1 , b = 1/2 , c = -1/2
Hence , we rewrite eqn.(1) as
![begin mathsize 14px style n space equals space 1 half cross times space l to the power of negative 1 end exponent space cross times T to the power of 1 divided by 2 end exponent cross times space m to the power of negative 1 divided by 2 end exponent end style](https://images.topperlearning.com/topper/tinymce/cache/5d14aae2c413aa361285d517f65b4f36.png)
![begin mathsize 14px style n space equals space fraction numerator 1 over denominator 2 space l end fraction square root of T over m end root end style](https://images.topperlearning.com/topper/tinymce/cache/8697631389605d4032241b0c0924cea1.png)
Answered by Thiyagarajan K | 14 Nov, 2023, 09:18: AM
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