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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
answered-by-expert Expert Answer
Let Frequency n begin mathsize 14px style proportional to end style la , where l is length of string and a is number .
 Let Frequency n begin mathsize 14px style proportional to end style Tb , where T is tension force and b is number .  
Let Frequency n begin mathsize 14px style proportional to end style mc , where m is linear mass density and c is number .  
 
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
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
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
 



 
 
 
 
 
 
 
 
 
 

Answered by Thiyagarajan K | 14 Nov, 2023, 09:18: AM

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