The frequency of stretched string depends upon length of string (l), tension in the string (t), and m per unit length (u) of the string. find the expression of vibration of string using method of dimension

Asked by tejpaljeet4 | 3rd Mar, 2021, 12:13: PM

Expert Answer:

Let frequency of vibration be ν , dimension of frequency of vibration = [ Mo Lo T-1 ]
 
Let length of string be l , dimension of l = [ Mo L To ]
 
Let Tension be T , dimension of Tension ( force ) =  [ M L T-2 ]
 
Let mass per unit length be m , dimension of m  = [ M L-1 ]
 
Let frequency of vibration ν =  lx Ty mz ..................... (1)
 
where x , y and z are to be determined from dimension analysis
 
If we apply dimensions on both side of eqn.(1), we get
 
[ Mo Lo T-1 ] = [ Mo L To ]x [ M L T -2 ]y  [ M L-1 ]z
 
By applying law of indices to the above expression we gte
 
y+z = 0
 
x +y-z = 0
 
 -2y = -1 
 
y = 1/2  ,  z = -1/2  , x = -1
 
 
Frequency of vibration ν is given as
 
begin mathsize 14px style nu space equals space k over l square root of T over m end root end style
where k is numerical constant without any dimension

Answered by Thiyagarajan K | 3rd Mar, 2021, 02:25: PM

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