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The frequency of vibration of string dep...

The frequency of vibration of string depends on the length `L` between the nodes, the tension `F` in the string and its mass per unit length `m`. Guess the expression for its frequency from dimensional analysis.

A

`2lsqrt((F)/(m))`

B

`(1)/(l)sqrt((m)/(F))`

C

`(1)/(2l)sqrt((m)/(F))`

D

`(1)/(2l)sqrt((F)/(m))`

Text Solution

Verified by Experts

The correct Answer is:
D
Let, `vpropF^(a)" "......(i)`
`propl^(b)" "......(ii)`
`propm^(c)" "......(iii)`
Combining the equations (i), (ii) and (iii) we get
`v=kF^(a)l^(b)m^(c)" ".......(iv)`
where k is a dimensionless constant of proportionally. In the equation (iv), substituting the dimensions of v, F, l and m, we get,
`[T^(-1)]=[MLT^(-2)]^(a)[L]^(b)[ML^(-1)]^(c)`
or `[M^(0)L^(0)T^(-1)]=[M^(a+c)L^(a+b-c)T^(-2a)]`
Comparing the dimensions of M, L and T on the two sides, we get
`-2a=-1ora=1//2`
`a+c=0orc=-1//2`
and `a+b-c=0orb=c-a=-1//2-1//2=-1`
In the equation (iv), putting the values of a, b and c we obtain
`v=kF^(1//2)l^(-1)m^(-1//2)orv=(k)/(l)sqrt((F)/(m))`
The constant of proportionally k is found to be equal to 1/2.
Therefore the above relation becomes
`:.v=(1)/(2l)sqrt((F)/(m))`
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