Question

assuming that the frequency r of a vibrating string may depend upon applied force length mass per unit length prove that r is directly proportional to 1/length root of force is divided by mass​

Answer 1

$\huge{\underline{\underline{\boxed{\sf{\green{Answer :- }}}}}}$

Dimension of frequency=[T−1]

Dimension of applied force,F=[MLT−2]

dimension of length,l=[L]

dimension of mass per unit length,m=[ML−1]

Now suppose,frequency =F^x L^y m^z

so,[T−1]= [MLT−2]^×[L]^y [ML−1]^z

[T−1]=[M]^(X+Z)[L]^(X+Y-Z)[T]^(-2X)

Compare both sides,

X+Z=0

X+Y-Z=0

-2X=-1 ⇒X=1/2

 $\small \underline {\sf{ So, \ Z \ = \ -1/2 \ and \ Y \ = \ -1}}$

Hence, frequency is directly proportional to 1/l(F/m)

Answer 2

Answer:

Dimension of frequency=[T−1]

Dimension of applied force,F=[MLT−2]

dimension of length,l=[L]

dimension of mass per unit length,m=[ML−1]

Now suppose,frequency =F^x L^y m^z

so,[T−1]= [MLT−2]^×[L]^y [ML−1]^z

[T−1]=[M]^(X+Z)[L]^(X+Y-Z)[T]^(-2X)

Compare both sides,

X+Z=0

X+Y-Z=0

-2X=-1 ⇒X=1/2

\small \underline {\sf{ So, \ Z \ = \ -1/2 \ and \ Y \ = \ -1}}

So, Z = −1/2 and Y = −1

Hence, frequency is directly proportional to 1/l(F/m)