A string of length ' l ' fixed at both ends vibrates in resonance with a tuning fork of frequency f ' at two successive values of tension T1 and T2 in the string. Find the specific mass (mass per unit length) of the string :

Question

A string of length ' l ' fixed at both ends vibrates in resonance with a tuning fork of frequency f ' at two successive values of tension T1 and T2 in the string. Find the specific mass (mass per unit length) of the string : (A) (T1 T2/f2 l2(√T1-√T2)2) (B) (2 T1 T2/f2 l2(√T1-√T2)2) (C) (T1 T2/2 f2 l2(√T1-√T2)2) (D) None of these

Tardigrade Admin · Accepted Answer

We use f=(n/2 l) √(T1/μ) ...(1) f=(n +1/2 l) √(T2/μ) ...(2) From equation (1) and (2) eliminate ' n ' to get μ. μ=(T1 T2/4 f2 l2(√T1-√T2)2)

Q. A string of length ' $l$ ' fixed at both ends vibrates in resonance with a tuning fork of frequency $f$ ' at two successive values of tension $T_{1}$ and $T_{2}$ in the string. Find the specific mass (mass per unit length) of the string :

$\frac{T _{1} T _{2}}{f ^{2} l ^{2} ( T _{1} - T _{2} ) ^{2}}$

$\frac{2 T _{1} T _{2}}{f ^{2} l ^{2} ( T _{1} - T _{2} ) ^{2}}$

$\frac{T _{1} T _{2}}{2 f ^{2} l ^{2} ( T _{1} - T _{2} ) ^{2}}$

None of these

We use
$f = \frac{n}{2 l} \frac{T _{1}}{μ}$ ...(1)
$f = \frac{n + 1}{2 l} \frac{T _{2}}{μ}$ ...(2)
From equation (1) and (2) eliminate ' $n$ ' to get $μ$ .
$μ = \frac{T _{1} T _{2}}{4 f ^{2} l ^{2} ( T _{1} - T _{2} ) ^{2}}$

Q. A string of length ' l ' fixed at both ends vibrates in resonance with a tuning fork of frequency f ' at two successive values of tension T1​ and T2​ in the string. Find the specific mass (mass per unit length) of the string :

f2l2(T1​​−T2​​)2T1​T2​​

f2l2(T1​​−T2​​)22T1​T2​​

2f2l2(T1​​−T2​​)2T1​T2​​