Question

A string vibrates with a frequency of $200Hz$. When its length is doubled and tension is altered, it begins to vibrate with a frequency of $300Hz$. The ratio of the new tension to the original tension is

• A. 9 : 1
• B. 1 : 9
• C. 3 : 1
• D. 1 : 3

Answer 1

Answer: (A)

$f=\frac{1}{2L}\sqrt{\frac{T}{\mu }}$
where $f$ is frequency, $T$ is Tension, $L=$ length and $\mu$ is constant $\therefore f\propto \frac{\sqrt{T}}{L}$
$\Rightarrow \frac{f_2}{f_1}=\left(\sqrt{\frac{T_2}{T_1}}\right)\left(\frac{L_1}{L_2}\right)$
Squaring on both sides and rearranging we get $\Rightarrow \frac{T_2}{T_1}=\frac{{\left(f_2{L}_2\right)}^2}{{\left(f_1{L}_1\right)}^2}$
$L_2=2L_1,f_1=200Hz,f_2=300Hz$
Substituting above values we get $\frac{T_2}{T_1}=\frac{9}{1}$