Question

A string vibrates with a frequency of $200Hz$. When its length is double 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)

$9:1$

$f=\frac{1}{2L}\sqrt{\frac{T}{\mu }}$

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

$\Rightarrow \frac{T_2}{T_1}=\frac{(f_2{L}_2{)}^2}{(f_1{L}_1{)}^2}$

$L_2=2L_1,f_1=200Hz,f_2=300Hz$

Substituting above values we get $\frac{T_2}{T_1}=\frac{9}{1}$