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

The correct option is A $9:1$

In case of vibrating string frequency is given by

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

Where $\mu =massperunitlength$

Case 1.

$f=200,length=L,tension=T_1\therefore 200=\frac{1}{2L}\sqrt{\frac{T_1}{\mu }}$

Case 2.

$f=300Hz,length=2L,tension=T_2\therefore 300=\frac{1}{2\left(2L\right)}\sqrt{\frac{T_2}{\mu }}$

take ratio we get

$\frac{200}{300}=\frac{2}{1}\times \sqrt{\frac{T_1}{T_2}}\therefore \sqrt{\frac{T_1}{T_2}}=\frac{1}{3}\therefore \frac{T_1}{T_2}=\frac{1}{9}or\frac{T_2}{T_1}=\frac{9}{1}$.