Question 14: Units , Dimensions and Error Analysis

Question

A sonometer string vibrates with a frequency of 400 Hz . When the length of the string is halved and the tension is altered, it begins to vibrate with a frequency of 200 Hz . The ratio of the new tension to the original tension in the string is:

Accepted Answer

Let $L_1$ and $T_1$ be the original length and tension, and $L_2=	frac12L_1$, $T_2$ the new ones. For a vibrating string,
$ f=\frac1{2L}\sqrt{\frac T\mu}\,, $
so
$ \frac{f_2}{f_1} =\frac{\frac1{2L_2}\sqrt{T_2/\mu}}{\frac1{2L_1}\sqrt{T_1/\mu}} =\frac{L_1}{L_2}\,\sqrt{\frac{T_2}{T_1}} =2\,\sqrt{\frac{T_2}{T_1}}. $
Plug in $f_1=400$Ã¢ÂÂHz and $f_2=200$Ã¢ÂÂHz:
$ \frac{200}{400} =2\sqrt{\frac{T_2}{T_1}} \quad\Longrightarrow\quad \frac12=2\sqrt{\frac{T_2}{T_1}} \quad\Longrightarrow\quad \sqrt{\frac{T_2}{T_1}}=\frac14 \quad\Longrightarrow\quad \frac{T_2}{T_1}=\frac1{16}. $
Answer: $1:16$.

Answer

$4: 1$

Answer

$16: 1$

Answer

$1: 4$

Answer

$1: 16$

Units , Dimensions and Error Analysis

A sonometer string vibrates with a frequency of 400 Hz . When the length of the string is halved and the tension is altered, it begins to vibrate with a frequency of 200 Hz . The ratio of the new tension to the original tension in the string is:

Choose the correct answer: