Question

What is the percentage change in the tension necessary in a sonometer of fixed length to produce a note one octave lower (half of original frequency) than before?

• A. $25\%$
• B. $50\%$
• C. $67\%$
• D. $75\%$

Answer 1

The correct option is D $75\%$

Velocity $v$ is proportional to square root of tension in the sonometer wire.
$v\alpha \sqrt{T}$
$\frac{v_1}{v_2}=\sqrt{\frac{T_1}{T_2}}$
$f\alpha v$
$\therefore \frac{v_1}{v_2}=2=\sqrt{\frac{T_1}{T_2}}$
$\Rightarrow T_2=\frac{T_1}{4}$
$\therefore \frac{T_1-T_2}{T_1}\times 100=\frac{T_1-\left(\frac{T_1}{4}\right)}{T_1}\times 100=75\%$