Question

A sonometer wire vibrates with a frequency of $500Hz$ when the tension is $T$${}_1$ and with a frequency of $400Hz$ when the tension is $T$${}_2$. Its frequency of vibration for a tension $(T_1-T_2)$ is (in Hz):

• A. $450$
• B. $300$
• C. $100$
• D. $210$

Answer 1

The correct option is B $300$

$f\alpha \sqrt{T}$

so,$f_1:f_2:f_3::\sqrt{T_1}:\sqrt{T_2}:\sqrt{T_3}$

or $\frac{f_1}{f_2}=\sqrt{\frac{T_1}{T_2}}or\frac{f_2}{f_3}=\sqrt{\frac{T_2}{T_3}}or\frac{f_1}{f_3}=\sqrt{\frac{T_1}{T_3}}$

$\frac{500}{400}=\sqrt{\frac{T_1}{T_2}}-\left(1\right)or\frac{400}{f}=\sqrt{\frac{T_2}{T_1-T_2}}-\left(2\right)or\frac{500}{f}=\sqrt{\frac{T_1}{T_1-T_2}}-\left(3\right)$

$500\sqrt{T_2}=400\sqrt{T_1}or400\sqrt{T_1-T_2}=f\sqrt{T_2}or500\sqrt{T_1-T_2}=f\sqrt{T_1}$

$25T_2=16T_{1}or400\times 400(T_1-T_1\frac{16}{25})=f^2\frac{16T_1}{25}$

$160000\left(\frac{9T_1}{25}\right)=f^2\frac{16T_1}{25}$

$90000=f^2$

f = 300 HZ.