The length of the wire between two ends of sonometer is 100 ~cm ....

The length of the wire between two ends of sonometer is

100 \mathrm{~cm}

. What should be the positions of two bridges below the wire so that the three segments of the wire have their fundamental frequencies in the ratio

1: 3: 5

Options

D. $\frac{1500}{23} \mathrm{~cm}, \frac{2000}{23} \mathrm{~cm}$

According to question Length of wire

=L=100 \mathrm{~cm}

And ratio of fundamental frequencies =1: 3: 5

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\begin{aligned}L_1: L_2: L_3 & =\frac{1}{1}: \frac{1}{3}: \frac{1}{5}\\ & =15: 5: 3 \end{aligned}

Let

x=

comman factor then

\begin{aligned}& 15 x+5 x+3 x=100 \\ & \Rightarrow \quad 2 x=100 \\ & \Rightarrow \quad x=\frac{100}{23}\\ & \therefore \quad L_1=15 \times \frac{100}{23}\\ & =\frac{1500}{23} \mathrm{~cm}\\ & \therefore \quad L_2=5 \times \frac{100}{23}\\ & =\frac{1500}{23} \mathrm{~cm}\\ & \end{aligned}

\begin{aligned} \therefore \quadL_3 & =3 \times \frac{100}{23}\\ & =\frac{300}{23} \mathrm{~cm} \end{aligned}

This shows that bridge should be placed from

A

\frac{1500}{23} \mathrm{~cm}

and

\frac{2000}{23} \mathrm{~cm}

respectively