Question

Two strings A and B of lengths, $L_A=80cm$ and $L_B=xcm$ respectively are used separately in a sonometer. The ratio of their densities $\left(\frac{d_A}{d_B}\right)$ is $0.81$. The diameter of B is one-half that of A. If the strings have the same tension and fundamental frequency, the value of $x$ is:

• A. $33$
• B. $102$
• C. $144$
• D. $130$

Answer 1

The correct option is C

$144$

Step 1: Given

Length of string A: $L_A=80cm$

Length of string B: $L_B=xcm$

Ratio of densities of string A to B: $\frac{d_A}{d_B}=0.81$

Ratio of diameters of string A to B: $\frac{D_A}{D_B}=2$

Step 2: Formula used

Linear density of a string is given by $\mu =dA=d\left(\pi r^2\right)$, where $d$ is the density, $A$ is the area and $r$ is the radius.

The frequency of a stretched string is $f=\frac{1}{2L}\sqrt{\frac{T}{\mu }}$, where $L$ is length of string, $T$ is tension in string and $\mu$ is linear density of string.

Step 3: Find the ratio of linear densities of string A to B

$$\begin{gathered} \frac{{\mu }_A}{{\mu }_B}=\frac{d_A}{d_B}\times \frac{\mathrm{\pi }{\left(\frac{{\mathrm{D}}_{\mathrm{A}}}{2}\right)}^2}{\mathrm{\pi }{\left(\frac{{\mathrm{D}}_{\mathrm{B}}}{2}\right)}^2} \\ =\frac{d_A}{d_B}\times {\left(\frac{{\mathrm{D}}_{\mathrm{A}}}{{\mathrm{D}}_{\mathrm{B}}}\right)}^2 \\ =0.81\times 2^2 \\ =3.24 \end{gathered}$$

Step 4: Find the ratio of lengths of the string

Find an expression for frequency of first spring by using the formula,

$f_A=\frac{1}{2L_A}\sqrt{\frac{T}{{\mu }_A}}$

Find an expression for frequency of first spring by using the formula,

$f_B=\frac{1}{2L_B}\sqrt{\frac{T}{{\mu }_B}}$

Calculate the ratio between the lengths of the springs by equating the frequencies, as the frequencies of both the strings are same.

$$\begin{gathered} \frac{1}{2L_A}\sqrt{\frac{T}{{\mu }_A}}=\frac{1}{2L_B}\sqrt{\frac{T}{{\mu }_B}} \\ \Rightarrow \frac{L_A}{L_B}=\frac{\frac{1}{2}\sqrt{\frac{T}{{\mu }_A}}}{\frac{1}{2}\sqrt{\frac{T}{{\mu }_B}}} \\ \Rightarrow \frac{L_A}{L_B}=\sqrt{\frac{{\mu }_B}{{\mu }_A}} \end{gathered}$$

Step 5: Find the lengths of the string

Substitute the values in the ratio of lengths obtained

$$\begin{gathered} \frac{80cm}{L_B}=\sqrt{\frac{1}{3.24}} \\ \Rightarrow L_B=80cm\times 1.8 \\ \Rightarrow L_B=80cm\times 1.8 \\ =144cm \end{gathered}$$

Hence, the ratio of the strings is $144cm$.