Question

Two strings of same material having lengths as $L$, $2L$ and radii $2r$, $r$ respectively, are vibrating in the fundamental mode. Tension applied to both the strings is same. The ratio of their respective fundamental frequencies is

• A. $4:3$
• B. $1:2$
• C. $1:1$
• D. $3:4$

Hint:

Changes in length and radius can cancel each other in string frequency problems.

Answer 1

The Correct Option is C

Step 1: Expression for fundamental frequency of a stretched string.
\[ f = \frac{1}{2L}\sqrt{\frac{T}{\mu}} \] where $\mu$ is mass per unit length.

Step 2: Mass per unit length relation.
\[ \mu = \rho \pi r^2 \] For same material, $\rho$ is constant.

Step 3: Calculate $\mu$ for both strings.
First string: $L,\, 2r$
\[ \mu_1 \propto (2r)^2 = 4r^2 \]
Second string: $2L,\, r$
\[ \mu_2 \propto r^2 \]

Step 4: Frequency ratio.
\[ \frac{f_1}{f_2} = \frac{L_2}{L_1}\sqrt{\frac{\mu_2}{\mu_1}} = \frac{2L}{L}\sqrt{\frac{r^2}{4r^2}} = 2 \times \frac{1}{2} = 1 \]

Step 5: Conclusion.
The ratio of fundamental frequencies is $1:1$.