Question

A string of length$70cm$ is stretched between two rigid supports. The resonant frequency for this string is found to be $420Hz$and $315Hz$. If there are no resonant frequencies between these two values, then what would be the minimum resonant frequency of this string?

• A. $10.5Hz$
• B. $1.05Hz$
• C. $105Hz$
• D. $1050Hz$

Answer 1

The correct option is C

$105Hz$

Step 1: Given data

Length of the string, $L=70cm$

Resonant frequency for the string, $f_1$and $f_2$are $420Hz,315Hz$ respectively.

Step 2: Find minimum resonant frequency

As, String fixed at both ends, $f=nv/2l$

Here, $n=$no. of loops

Now, $315=nv/2l.....................................................\left(1\right)$

$420=(n+1)v/2l...............................................\left(2\right)$

On, dividing equation (1) by equation (2), we get

$$\begin{gathered} (n+1)/n=420/315 \\ 315n+315=420n \\ n=315/(420–315) \\ =3 \end{gathered}$$

$$\begin{gathered} \Rightarrow 315=3v/2l \\ \Rightarrow v/2l=315/3 \\ \Rightarrow v/2l=105 \end{gathered}$$

Therefore, minimum resonant frequency, $f=105Hz$

Hence, option (C) is correct.