Animated Solution for Physics - Waves: A wire of length $L$ and mass per unit length $6.0\times 10^{-3}{\text{ kgm}}^{-1}$ is put under tension of $540\text{ N}$. Two consecutive frequencies that it resonates at are : $420\text{ Hz}$ and $490\text{ Hz}$. Then, $L$ in metres is

Imagine a guitar string pulled tightly between two fixed points. When you pluck it, the string vibrates, creating a beautiful standing wave. The frequencies at which this string naturally wants to vibrate are called its resonant frequencies or harmonics.

For a string of length $L$, fixed at both ends, the resonant frequencies are not just random numbers. They follow a strict, elegant mathematical rule. The $n$-th resonant frequency is given by the formula:

Here, $T$ is the tension in the string, $\mu$ is the linear mass density (mass per unit length), and $n$ is an integer ($1,2,3,\dots$) representing the harmonic number. The fundamental frequency, or the first harmonic, occurs when $n=1$.

In our problem, we are given two consecutive resonant frequencies: $420\text{ Hz}$ and $490\text{ Hz}$. Because they are consecutive, if the first one is the $n$-th harmonic, the next one must be the $(n+1)$-th harmonic.

Let's write down the equations for these two frequencies:

Now, here is where the magic happens. If we subtract the $n$-th frequency from the $(n+1)$-th frequency, the terms involving $n$ completely cancel out!

Notice that the result is exactly the formula for the fundamental frequency ($f_1$). This is a powerful shortcut: The difference between any two consecutive harmonic frequencies of a string fixed at both ends is always equal to its fundamental frequency.

So, we can easily find the fundamental frequency:

Now that we know the fundamental frequency is $70\text{ Hz}$, we can use it to find the length of the string $L$. We are given the tension $T=540\text{ N}$ and the linear mass density $\mu =6.0\times 10^{-3}\text{ kg/m}$.

Let's plug these values into our fundamental frequency equation:

First, let's simplify the term inside the square root. Dividing $540$ by $6.0\times 10^{-3}$ is the same as $540\times \frac{1000}{6}$, which gives $90000$.

The square root of $90000$ is exactly $300$. So our equation simplifies beautifully to:

Now, it's just a matter of simple algebra to isolate $L$:

Calculating the decimal value, we get:

Looking at the options provided in the question, the closest value is $2.1\text{ m}$. Thus, the correct option is (b).