Animated Solution for Physics - Waves: A string of length $1\text{ m}$ and mass $2\times 10^{-5}\text{ kg}$ is under tension $T$. when the string vibrates, two successive harmonics are found to occur at frequencies $750\text{ Hz}$ and $1000\text{ Hz}$. The value of tension $T$ is _______ Newton.

Have you ever plucked a guitar string and wondered what exactly is happening in that blur of motion? The physics of music is fundamentally the physics of standing waves. When you pluck a string, it doesn't just vibrate randomly; it sets up a highly organized pattern of nodes (points of zero movement) and antinodes (points of maximum movement).

In this beautiful problem from JEE Advanced, we are given a string of length $L=1\text{ m}$ and an incredibly light mass of $m=2\times 10^{-5}\text{ kg}$. This string is held taut under some unknown tension $T$. Our mission is to find this tension by analyzing the frequencies at which the string naturally wants to vibrate—its resonant frequencies or harmonics.

Before we dive into the numbers, let's establish the ground rules. The speed of a transverse wave on a stretched string is governed by two physical properties: how tight the string is (the tension $T$) and how heavy it is per unit length (the linear mass density $\mu$).

First, we calculate the linear mass density:

For a string fixed at both ends, the allowed wavelengths are restricted by the boundary conditions. The ends cannot move, so they must be nodes. This leads to the classic formula for the frequency of the $n$-th harmonic:

Fundamental Frequency

The problem states that two successive harmonics occur at $750\text{ Hz}$ and $1000\text{ Hz}$. Let's call them the $n$-th and $(n+1)$-th harmonics.

We could set up a ratio to find $n$, but there is a much more elegant shortcut. Notice that the frequencies of the harmonics are simply integer multiples of the fundamental frequency ($f_1$).

If we subtract the frequency of the $n$-th harmonic from the $(n+1)$-th harmonic, the $n$ terms cancel out beautifully:

Using this powerful insight, we can instantly find $f_1$:

Now that we have the fundamental frequency, we can plug it back into our master equation for $n=1$:

Substituting our known values:

Let's carefully unravel this equation to isolate $T$. First, multiply both sides by 2 to clear the denominator:

To eliminate the square root, we square both sides. Don't let the large numbers intimidate you; $500^2$ is simply $25$ followed by four zeros:

Finally, multiply by the linear mass density to solve for the tension:

Notice how perfectly the numbers align. $250000$ is $2.5\times 10^5$. When multiplied by $10^{-5}$, the powers of ten cancel out completely:

By understanding the physical meaning behind the formulas—specifically that the gap between harmonics is the fundamental frequency—we bypassed tedious simultaneous equations and arrived at the solution with elegance and speed.