Question

A string of length $1m$ and mass $5g$ is fixed at both ends. The tension in the string is $8.0N$ . The string is set into vibration using an external vibrator of frequency $100Hz$ . The separation between successive nodes on the string is close to ____?
(A) $16.6cm$
(B) $20.0cm$
(C) $10.0cm$
(D) $33.3cm$

Answer 1

Hint : We need to first find the velocity of the wave on the string with the tension in the string and the mass of the string given in the question. Then using that value of velocity we can find the wavelength, where the separation between two nodes will be half of the wavelength.

Formula Used : In this solution we will be using the following formula,
$\Rightarrow V = \sqrt {\dfrac{T}{m}}$
where $V$ is the velocity of the wave,
$T$ is the tension in the string
$m$ is the mass of the string
$\Rightarrow V = f\lambda$
where $f$ is the frequency and $\lambda$ is the wavelength of the wave.

Complete step by step answer
In the question we are provided the mass of a string which is fixed at both the ends and the tension in the string. Now when the string is set into vibration, then we can find the velocity of the wave on the string. So we can use the formula,
$\Rightarrow V = \sqrt {\dfrac{T}{m}}$
In the question we are given $T = 8N$ and the mass of the string as, $m = 5g$ . So we can write this in SI unit as $m = \dfrac{5}{{1000}}kg$
Now by substituting these values in the formula we get,
$\Rightarrow V = \sqrt {\dfrac{8}{{\dfrac{5}{{1000}}}}}$
So we get this as,
$\Rightarrow V = \sqrt {\dfrac{{8 \times 1000}}{5}}$
On calculating we get,
$\Rightarrow V = \sqrt {1600}$
Hence the velocity of the wave is, $V = 40m/s$
Now using this velocity we can find the wavelength of the wave as the velocity is the product of the frequency and the wavelength.
$\Rightarrow V = f\lambda$
Therefore, we can also write this as,
$\Rightarrow \lambda = \dfrac{V}{f}$
We are given in the question $f = 100Hz$
So substituting the values we get,
$\Rightarrow \lambda = \dfrac{{40}}{{100}}$
So we get the wavelength of the wave on the string as, $\lambda = 0.4m$
Now in a wave, two nodes are present in a single wavelength. Hence the distance between two nodes will be half the wavelength. So separation between the two nodes, $D = \dfrac{\lambda }{2}$
Substituting the values we have,
$\Rightarrow D = \dfrac{{0.4}}{2} = 0.2m$
Now we can write this in centimeter as, $D = 0.2 \times 100cm = 20.0cm$
Hence option (B) is correct.

Note
The velocity of a wave on a string is determined by the tension and the mass of the string per unit length. Since in this given question, the string is of unit length, so the mass of the string per unit length is the same as the mass of the string.