Question

A 20cm long string having a mass of 1.0kg, if fixed at both the ends. The tension in the string is 0.5N. The string is set into vibrations using an external vibrator of frequency 100Hz. Find the separation (in cm) between the successive nodes of the string.

Answer 1

Hint: To solve this problem, use the formula for velocity of wave on a string. Substitute the values and calculate the velocity of the wave on the string. Then, use the relation between velocity, wavelength and frequency. Values of frequency and velocity we have, thus obtain the value of the wavelength. The separation between successive nodes is the half of the frequency of the wave. Thus, divide the calculated frequency by 2. The obtained answer is the separation between two successive nodes of the string.

Formula used:
$v = \sqrt {\dfrac {T}{\mu}}$
$\mu = \dfrac {m}{l}$
$ \lambda= \dfrac{v}{f}$
$d= \dfrac {\lambda}{2}$

Complete answer:
Given: Length of the string (l)= 20cm = $ 20 \times {10}^{-2}m$
            Mass of the string (m)= 1.0kg= $1 \times {10}^{-3} g$
            Tension in the string (T)= 0.5N
            Frequency (f)= 100Hz
Velocity of a wave in a string is given by,
$v = \sqrt {\dfrac {T}{\mu}}$ …(1)
Where, $\mu$ is the mass per unit length

Mass per unit length can be calculated by,
$\mu = \dfrac {m}{l}$
Substituting the values in above equation we get,
$\mu = \dfrac {1 \times {10}^{-3}}{20 \times {10}^{-2}}$
$\Rightarrow \mu = 0.5 \times {10}^{-2} kg{m}^{-1}$
Now, substituting these values in the equation. (1) we get,
$v=\sqrt {\dfrac {0.5}{0.5 \times {10}^{-2} }}$
$\Rightarrow v = \sqrt {{10}^{2}}$
$\Rightarrow v = 10 m{s}^{-1}$
We know, relation between frequency and wavelength is given by,
$Wavelength = \dfrac{Velocity}{Frequency}$
$\Rightarrow \lambda= \dfrac{v}{f}$
Substituting the values we get,
$\lambda = \dfrac {10}{100}$
$\Rightarrow \lambda = 0.1m$
$\rightarrow \lambda= 10cm$

We know, Separation between two successive nodes is given by,
$d= \dfrac {\lambda}{2}$
Substituting the values we get,
$d= \dfrac {10}{2}$
$\Rightarrow d= 5cm$
Hence, the separation between the successive nodes on the string is 5 cm.

Note:
Students should not get confused between nodes and antinodes & crest and trough. While discussing the motion of a travelling wave, the point of maximum positive displacement is called the crest while the point of maximum negative displacement is called trough. The nodes are not actually the part of the wave. These are just the unique points located inside a medium which makes the wave pattern.