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Wave Motion and Waves on a String — Physics STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 SciencePhysicsWave Motion and Waves on a String1 Mark
Question
A sonometer wire supports a 4kg load and vibrates in fundamental mode with a tuning fork of frequency 416Hz. The length of the wire between the bridges is now doubled. In order to maintain fundamental mode, the load should be changed to:
  1. 1kg
  2. 2kg
  3. 8kg
  4. 16kg.
✓

Answer

  1. 16kg.

Explanation:

According to the relation of the fundamental frequency of a string

$\nu=\frac{1}{2\text{l}}\sqrt{\frac{\text{F}}{\mu}}$

Where l is the length of the string

F is the tension

$\mu$ is the linear mass density of the string

We know that v1 = 416Hz, l1 = l and l2 = 2l

Also, m1 = 4kg and m2 = ?

$\nu_1=\frac{1}{2\text{l}_1}\sqrt{\frac{\text{m}_1\text{g}}{\mu}}\ \dots(1)$

$\nu_2=\frac{1}{2\text{l}_2}\sqrt{\frac{\text{m}_2\text{g}}{\mu}}\ \dots(2)$

So, in order to maintain the same fundamental mode

$\nu_1=\nu_2$

squaring both sides of equations (1) and (2) and then equating

$\frac{1}{4\text{l}^2}\frac{4\text{g}}{\mu}=\frac{1}{16\text{l}^2}\frac{\text{m}_2\text{g}}{\mu}$

$\Rightarrow\text{m}_2=16\text{kg}$

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