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Wave Motion and Waves on a String — Physics STD 11 Science — Question

Gujarat BoardEnglish MediumSTD 11 SciencePhysicsWave Motion and Waves on a String1 Mark
MCQ
Two strings A and B, made of same material, are stretched by same tension. The radius of string A is double of the radius of B. A transverse wave travels on A with speed $\nu_\text{A}$ and on B with speed $\nu_\text{B}.$ The ratio $\frac{\nu_\text{A}}{\nu_\text{B}}$ is:
  • ✓
    $\frac{1}{2}$
  • B
    $2$
  • C
    $\frac{1}{4}$
  • D
    $4.$
✓

Answer

Correct option: A.
$\frac{1}{2}$
wave speed is given by $\nu=\sqrt{\frac{\text{T}}{\mu}}$
Where
T is the tension in the string
v is the speed of the wave
$\mu$ Is the mass per unit length of the stnng
$\mu=\frac{\text{M}}{\text{L}}=\rho\frac{\text{V}}{\text{L}}=\rho\frac{(\text{AL})}{\text{L}}$
Where
Mis the mass of the stnng. which can be written as pV
L Is the length of the string
$=\rho(\pi\text{r}^2)=\rho\Big(\pi\frac{\text{D}^2}{4}\Big)$
$\therefore\nu=\sqrt{\frac{\text{T}}{\rho\pi\frac{\text{D}^2}{4}}}=\frac{2}{\text{D}}\sqrt{\frac{\text{T}}{\rho\pi}}$
Where Dis the diameter of the string.
Thus, $\text{V}\propto\frac{1}{\text{D}}$
Since, $\text{r}_\text{A}=2\text{r}_\text{B}$
$\text{v}_\text{A}\propto\frac{1}{2\text{r}_\text{A}}\propto\frac{1}{2\times2\text{r}_\text{B}}\ \dots(1)$
$\text{v}_\text{B}\propto\frac{1}{2\text{r}_\text{B}}\ \dots(2)$
From Equations (1) and (2) we get
$\frac{\text{v}_\text{A}}{\text{v}_\text{B}}=\frac{1}{2}$

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