Question:

Two uniform strings $A$ and $B$ made of steel are made to vibrate under the same tension. If the first overtone of $A$ is equal to the second overtone of $B$ and if the radius of $A$ is twice that of $B$, the ratio of the lengths of the strings is

Updated On: Jun 7, 2022
  • 1:02
  • 1:03
  • 1:04
  • 1:06
Hide Solution
collegedunia
Verified By Collegedunia

The Correct Option is B

Solution and Explanation

First overtone of string $A$
= Second overtone of string $B$
$\Rightarrow $ 2nd harmonic of $A$ = 3rd harmonic of $B $
$\upsilon_A = \upsilon_B$
For a string $\upsilon = \frac{p}{l D} \sqrt{\frac{T}{\pi \rho}}$
For 2nd harmonic, $p = 2$, for 3rd harmonic,
$p = 3$
$\therefore \:\: \upsilon_A = \frac{2}{l_A D_A} \sqrt{\frac{T}{\pi \rho}}$ and $\upsilon_B = \frac{3}{l_B D_B} \sqrt{\frac{T}{\pi \rho}}$
$\frac{2}{l_A D_A} \sqrt{\frac{T}{\pi \rho}} = \frac{3}{l_B D_B} \sqrt{\frac{T}{\pi \rho}}$ (from eqn (i))
$\therefore \: \frac{2}{l_A D_A} = \frac{3}{l_B \frac{D_A}{2} } \Rightarrow \frac{l_A}{l_B} = \frac{2}{6} = \frac{1}{3} [ \because \, D_A = 2 D_B ] $
Was this answer helpful?
0
0

Concepts Used:

Waves

Waves are a disturbance through which the energy travels from one point to another. Most acquainted are surface waves that tour on the water, but sound, mild, and the movement of subatomic particles all exhibit wavelike properties. inside the most effective waves, the disturbance oscillates periodically (see periodic movement) with a set frequency and wavelength.

Types of Waves:

Transverse Waves -

Waves in which the medium moves at right angles to the direction of the wave.

Examples of transverse waves:

  • Water waves (ripples of gravity waves, not sound through water)
  • Light waves
  • S-wave earthquake waves
  • Stringed instruments
  • Torsion wave

The high point of a transverse wave is a crest. The low part is a trough.

Longitudinal Wave -

A longitudinal wave has the movement of the particles in the medium in the same dimension as the direction of movement of the wave.

Examples of longitudinal waves:

  • Sound waves
  • P-type earthquake waves
  • Compression wave