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A Steel Wire of Mass 4⋅0 G and Length 80 Cm is Fixed at the Two Ends. the Tension in the Wire is 50 N. Find the Frequency and Wavelength of the Fourth Harmonic of the Fundamental. - Physics

Sum

A steel wire of mass 4⋅0 g and length 80 cm is fixed at the two ends. The tension in the wire is 50 N. Find the frequency and wavelength of the fourth harmonic of the fundamental.

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Solution

Given:
Mass of the steel wire = 4.0 g
Length of the steel wire = 80 cm = 0.80 m
Tension in the wire = 50 N
Linear mass density (m)

\[= \left( \frac{4}{80} \right)  g/cm   = 0 . 005  kg/m\]

\[\text{ Wave  speed, }  \nu = \sqrt{\left( \frac{T}{m} \right)}\] 

\[ = \sqrt{\left( \frac{50}{0 . 005} \right)} = 100  m/s\]

\[\text{ Fundamental  frequency  ,}    f_o  = \frac{1}{2L}\sqrt{\left( \frac{T}{m} \right)}\] 

\[       = \frac{1}{2 \times 0 . 8} \times \sqrt{\left( \frac{50}{0 . 005} \right)}\] 

\[       = \frac{100}{2 \times 0 . 8} = 62 . 5  Hz\] 

\[\text { First  harmonic = 62 . 5  Hz }\] 

If   f_4  =frequency  of  the  fourth  harmonic:

\[ \Rightarrow  f_4  = 4 f_0  = 62 . 5 \times 4\] 

\[ \Rightarrow  f_4  = 250  Hz\] 

\[\text{ Wavelength  of  thefourth  harmonic,}    \lambda_4  = \frac{\nu}{f_4} = \frac{100}{250}\] 

\[ \Rightarrow  \lambda_4  = 0 . 4  m = 40  cm\]

  Is there an error in this question or solution?
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APPEARS IN

HC Verma Class 11, 12 Concepts of Physics 1
Chapter 15 Wave Motion and Waves on a String
Q 36 | Page 326
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