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Chapter 17: Q. 37 (page 484)

A string vibrates at its third-harmonic frequency. The amplitude at a point 30 cm from one end is half the maximum amplitude. How long is the string?

Short Answer

Expert verified

The length of the string is 5.40 m

Step by step solution

01

Write the given information 

Third-harmonic frequency, n= 3
The amplitude of the wave at x=30cm is Amax/2

02

To determine the length of the string

The general expression of the amplitude of the wave at any distance x
A(x)=Amaxsin2πxλ

According to the question
Amax2=Amaxsin2π(30cm)λsin60πλ=12⇒60πλ=π6⇒λ=360cm

Since it is three harmonic vibrations, there would be three loops with the length λ/2.
Therefore, the total length of the string is given by
L=33602=540cm
Thus the length of the string is 5.40 m

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