Question

A string of length l is fixed at both ends. H is vibrating in its 3rd overtone with maximum amplitude a. What is the amplitude at a distance l/3 from one end?

• A. $\frac{\sqrt{3}}{2}a$
• B. $0$
• C. $a$
• D. $\frac{a}{2}$

Answer 1

The correct option is A

$\frac{\sqrt{3}}{2}a$

Step 1:Given:

$x=\frac{l}{3}$

$2A=a$

$n=3$

Step 2:Find:

Amplitude.

Step 3:Calculate Amplitude:

$$\begin{gathered} y=2ASin\left(\frac{\left(n+1\right)\pi x}{l}\right)\cos \omega t \\ =aSin\left(\frac{\left(3+1\right)\pi \times {\displaystyle \frac{l}{3}}}{l}\right)\cos \omega t \\ =aSin\left(\frac{4\pi }{3}\right)\cos \omega t \\ =-a\frac{\sqrt{3}}{2}\cos \omega t \end{gathered}$$

Hence, Option(A) is correct.