Question

A closed organ pipe and an open pipe of same length produce 4 beats when they are set onto vibrations simultaneously. If the length of each of them were twice their initial lengths, the number of beats produced will be

• A. 2
• B. 4
• C. 1
• D. 8

Answer 1

The correct option is A 2

Let, the length of open organ pipe and close organ pipe is L.

Now, for open organ pipe the length is

$L=\frac{n\lambda }{2}$

$L=\frac{nv}{2{\nu }_o}$ ............................($\because \lambda =\frac{v}{\nu }$)

Therefore

${\nu }_o=\frac{nv}{2L}$

Also, for close organ pipe the length is

$L=(2n+1)\frac{\lambda }{4}$

$L=(2n+1)\frac{v}{4{\nu }_c}$ ............................($\because \lambda =\frac{v}{\nu }$)

Therefore

${\nu }_c=(2n+1)\frac{v}{4L}$

As per the problem

${\nu }_c-{\nu }_o=4$

$(2n+1)\frac{v}{4L}-\frac{nv}{2L}=4$

$\frac{(2nv+v)-2nv}{4L}=4$

$\frac{v}{4L}=4$

$\frac{1}{2}\frac{v}{4L}=\frac{1}{2}\left(4\right)$ ........................( length of pipes is doubled)

$\frac{v}{8L}=2$