Question

In a pipe open at both ends, '$n_1$; and '$n_2$' be the frequencies corresponding to vibrating lengths '$l_1$' and '$l_2$' respectively. The end correction is

• A. $\frac{n_1{l}_1-n_2{l}_2}{2(n_1-n_2)}$
• B. $\frac{n_2{l}_2-n_1{l}_1}{2(n_2-n_1)}$
• C. $\frac{n_2{l}_2-n_1{l}_1}{2(n_1-n_2)}$
• D. $\frac{n_1{l}_1-n_2{l}_2}{(n_1-n_2)}$

Answer 1

The correct option is C $\frac{n_2{l}_2-n_1{l}_1}{2(n_1-n_2)}$

Let the end correction at each side of open pipe be $e$.

An open pipe has end correction on both sides.

Thus frequency in open pipe corresponding to $l_1$, $n_1=\frac{v}{2(l_1+2e)}$

Similarly,frequency in open pipe corresponding to $l_2$, $n_2=\frac{v}{2(l_2+2e)}$

Dividing both equations we get, $\frac{n_1}{n_2}=\frac{l_2+2e}{l_1+2e}$

Or $n_1{l}_1+2e{n}_1=n_2{l}_2+2e{l}_2$

Or $2e(n_1-n_2)=n_2{l}_2-n_1{l}_1$

$⟹$ $e=\frac{n_2{l}_2-n_1{l}_1}{2(n_1-n_2)}$