Question

In an organ pipe whose one end is at $x=0$ , the pressure is expressed by $\text{p}={\text{p}}_{0}cos\frac{3\pi \text{x}}{2}sin300\pi \text{t}$ where x is in meter and $t$ in seconds. The organ pipe can be

• A. Closed at one end, open at another with length $=0.5m$
• B. Open at both ends, length $=1m$
• C. Closed at both ends, length $=2m$
• D. Closed at end, open at another with length $=\frac{2}{3}\text{m}$

Answer 1

Answer: (C)

For $\text{x}=\lambda$
$\frac{3\pi \text{x}}{2}=2\pi$
$\Rightarrow \lambda =\frac{4}{3}$
at x = 0
there is a pressure antinode. So first end must be close and in case of close at bot end
$l=\frac{\text{n}\lambda }{2}\Rightarrow l=\frac{2\text{n}}{3}$
for close at one end
$l=\left(2\text{n}+1\right)\frac{\lambda }{4}=\frac{\left(2\text{n}+1\right)}{3}$
So for close at one end possible lengths are
$\frac{\left(2\text{n}+1\right)}{3}=\frac{1}{3}\text{, }1\text{, }\frac{5}{3}\text{, }\frac{7}{3}$
Open at both ends is not possible for close at both end possible length are
$\frac{2\text{n}}{3}=\frac{2}{3}\text{, }\frac{4}{3}\text{, }2\text{, }\frac{8}{3}$ ...... so only possibility is (3)