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Superposition of Waves — Physics STD 12 Science — Question

Maharashtra BoardEnglish MediumSTD 12 SciencePhysicsSuperposition of Waves1 Mark
MCQ
The closed and open pipes have same length. When they are vibrating simultaneously in first overtone, produce three beats. The length of open pipe is made $\frac{1}{3}^{\text {rd }}$ and closed pipe is made three times the original, the number of beats produced will be
  • A
    8
  • B
    14
  • ✓
    17
  • D
    20
✓

Answer

Correct option: C.
17
(c) : For open organ pipe, first overtone $v_1=\frac{v}{L}$ For closed organ pipe, first overtone $v_1^{\prime}=\frac{3 v}{4 L}$
Given : Beats $=3 Hz$
$
v_1-v_1^{\prime}=3=\frac{v}{L}-\frac{3 v}{4 L}=3=\frac{v}{4 L}=3 \Rightarrow \frac{v}{L}=12
$
For open organ pipe, fundamental frequency $v=\frac{v}{2 L^{\prime}}$ Given $L^{\prime}=\frac{L}{3} \quad \therefore v=\frac{v}{2\left(\frac{L}{3}\right)}=\frac{3 v}{2 L}$
For closed organ pipe, fundamental frequency $v^{\prime}=\frac{v}{4 L^{\prime}}$ Given $L^{\prime}=3 L \quad \therefore \quad v^{\prime}=\frac{v}{4(3 L)}=\frac{v}{12 L}$
$\therefore \quad$ Beat produced $=v-v^{\prime}$
$
=\frac{3 v}{2 L}-\frac{v}{12 L}=\frac{17 v}{12 L}=\frac{17}{12} \cdot 12=17
$

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