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A glass tube is open at both the ends. A tuning fork of frequency $f$ resonates with the air column inside the tube. Now the tube is placed vertically inside water so that half the length of the tube is filled with water. Now the air column inside the tube is in unison with another fork of frequency $f^{\prime}$. Then
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The correct answer is:
$f^{\prime}=f$
For fundamental frequency of vibration of air column
$$
l=\frac{\lambda}{2} \Rightarrow \lambda=2 l
$$
Frequency, $f=\frac{v}{\lambda}$
$\Rightarrow \quad f=\frac{v}{2 l}$
When tube is placed inside water with half length immersed in water then
Frequency, $f^{\prime}=\frac{v}{\lambda^{\prime}}=\frac{v}{2 l}$
From Eqs. (i) and (ii), we get
$$
f^{\prime}=f
$$
$$
l=\frac{\lambda}{2} \Rightarrow \lambda=2 l
$$
Frequency, $f=\frac{v}{\lambda}$
$\Rightarrow \quad f=\frac{v}{2 l}$
When tube is placed inside water with half length immersed in water then
Frequency, $f^{\prime}=\frac{v}{\lambda^{\prime}}=\frac{v}{2 l}$
From Eqs. (i) and (ii), we get
$$
f^{\prime}=f
$$
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