Question

In the experiment for the determination of the speed of sound in air using the resonance column method, the length of the air column that resonates in the fundamental mode, with a tuning fork is $0.1\mathrm{m}$ When this length is changed to $0.35\text{m}$ the same tuning fork resonates with the first overtone. Calculate the end correction

• A. $0.012\text{m}$
• B. $0.025\text{m}$
• C. $0.05\text{m}$
• D. $0.024\text{m}$

Answer 1

The correct option is B

$0.025\text{m}$

Step 1: Given that and simplify

Let the end correction in a closed organ pipe be $∆\text{l}$

Length of the air column, ${\mathrm{l}}_1=0.1\mathrm{m}$

Step 2: Calculate the fundamental mode

Apply the formula of the fundamental mode

$$\begin{gathered} \Rightarrow {\mathrm{f}}_1=\frac{\mathrm{v}}{4\left({\mathrm{l}}_1+∆\mathrm{l}\right)} \\ \Rightarrow {\mathrm{f}}_1=\frac{\mathrm{v}}{4\left(0.1+∆\mathrm{l}\right)}.......\left(1\right) \end{gathered}$$

Step 3: Calculate the value of the first overtone

Length of air column for the first overtone, ${\mathrm{l}}_2=0.35\mathrm{m}$

$$\begin{gathered} \Rightarrow {\mathrm{f}}_1=\frac{3\mathrm{v}}{4\left({\mathrm{l}}_2+∆\mathrm{l}\right)} \\ \Rightarrow {\mathrm{f}}_1=\frac{3\mathrm{v}}{4\left(0.35+∆\mathrm{l}\right)}......\left(2\right) \end{gathered}$$

Step 4: Calculate the value of the end correction

According to the question

$\begin{array}{rcl}& \Rightarrow & \mathrm{Equation}\left(1\right)=\mathrm{Equation}\left(2\right)\\ & \Rightarrow & \frac{\mathrm{v}}{4\left(0.1+∆\mathrm{l}\right)}=\frac{3\mathrm{v}}{4\left(0.35+∆\mathrm{l}\right)}\end{array}$

On solving, we get

$\begin{array}{rcl}0.35+∆\mathrm{l}& =& 0.3+∆\mathrm{l}\\ ∆\mathrm{l}& =& 0.025\mathrm{m}\end{array}$

The end correction is $\begin{array}{l}0.025\mathrm{m}\end{array}$.

Hence, option $\left(B\right)$ is correct.