The Question is from NEET guide , and it was created by the author NAGA SAI

Question

While measuring the speed of sound by performing a resonance column experiment, a student gets the first resonance condition at a column length of $18{\rm{ }}cm$ during winter. Repeating the same experiment during summer, she measures the column length to be $x{\rm{ }}cm$ for the second resonance. Then

NAGA SAI · Accepted Answer

## CORE INFORMATION
✓ **Answer**: B
## CONCEPT FRAMEWORK
**Primary Concept**: Resonance in a closed pipe and the temperature dependence of the speed of sound.
**Related Topics**: Standing waves, wavelength, frequency.
**Prerequisites**: Understanding of resonance conditions in a closed pipe, relationship between speed of sound, frequency and wavelength, and the effect of temperature on the speed of sound.
## SOLUTION PATHWAY
**Given Information**:
    * First resonance at 18 cm in winter.
    * Second resonance at x cm in summer.
    * The speed of sound increases with temperature.
**Method & Steps**:
    1. **Resonance condition in a closed pipe:** For the first resonance, the length of the air column (L₁) is approximately λ/4, and for the second resonance, the length of the air column (L₂) is approximately 3λ/4. Where λ is the wavelength of the sound wave.
        Reasoning: The resonance occurs when the air column forms a standing wave. The first resonance corresponds to the shortest air column length that creates a standing wave, and the second resonance occurs when the next possible standing wave is formed in the air column.
        Formula/Rule: $L_1 \approx \frac{\lambda}{4}$, $L_2 \approx \frac{3\lambda}{4}$
    2. **Wavelength in winter:** Let λ_w be the wavelength in winter. Then 18 cm ≈ λ_w/4. Thus, λ_w ≈ 72 cm.
        Working: Substituting the given value for L_1 into the formula.
        Key Point: This is the approximate relationship for resonance.
    3. **Temperature effect on speed of sound:** The speed of sound increases with temperature. Since summer is warmer than winter, the speed of sound and therefore the wavelength will be larger in the summer.
        Reasoning: The speed of sound in air is directly proportional to the square root of temperature.
       Formula/Rule: $v = f\lambda$, v increases with temperature
    4. **Wavelength in summer:** Let λ_s be the wavelength in summer. Since the speed of sound is greater in summer, λ_s > λ_w. Let's denote the length for second resonance as L_2 = x cm. Then x cm ≈ 3λ_s/4. Since λ_s > λ_w, 3λ_s/4 > 3λ_w/4.
        Working: Applying the condition that speed of sound and wavelength increases in summer
        Key Point: The second resonance is at a length approximately equal to 3λ/4.
    5. **Finding x:** Since x ≈ 3λ_s/4 and λ_s > 72 cm, 3λ_s/4 > 3(72cm)/4=54cm . Therefore x > 54 cm.
        Result: x > 54 cm
        Verification: This matches the correct answer option.
 ## OPTION ANALYSIS
[A] :x: This is incorrect. The length for the second resonance will be larger than 18cm and also significantly larger than 18 cm.
[B] ✓: This is the correct option. The second resonance length is greater than 54cm since the wavelength has increased in summer.
[C] :x: This is incorrect. The second resonance will be greater than 54 cm.
[D] :x: This is incorrect. The second resonance length is greater than 54cm.
 ## LEARNING AIDS
**Common Mistakes**:
    1. **Confusing first and second resonance:** Misinterpreting which resonance corresponds to which length.
        How to avoid: Remember the first resonance corresponds to λ/4 and second to 3λ/4.
    2. **Ignoring the temperature effect:** Failing to realize the speed of sound increases with temperature.
        How to avoid: Know the relationship between temperature and the speed of sound.
**Quick Tips**:
    1.  The length for second resonance is 3 times that of the first resonance at the same temperature.
    2.  The speed of sound increases with temperature, therefore the wavelength also increases at the same frequency.

NAGA SAI · Answer

$\begin{array}{*{20}{l}}
{18{\rm{ }} > x}
\end{array}$

NAGA SAI · Answer

The correct Option is "$\begin{array}{*{20}{l}}
{X{\rm{ }} > 54}
\end{array}$"

NAGA SAI · Answer

$\begin{array}{*{20}{l}}
{54{\rm{ }} > {\rm{ }}x{\rm{ }} > 36}
\end{array}$

NAGA SAI · Answer

$\begin{array}{*{20}{l}}
{36{\rm{ }} > {\rm{ }}x{\rm{ }} > {\rm{ }}18}
\end{array}$

NEET Questions

Select Subject

Question Tags

Question Tags

Question Tags

Question Tags

Question Tags

Question Tags

Question Tags

Question Tags

Question Tags

Question Tags

NEET Subjects

Our Bots

Company