In a resonance experiment, two air columns (closed at one end) of 100 cm and 120 cm long, give 15 beats per second when each one is sounding in the respective fundamental modes. The velocity of sound in the air column is :
Show Hint
- 335 m/s
- 370 m/s
- 340 m/s
- 360 m/s
The Correct Option is D
Approach Solution - 1
Approach Solution -2
In this problem, we are dealing with resonance in two air columns that are closed at one end. The lengths of these air columns are 100 cm and 120 cm. They both produce sounds in their respective fundamental modes and produce 15 beats per second. We need to find the velocity of sound in these air columns.
To solve this, we start by using the formula for the frequency of a closed-end air column in its fundamental mode:
\(f = \frac{v}{4L}\)
where:
- \(v\) is the velocity of sound in air.
- \(L\) is the length of the air column.
Let the frequencies of the air columns be \(f_1\) and \(f_2\) for lengths 100 cm and 120 cm respectively. Then, we have:
- \(f_1 = \frac{v}{4 \times 100} = \frac{v}{400}\)
- \(f_2 = \frac{v}{4 \times 120} = \frac{v}{480}\)
Given that they produce 15 beats per second, the difference in frequency is:
\(|f_1 - f_2| = 15\)
Substituting the expressions for \(f_1\) and \(f_2\):
\(\left|\frac{v}{400} - \frac{v}{480}\right| = 15\)
By simplifying the difference:
\(\frac{v}{400} - \frac{v}{480} = 15\)
Find a common denominator (2400 in this case) to combine the fractions:
\(\frac{6v}{2400} - \frac{5v}{2400} = 15\)
This simplifies to:
\(\frac{v}{2400} = 15\)
Multiplying through by 2400 gives:
\(v = 15 \times 2400 = 36000 \; \text{cm/s} = 360 \; \text{m/s}\)
Therefore, the velocity of sound in the air columns is 360 m/s. This corresponds to the given option: 360 m/s.
Top JEE Main Physics Questions
- In a hydrogen like atom, when an electron jumps from the $M$ - shell to the $L$ - shell, the wavelength of emitted radiation is $\lambda$. If an electron jumps from $N$-shell to the $L$-shell, the wavelength of emitted radiation will be :
- Two masses $m$ and $\frac{m}{2}$ are connected at the two ends of a massless rigid rod of length $l$. The rod is suspended by a thin wire of torsional constant $k$ at the centre of mass of the rod-mass system(see figure). Because of torsional constant $k$, the restoring torque is $\tau =k\theta$ for angular displacement $\theta$. If the rod is rota ted by $\theta_0$ and released, the tension in it when it passes through its mean position will be:
A black body is at a temperature of 2880 K. The energy of radiation emitted by this body with wavelength between 499 nm and 500 nm is U1, between 999 nm and 1000 nm is U2 and between 1499 nm and 1500 nm is U3. The Wien's constant, b = 2.88×106 nm-K. Then,
- $I _{ CM }$ is the moment of inertia of a circular disc about an axis (CM) passing through its center and perpendicular to the plane of disc $I_{A B}$ is it's moment of inertia about an axis $AB$ perpendicular to plane and parallel to axis $CM$ at a distance $\frac{2}{3} R$ from center Where $R$ is the radius of the dise The ratio of $I _{ AB }$ and $I _{ CM }$ is $x: 9$ The value of $x$ is______
- A nucleus disintegrates into two smaller parts, which have their velocities in the ratio $3: 2$ The ratio of their nuclear sizes will be $\left(\frac{x}{3}\right)^{\frac{1}{3}}$ The value of ' $x$ ' is:-
Top JEE Main Sound Wave Questions
- In a closed organ pipe, the frequency of the fundamental note is 30 Hz. A certain amount of water is now poured in the organ pipe so that the fundamental frequency is increased to 110 Hz. If the organ pipe has a cross-sectional area of 2 cm\(^2\), the amount of water poured in the organ tube is \( \_\_\_\_\_\_ \) g. (Take speed of sound in air as 330 m/s)
Two resistances of 100Ω and 200Ω are connected in series with a battery of 4V and negligible internal resistance. A voltmeter is used to measure voltage across the 100Ω resistance, which gives a reading of 1V. The resistance of the voltmeter must be _____ Ω.
- A closed and an open organ pipe have same lengths. If the ratio of frequencies of their seventh overtones is \(\left(\frac{a-1}{a}\right)\), then the value of \(a\) is ______.
- Two open organ pipes of length $60 \, \text{cm}$ and $90 \, \text{cm}$ resonate at $6^\text{th}$ and $5^\text{th}$ harmonics respectively. The difference of frequencies for the given modes is ____ $\text{Hz}$.
(Velocity of sound in air $= 333 \, \text{m/s}$) - In a sound wave, the displacement of the air particles follows the equation \( y = A \cos(kx - \omega t) \). What is the wave velocity?
Top JEE Main Questions
- In a hydrogen like atom, when an electron jumps from the $M$ - shell to the $L$ - shell, the wavelength of emitted radiation is $\lambda$. If an electron jumps from $N$-shell to the $L$-shell, the wavelength of emitted radiation will be :
- Two masses $m$ and $\frac{m}{2}$ are connected at the two ends of a massless rigid rod of length $l$. The rod is suspended by a thin wire of torsional constant $k$ at the centre of mass of the rod-mass system(see figure). Because of torsional constant $k$, the restoring torque is $\tau =k\theta$ for angular displacement $\theta$. If the rod is rota ted by $\theta_0$ and released, the tension in it when it passes through its mean position will be:
What will be the equilibrium constant of the given reaction carried out in a \(5 \,L\) vessel and having equilibrium amounts of \(A_2\) and \(A\) as \(0.5\) mole and \(2 \times 10^{-6}\) mole respectively?
The reaction : \(A_2 \rightleftharpoons 2A\)- The value of is
- The number of terms of an is even. The sum of the odd terms is and of the even terms is . The last term exceeds the first by . Then the number of terms in the series is ______.