Calculating Initial Beat Frequency
The initial number of beats observed is 8 in a time interval of 2 seconds. The initial beat frequency ($f_{beat1}$) is calculated as:
$f_{beat1} = \frac{\text{Number of beats}}{\text{Time interval}} = \frac{8}{2\text{ s}} = 4\text{ Hz}$
Determining Possible Original Frequencies
The beat frequency is the absolute difference between the frequencies of two tuning forks ($f_A$ and $f_B$). Given the frequency of tuning fork B ($f_B$) is $380\text{ Hz}$ and the initial beat frequency ($f_{beat1}$) is $4\text{ Hz}$, we have:
$|f_A - f_B| = f_{beat1}$
Substituting the known values:
$|f_A - 380\text{ Hz}| = 4\text{ Hz}$
This equation yields two possible values for the original frequency of tuning fork A ($f_A$):
- Case 1: $f_A - 380\text{ Hz} = 4\text{ Hz} \implies f_A = 384\text{ Hz}$
- Case 2: $f_A - 380\text{ Hz} = -4\text{ Hz} \implies f_A = 376\text{ Hz}$
Analyzing the Effect of Loading on Frequency
Loading a tuning fork with wax adds mass, which slightly decreases its natural frequency. Let the new frequency of tuning fork A after loading be $f_A'$. Therefore, $f_A' < f_A$.
After loading, the new beat frequency ($f_{beat2}$) is observed as 4 beats in 2 seconds:
$f_{beat2} = \frac{4}{2\text{ s}} = 2\text{ Hz}$
The new relationship between the frequencies is:
$|f_A' - f_B| = f_{beat2}$
Substituting the known values:
$|f_A' - 380\text{ Hz}| = 2\text{ Hz}$
This equation yields two possible values for the new frequency $f_A'$:
- $f_A' - 380\text{ Hz} = 2\text{ Hz} \implies f_A' = 382\text{ Hz}$
- $f_A' - 380\text{ Hz} = -2\text{ Hz} \implies f_A' = 378\text{ Hz}$
Selecting the Correct Original Frequency
We need to determine which of the initial possibilities for $f_A$ ($384\text{ Hz}$ or $376\text{ Hz}$) is consistent with the frequency decrease upon loading and the observed change in beat frequency.
- Scenario 1: If the original frequency $f_A = 384\text{ Hz}$. Loading fork A causes its frequency to decrease to $f_A'$. The possible new frequencies are $f_A' = 382\text{ Hz}$ or $f_A' = 378\text{ Hz}$. Both $382\text{ Hz}$ and $378\text{ Hz}$ are less than $384\text{ Hz}$, which is consistent with the effect of loading. The beat frequency correctly decreased from $4\text{ Hz}$ to $2\text{ Hz}$.
- Scenario 2: If the original frequency $f_A = 376\text{ Hz}$. Loading fork A causes its frequency to decrease to $f_A'$. The possible new frequencies are $f_A' = 382\text{ Hz}$ or $f_A' = 378\text{ Hz}$. However, both $382\text{ Hz}$ and $378\text{ Hz}$ are greater than the original frequency $376\text{ Hz}$. This contradicts the physical principle that loading a tuning fork *decreases* its frequency. Thus, the original frequency cannot be $376\text{ Hz}$.
Therefore, the original frequency of tuning fork A must be $384\text{ Hz}$.
Similar Questions
- The displacement of a particle, executing simple harmonic motion with time period $T$, is expressed as $x(t) = A\sin\omega t$, where $A$ is the amplitude. The maximum value of potential energy of this oscillator is found at $t = T/2\beta$. The value of $\beta$ is ________.
A simple pendulum has a bob with mass $m$ and charge $q$. The pendulum string has negligible mass. When a uniform and horizontal electric field $\vec{E}$ is applied, the tension in the string changes. The final tension in the string, when pendulum attains an equilibrium position is _________.
(g: acceleration due to gravity)In an open organ pipe $\nu_3$ and $\nu_6$ are $3^{\text{rd}}$ and $6^{\text{th}}$ harmonic frequencies, respectively. If $\nu_6 - \nu_3 = 2200 \text{ Hz}$ then length of the pipe is _________ mm.
(Take velocity of sound in air is $330 \text{ m/s}$.)- Using a simple pendulum experiment g is determind by measuring its time period T. Which of the following plots represent the correct relation between the pendulum length L and time period T ?
- A cylindrical block of mass $M$ and area of cross section $A$ is floating in a liquid of density $\rho$ and with its axis vertical. When depressed a little and released the block starts oscillating. The period of oscillation is ______.
- The kinetic energy of a simple harmonic oscillator is oscillating with angular frequency of $176\text{ rad/s}$. The frequency of this simple harmonic oscillator is __________ $\text{Hz}$. $\left[ \text{take } \pi = \frac{22}{7} \right]$
- A simple pendulum of string length 30 cm performs 20 oscillations in 10 s. The length of the string required for the pendulum to perform 40 oscillations in the same time duration is ___________ cm. [Assume that the mass of the pendulum remains same.]
- The velocity of sound in air is doubled when the temperature is raised from $0^\circ\text{C}$ to $\alpha^\circ\text{C}$. The value of $\alpha$ is ________.
- A particle is executing simple harmonic motion. Its amplitude is $A$ and time period is 5 sec. The time required by it to move from $x = A$ to $x = \frac{A}{\sqrt{2}}$ is _________ sec.
- Match List - I with List - II.
List - I List - II A. $\sin^{2} \omega t$ I. Periodic with time period $T=\frac{\pi}{\omega}$ but not simple harmonic motion (SHM) B. $\sin^{3} (2\omega t)$ II. Periodic with time period $T=\frac{2\pi}{\omega}$ but Not SHM C. $\sin (\omega t) + \cos(\pi \omega t)$ III. Periodic with time period $T=\frac{\pi}{\omega}$ and SHM D. $\cos \omega t + \cos 2\omega t$ IV. Non-periodic
Choose the correct answer from the options given below :
Important Questions from Oscillations and Waves
- The displacement of a particle, executing simple harmonic motion with time period $T$, is expressed as $x(t) = A\sin\omega t$, where $A$ is the amplitude. The maximum value of potential energy of this oscillator is found at $t = T/2\beta$. The value of $\beta$ is ________.
A simple pendulum has a bob with mass $m$ and charge $q$. The pendulum string has negligible mass. When a uniform and horizontal electric field $\vec{E}$ is applied, the tension in the string changes. The final tension in the string, when pendulum attains an equilibrium position is _________.
(g: acceleration due to gravity)In an open organ pipe $\nu_3$ and $\nu_6$ are $3^{\text{rd}}$ and $6^{\text{th}}$ harmonic frequencies, respectively. If $\nu_6 - \nu_3 = 2200 \text{ Hz}$ then length of the pipe is _________ mm.
(Take velocity of sound in air is $330 \text{ m/s}$.)- Using a simple pendulum experiment g is determind by measuring its time period T. Which of the following plots represent the correct relation between the pendulum length L and time period T ?
- A cylindrical block of mass $M$ and area of cross section $A$ is floating in a liquid of density $\rho$ and with its axis vertical. When depressed a little and released the block starts oscillating. The period of oscillation is ______.
