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 $384\text{ Hz}$.