Q.A tuning fork $A$ of frequency $256\ \text{Hz}$ produces $4$ beats per second when sounded together with another tuning fork $B$ of unknown frequency. When fork $B$ is loaded with a little wax (which lowers its frequency slightly), the number of beats per second decreases to $2$. Find the original (unloaded) frequency of fork $B$.

Given: $f_A=256\ \text{Hz}$, original beat frequency $=4$ beats/s, so $f_B=256+4=260\ \text{Hz}$ or $f_B=256-4=252\ \text{Hz}$ -- two possibilities before the loading clue is used.

Loading a tuning fork with wax always LOWERS its frequency slightly (added mass increases inertia, lowering the natural frequency). The beat frequency is then observed to DECREASE, from $4$ to $2$ beats/s, once $B$ is loaded.

If $f_B$ had originally been $252\ \text{Hz}$ (below $f_A$), loading $B$ would lower it further away from $256\ \text{Hz}$, INCREASING the gap $|f_A-f_B|$ and hence increasing, not decreasing, the beat frequency -- contradicting what is observed. …

List both candidate frequencies ($f_A\pm4$), then use the fact that loading only ever lowers frequency, together with the observed DECREASE in beats, to determine which candidate must be c …

• Picking $f_B=252\ \text{Hz}$ without checking against the loading clue (that value would make the beats INCREASE on loading, not decrease). …