Calculate the frequency of the tuning fork which when vibrated with the tuning fork of frequency 256 Hz produces 6 beat/s and when vibrated with a tuning fork having frequency 253 Hz produces 3 beat/s.

To find the frequency of the unknown tuning fork, we can use the information about the beats produced when it vibrates with two known frequencies: 256 Hz and 253 Hz.

Understanding Beats

Beats occur when two sound waves of slightly different frequencies interfere with each other. The beat frequency is the absolute difference between the two frequencies.

Calculating the Unknown Frequency

Let the frequency of the unknown tuning fork be represented as f.

• When f is paired with 256 Hz, the beat frequency is 6 beats/s:
• |f - 256| = 6
• When f is paired with 253 Hz, the beat frequency is 3 beats/s:
• |f - 253| = 3

Setting Up Equations

From the first condition:

• f - 256 = 6 → f = 262
• 256 - f = 6 → f = 250

From the second condition:

• f - 253 = 3 → f = 256
• 253 - f = 3 → f = 250

Finding the Valid Frequency

The possible frequencies from the equations are 262 Hz, 250 Hz, and 256 Hz. However, only 250 Hz satisfies both conditions:

• For 256 Hz: |250 - 256| = 6
• For 253 Hz: |250 - 253| = 3

Final Answer

The frequency of the unknown tuning fork is 250 Hz.