A trumpet player, using her ear, hears 5 beats per second when she plays a note on her trumpet and simultaneously sounds a 440 Hz tuning fork. After pulling her tuning valve out to slightly increase the length of her trumpet, she hears 3 beats per second against the tuning fork. What was the trumpet players' initial frequency?

Explanation:

The given  frequency of tuning fork is 440 Hz . It produces 5 beats with trumpet wire . That means , the frequency of wire can be 440 ± 5

It can be either 445 or 435

Now the length of wire is increased , by which its frequency decreases . Because frequency is inversely proportional to length of wire .

If we decrease the frequency in 435 , the difference between tuning fork frequency and wire frequency will become greater than 5 even . So it cannot produce 3 beats with it .

If we decrease frequency from 445 , it can become 443 Hz . It gives 3 beat with the tuning fork as given .

Thus the initial frequency of wire is 445 Hz

force required to detach it is given by

charges are taken to be fundamental charge

now we have

so above is the minimum distance at which it will detach