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Two tuning forks with natural frequencie...

Two tuning forks with natural frequencies of `340 Hz` each move relative to a stationary observer. One fork moves away form the observer, while the other moves towards him at the same speed. The observer hears beats of frequency `3 Hz`. Find the speed of the tuning fork.

Text Solution

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Let `upsilon_(s)` be speed of each tuning fork relative to the observer.
For the fork moving away from observer, apparent frequency is `v_(1)=(upsilonv)/(upsilon+upsilon_(s))`
and for fork moving towards the observer, apparent frequency is `v_(2)=(upsilonv)/(upsilon-upsilon_(s))`
Number os beats hear `//` sec., `n=v^(2)-v^(1)`
`3=(upsilonv)/(upsilon-upsilon_(s))-(upsilonv)/(upsilon+upsilon_(s))=(2upsilonvupsilon_(s))/(upsilon^(2)-upsilon_(s)^(2))=(2upsilonvupsilon_(s))/(upsilon^(2)(1-(upsilon_(s)^(2))/(upsilon^(2))))=(2vupsilon_(s))/(upsilon)(1-(upsilon_(s)^(2))/(upsilon^(2)))^(-1)=(2vupsilon_(s))/(upsilon)` `( :' upsilon_(s)^(2)lt ltupsilon^(2))`
`upsilon_(s)=(3upsilon)/(2v)=(3xx340)/(2xx340)=1.5m//s`
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