A tuning fork `P` of unknown frequency gives `7` beats in `2` seconds with another tuning fork `Q` is moved towards a wall with a speed of `5 m//s`, it gives `5` beats per second for an observer located left to it. On filling, `P` gives `6` beat per second with `Q`. The frequency (in Hz) of `P` is given by `(80 xx alpha)+beta, (alpha, beta in I, 0 le alpha, beta le 9)` then find the value of `alpha + beta`. Assume speed of sound `= 332 m//s`.

Answer

9

Explanation

Let `f_(1)` and `f_(2)` be the frequency of tuning forks `P` and `Q`, then `I f_(1)-f_(2) I=7//2`
Apparent frequency for `O` corresponding to signal directly coming from `Q=f_(2)((v)/(v+v_(q)))`
Apparent frequency of the echo `= f_(2)((v)/(v-v_(q))):. Delta f_(2)-f_(2)[(2v_(q)v)/(v^(2)-v_(q)^(2))]`
Since, `Delta f_(2) = 5` (given) `:. f_(2) =163.5 Hz` Now, `f_(1) = 163.5 pm 3.5 = 167` or `160 Hz`, when `P` is filled, its frequency will increase, since it si given that filled `P` gives greater number of beats with `Q`. It implies that `f_(1)` must be `167 Hz`.