Answer
(a) The fundamental frequency of this string is $260~Hz$
(b) The total mass of the string is 2.8 grams.
Work Step by Step
(a) Since there are no other resonant frequencies between $780~Hz$ and $1040~Hz$, these two frequencies have the form $nf = 780~Hz$ and $(n+1)~f = 1040~Hz$, for some integer $n$, and where $f$ is the fundamental frequency.
We can find $f$:
$f = (n+1)~f - n~f = 1040~Hz-780~Hz = 260~Hz$
The fundamental frequency of this string is $260~Hz$
(b) We can find the wave speed in the string:
$v = \lambda~f$
$v = (2L)~f$
$v = (2)(1.6~m)(260~Hz)$
$v = 832~m/s$
We can find the total mass of the string:
$v = \sqrt{\frac{F}{m/L}}$
$v = \sqrt{\frac{F~L}{m}}$
$v^2 = \frac{F~L}{m}$
$m = \frac{F~L}{v^2}$
$m = \frac{(1200~N)(1.6~m)}{(832~m/s)^2}$
$m = 0.0028~kg$
$m = 2.8~grams$
The total mass of the string is 2.8 grams.
