Answer
(a) The finger should be placed a distance of 45.0 cm from the bridge.
(b) We would not be able to play a G4 note (392 Hz) on this string.
Work Step by Step
(a) We can write an expression for the frequency of A4.
$f_A = \frac{v}{2L}$
Let $x$ be the distance from the bridge to the finger. We can write an expression for the frequency of D5.
$f_D = \frac{v}{2x}$
We can divide the first equation by the second equation.
$\frac{f_A}{f_D} = \frac{\frac{v}{2L}}{\frac{v}{2x}}$
$x = \frac{f_A~L}{f_D}$
$x = \frac{(440~Hz)(60.0~cm)}{587~Hz}$
$x = 45.0~cm$
The finger should be placed a distance of 45.0 cm from the bridge.
(b) Without re-tuning, the lowest frequency we can play on this string is 440 Hz. If we place our finger anywhere on the string, the frequency can only increase. Since we can not play lower frequencies than 440 Hz, we would not be able to play a G4 note (392 Hz) on this string.
