The portion of the string of a certain musical instrument between the bridge and upper end of the finger board (that part of the string that is free to vibrate) is 60.0 cm long, and this length of the string has mass 2.00 g. The string sounds an A4 note (440 Hz) when played. (a) Where must the player put a finger (what distance x from the bridge) to play a D 5 note (587 Hz)? (See ?Fig. E15.45?.) For both the A4 and D5 notes, the string vibrates in its fundamental mode. (b) Without retuning, is it possible to play a G4 note (392 Hz) on this string? Why or why not?
Solution 47E Introduction The length of the string to produce the A4 note is given, we have to calculate the required length to produce the D5 note. Step 1 Let us consider that the speed of the wave in the string is v . Now the frequency of A4 node is f = 440 Hz and the frequency of the D5 note is f = 587 Hz. 1 2 Now let us consider that the wavelength of the A4 and D5 notes are and1 . Henc2 the speed is given by v = f 1 1 f 2 2.(1) Now, the length of the string for L =160 cm and the length of the string for D5 note is L . We2 also know that, for the fundamental note, the wavelength is given by 1 2L a1d = 22 2 Hence from the equation (1) we can write that 2L f = 2L f 1 1L f 2(440 Hz)(60 cm) L 2 f 1= (587 Hz) 45.0 Hz 2 So the player must put the finger at 45.0 cm away.