Physics quiz question: comparing piano string wave speeds

Physics 205A Quiz 6, fall semester 2018
Cuesta College, San Luis Obispo, CA

A D₃ piano string[*] has linear mass density of 7.8 g/m, and is stretched with a tension of 626 N. A thicker A₀ piano string has a linear mass density of 291 g/m, and is stretched with a greater tension of 1,350 N. The __________ has a faster transverse wave speed.
(A) D₃ piano string.
(B) A₀ piano string.
(C) (There is a tie.)
(D) (Not enough information is given.)

[*] A. Stulov, "Physical Modelling of the Piano String Scale," Applied Acoustics, vol. 69 (2008) pp. 977–984, cs.ioc.ee/~stulov/appl08.pdf.

Correct answer (highlight to unhide): (A)

The speed v of transverse waves along the piano string depends on the tension F and the linear mass density (mass per unit length) (m/L):

v = √(F/(m/L)).

For the D₃ piano string (linear mass density is 7.8 g/m = 7.8×10^–3 kg/m) the wave speed is:

v = √((626 N)/(7.8×10^–3 kg/m)) = 283.2956234332... m/s,

or to two significant figures, 2.8×10² m/s.

The A₀ piano string (linear mass density is 291 g/m = 0.291 kg/m) the wave speed is:

v = √((1,350 N)/(0.291 kg/m)) = 68.111491378... m/s,

or to two significant figures, 68 m/s.

Thus the D₃ piano string has a faster transverse wave speed than the A₀ piano string.

Sections 70854, 70855
Exam code: quiz06POr7
(A) : 44 students
(B) : 6 students
(C) : 2 students
(D) : 0 students

Success level: 85%
Discrimination index (Aubrecht & Aubrecht, 1983): 0.42