Physics quiz question: shortening length of guitar string

Physics 205A Quiz 6, fall semester 2014
Cuesta College, San Luis Obispo, CA

Cf. Giambattista/Richardson/Richardson, Physics, 2/e, Conceptual Question 11.4, Multiple-Choice Question 11.5

For a standard guitar the B3 string vibrates at a fundamental frequency of 247 Hz with a tension of 40.5 N. The string is 0.648 m long[*]. A capo is a device that is attached to this guitar to shorten the length of the B3 string, which would then have a fundamental frequency __________ 247 Hz.
(A) lower than.
(B) equal to.
(C) higher than.
(D) (Not enough information is given.)

[*] "Gauge: 8-38 Standard Set," jemsite.com/forums/f21/string-tension-charts-73846.html.

Correct answer (highlight to unhide): (C)

The speed of waves along the string is given by:

v = √(F/(m/L)).

Shortening the length of the string that vibrates would not change the linear mass density m/L (as there would be proportionally less mass for a shorter string length), and since the tension F would also be unchanged, then the speed v of waves that travel along the string remains constant.

The fundamental frequency of a standing wave on a string is given by:

f₁ = v/(2·L),

and since decreasing the length L of the string, while leaving the wave speed v unchanged, would result in an increase in the f₁ frequency.

Sections 70854, 70855
Exam code: quiz06eAg7
(A) : 4 students
(B) : 3 students
(C) : 57 students
(D) : 0 students

Success level: 89%
Discrimination index (Aubrecht & Aubrecht, 1983): 0.33