Physics quiz question: standing wave string tension

Physics 5A Quiz 6, Fall Semester 2007
Cuesta College, San Luis Obispo, CA

Cf. Giambattista/Richardson/Richardson, Physics, 1/e, Problem 11.48

[3.0 points.] Consider a guitar's A-string of a length 0.80 m and is stretched to a tension of 40 N. It vibrates at a fundamental frequency of 220 Hz. If the tension in the string is increased by a factor of 2.0, what happens to the fundamental frequency of this string?
(A) It increases by a factor of 1.4.
(B) It increases by a factor of 2.0.
(C) It increases by a factor of 4.0.
(D) (The fundamental frequency remains the same.)

Correct answer: (A)

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

f_1 = v/(2*L),

where the speed of waves along the string depends on the tension F and linear mass density mu:

v = sqrt(F/mu).

Increasing the tension F by a factor of 2.0 increases the wave speed by a factor of sqrt(2.0) = 1.4, which increases the fundamental frequency of the string by that same 1.4 factor.

Student responses
Sections 0906, 0907
(A) : 7 students
(B) : 7 students
(C) : 3 students
(D) : 4 students