Physics clicker question: standing wave on tightened string

Physics 205A, Spring Semester 2009
Cuesta College, San Luis Obispo, CA

Cf. Giambattista/Richardson/Richardson, Physics, 2/e, Problem 11.46

Students were asked the following clicker question (Classroom Performance System, einstruction.com) at the end of their learning cycle:

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, the fundamental frequency of this string increases by a factor of:
(A) 1.4.
(B) 2.0.
(C) 4.0.
(D) (The fundamental frequency remains the same.)
(E) (I'm lost, and don't know how to answer this.)

Sections 30880, 30881
(A) : 19 students
(B) : 7 students
(C) : 4 students
(D) : 0 students
(E) : 1 student

This question was asked again after displaying the tallied results with the lack of consensus, with the following results. No comments were made by the instructor, in order to see if students were going to be able to discuss and determine the correct answer among themselves.

Sections 30880, 30881
(A) : 21 students
(B) : 1 student
(C) : 3 students
(D) : 0 students
(E) : 0 students

Correct answer: (A)

The speed v of transverse waves along a string is:

v = sqrt(F/mu),

where F is the tension in the string, while mu is the linear mass density (mass per unit length) of the string.

The fundamental frequency f_1 for a standing wave on a string is given by:

f_1 = v/(2*L),

where L is the length of the rope. Thus doubling the tension F would increase the wave speed v by a factor of sqrt(2) = 1.4, and thus the fundamental frequency f_1 would also increase by the same 1.4 factor.

Pre- to post- peer-interaction gains:
pre-interaction correct = 61%
post-interaction correct = 84%
Hake, or normalized gain = 59%