Physics midterm question: decreasing the fundamental frequency

Physics 205A Midterm 2, fall semester 2011
Cuesta College, San Luis Obispo, CA

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

A cello's C-string[*],[**] has a length of 0.70 m and is stretched to a tension of 140 N. It vibrates at a fundamental frequency of 65.4 Hz. __________ would decrease the fundamental frequency of this cello string.
(A) Pressing a finger on the string, such that a shorter length of it vibrates.
(B) Increasing the tension of the string.
(C) (Either of the above choices.)
(D) (Neither of the above choices.)

[*] cello.org/Newsletter/Articles/tipbook/tipbook.htm
[**] rdebey.com/string_tension.htm.

Correct answer (highlight to unhide): (D)

The speed of waves along the string is given by:

v = √(F/(m/L)).

Using a shorter length of string would not change the tension F, and would neither change the linear mass density (m/L)--while L would decrease, the mass m would also proportionally decrease as well, keeping the ratio (m/L) constant. Thus the speed v would remain unchanged.

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

f₁ = v/(2·L).

Then using a shorter length of string would increase the fundamental frequency f₁ (as the speed v would remain constant).

As for increasing the tension in the string, since:

v = √(F/(m/L)),

the linear mass density (m/L) would remain constant, such that the speed v would increase. Then for the fundamental frequency of a standing wave on a string:

f₁ = v/(2·L),

increasing the speed v would then increase the fundamental frequency f₁ (as L would remain constant).

Thus neither pressing a finger on the string (such that a shorter length of it vibrates), nor increasing the tension of the string would decrease the fundamental frequency f₁.

Sections 70854, 70855
Exam code: midterm02fR3q
(A) : 7 students
(B) : 7 students
(C) : 9 students
(D) : 30 students

Success level: 55%
Discrimination index (Aubrecht & Aubrecht, 1983): 0.22