Physics final exam question: longer string, more tension fundamental standing wave

Physics 205A Final Exam, fall semester 2016
Cuesta College, San Luis Obispo, CA

A Physics 205A student builds a standing wave experiment with a mass hanging over a pulley to create tension in a string, which has a fundamental frequency of 20 Hz. The length of string used is increased by a factor of two, and the amount of mass hanging off of the string is also increased by a factor of two. (Ignore stretching in the string.) Discuss why the longer string with a greater hanging mass will have a fundamental frequency lower than 20 Hz. Explain your reasoning using the properties of wave speeds, and standing waves.

Solution and grading rubric:

• p:
  Correct. Understands that:
  1. the fundamental standing wave frequency f₁ depends on the wave speed v (set by the tension) and the physical length L of string between the ends; and
  2. the hanging mass increases by a factor of two, increasing the tension F in the string by a factor of, which increases the wave speed v by a factor of √2, and thus increases the fundamental standing wave frequency f₁ by a factor of √2; and
  3. doubling the length L between the ends halves the fundamental standing wave frequency f₁; such that the overall change in the fundamental standing wave frequency f₁ will be such that it is lower, by a factor of (√2)/2, or 0.70 times the original value of f₁.
• r:
  As (p), but argument indirectly, weakly, or only by definition supports the statement to be proven, or has minor inconsistencies or loopholes.
• t:
  Nearly correct, but argument has conceptual errors, or is incomplete. Only recognizes one of the arguments (2)-(3) as affecting f₁.
• v:
  Limited relevant discussion of supporting evidence of at least some merit, but in an inconsistent or unclear manner. Some garbled attempt at applying standing wave frequency parameters.
• x:
  Implementation/application of ideas, but credit given for effort rather than merit. Approach other than applying standing wave frequency parameters.
• y:
  Irrelevant discussion/effectively blank.
• z:
  Blank.
  

Grading distribution:
Sections 70854, 70855, 73320
Exam code: finali0w4
p: 11 students
r: 3 students
t: 27 students
v: 7 students
x: 0 students
y: 0 students
z: 1 student

A sample "p" response (from student 0424):