Physics final exam problem: changing string tension and standing wave mode

Physics 205A Final Exam, fall semester 2011
Cuesta College, San Luis Obispo, CA

Cf. Giambattista/Richardson/Richardson, Physics, 2/e, Problem 11.49, Review & Synthesis 14 (p. 454).

[20 points.] A string with a length of 0.80 m between supports is stretched by a 2.3 kg mass hanging from one end. A device oscillates the n = 3 mode at 84 Hz. 




Find the mass that should hang off of this string such that the same 84 Hz frequency vibrates the n = 2 mode. Show your work and explain your reasoning.

Solution and grading rubric:

• p = 20/20:
  Correct. Mass of the string is not provided. However:
  
  f₃ = 84 Hz = ((3/(2*L))*sqrt(m*g/µ),
  
  where m = 2.3 kg, µ is the linear mass density of the string; and:
  
  f₂' = 84 Hz = ((2/(2*L))*sqrt(m'*g/µ),
  
  where m' is the new mass that hangs off of the string when vibrating in the n = 2 mode. Equating:
  
  f₂' = f₃,
  
  and cancelling common factors results in:
  
  m' = m*(3/2)^2 = 5.2 kg.
  
  And/or may have solved for quantities such as wave speed along each string, linear mass density, etc., in intermediate steps.
  
• r = 16/20:
  Nearly correct, but includes minor math errors.
  
• t = 12/20:
  Nearly correct, but approach has conceptual errors, and/or major/compounded math errors. Typically calculates linear mass density µ as (2.3 kg)/(0.80 m) = 5.6 kg/m, and/or confounds µ with mass m, tension force F with frequency f, etc., but still has systematic attempt at finding linear mass density from original n = 3 case, and then feeds (erroneous) µ and/or other factors into the n = 2 case, along with other algebraic or nomenclature errors.
  
• v = 8/20:
  Implementation of right ideas, but in an inconsistent, incomplete, or unorganized manner. Some attempt at manipulating standing wave frequency equations.
  
• x = 4/20:
  Implementation of ideas, but credit given for effort rather than merit.
  
• y = 2/20:
  Irrelevant discussion/effectively blank.
  
• z = 0/20:
  Blank.
  

Grading distribution:
Sections 70854, 70855
Exam code: finalB0xs
p: 9 students
r: 5 students
t: 23 students
v: 6 students
x: 1 student
y: 2 students
z: 2 students

A sample "p" response (from student 1124):
A sample "t" response (from student 3389):
Kudos (from student 0415):