Physics midterm problem: string standing waves

Physics 205A Midterm 2, fall semester 2009
Cuesta College, San Luis Obispo, CA

Cf. Giambattista/Richardson/Richardson, Physics, 2/e, Problem 11.49

A string is attached with a length of 0.80 m between supports and is stretched by a 4.5 kg hanging mass at one end. A function generator oscillates the string at its fundamental frequency of 170 Hz. Find (a) the linear mass density of the string, and (b) the mass that should hang off of the string such that the same 170 Hz frequency vibrates the n = 3 mode (as shown below). Show your work and explain your reasoning using the properties of wave speeds, periodic waves, and standing waves.


Solution and grading rubric:

• p:
  Correct. Mass of the string is not provided. However, f₁ = 170 Hz, L = 0.80 m, such that v = 272 m/s. With v and tension F = m·g = 44.1 N (where m is the hanging mass, not the string mass), linear mass density = 6.0×10⁴ kg/m. Then with a new situation, f₃ = 170 Hz = 3·f_1,new , and with L and the linear mass density the same as before, then the new hanging mass m_new = 0.50 kg. Or argues that for frequency to remain at 170 Hz, while n increases by a factor of three, the new wave speed must be reduced by a factor of three, such that the tension and the hanging mass must be reduced by a factor of nine.)
  
• r:
  Nearly correct, but includes minor math errors. Correctly finds the linear mass density of the string (or may have omitted a factor of g = 9.80 m/s²), but instead has wave speed increased by a factor of three, and thus the hanging mass increases by a factor of nine, or 40.5 kg (or similar increase).
  
• t:
  Nearly correct, but approach has conceptual errors, and/or major/compounded math errors. Typically finds linear mass density as (4.5 kg)/(0.80 m) = 5.6 kg/m, or confounds mu with mass, velocity with frequency, etc., but still has systematic attempt at finding linear mass density from original n = 1 case, and then feeds (erroneous) linear mass density into the n = 3 case, along with other algebraic or nomenclature errors.
  
• v:
  Implementation of right ideas, but in an inconsistent, incomplete, or unorganized manner. Involves mass-spring or pendulum period equations.
  
• x:
  Implementation of ideas, but credit given for effort rather than merit.
  
• y:
  Irrelevant discussion/effectively blank.
  
• z:
  Blank.
  

Grading distribution:
Sections 70854, 70855
p: 4 students
r: 11 students
t: 23 students
v: 10 students
x: 1 student
y: 0 students
z: 0 students

A sample "p" response (from student 2323):