Physics quiz question: dimensional analysis of mass on spring

Physics 205A Quiz 1, fall semester 2011
Cuesta College, San Luis Obispo, CA

Cf. Giambattista/Richardson/Richardson, Physics, 2/e, Comprehensive Problem 1.85

Consider three physical quantities:

Frequency f, measured in units of s^–1.
Mass m, measured in units of kg.
Spring constant k, measured in units of kg/s².

What is the dependence of f on m and k?
(A) f ∝ m^-3/2*k^-1/2.
(B) f ∝ m^-1/2*k^1/2.
(C) f ∝ m^1/2*k^1/2.
(D) f ∝ m^1/2*k^3/2.

Correct answer: (B)

Starting with a prototype equation with arbitrary exponents p and q:

f ∝ m^p * k^q,

then substituting in their respective units and gathering like terms:

(s)^-1 = (kg)^p * (kg/s²)^q,
s^-1 = kg^(p+q) * s^-2q.

Reading off exponents for each units:

[s]: -1 = -2q,
[kg]: 0 = p + q,

then q = (1/2), and p = (-1/2).

Sections 70854, 70855
Exam code: quiz01unI7
(A) : 24 students
(B) : 7 students
(C) : 13 student
(D) : 13 students

Success level: 40%
Discrimination index (Aubrecht & Aubrecht, 1983): 0.51