Physics quiz question: dimensional analysis of pendulum period

Physics 205A Quiz 1, Fall Semester 2008
Cuesta College, San Luis Obispo, CA

Cf. Giambattista/Richardson/Richardson, Physics, 1/e, Problem 1.30

[Version 1]

[3.0 points.] Using dimensional analysis, how does the period T of a pendulum (measured in seconds) depend on some, or all of the following properties: m is the mass of the pendulum bob, L is the length of the pendulum string, and g is the gravitational field strength (measured in units of m/s^2)?
(A) T = L^(1/2) * g^(-1/2).
(B) T = m^(1/2) * L^(-1/2).
(C) T = g^(1/2) * m^(-1/2).
(D) T = g^(1/2) * L^(1/2) * m^(-1/2).

Correct answer: (A)

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

T = m^(p) * L^(q) * g^(r),

then substituting in their respective units and gathering like terms (here m is for meters, and not for mass):

s = kg^(p) * m^(q) * (m/s^2)^(r);
s = kg^(p) * m^(q+r) * s^(-2*r).

By inspection, p = 0, as there are no kilogram units on the left side of the equation. Thus r = -1/2, in order to match the single unit of seconds on the left side of the equation with the units of seconds raised to the -2*r power on the right side. Thus for the sole remaining meters unit, 0 = q + r, where r = -1/2 gives q = 1/2, giving the completed result T = L^(1/2) * m^(-1/2).

Student responses
Sections 70854, 70855
(A) : 9 students
(B) : 5 students
(C) : 6 students
(D) : 6 students

[Version 2]

[3.0 points.] Using dimensional analysis, how does the period T of a pendulum (measured in seconds) depend on some, or all of the following properties: m is the mass of the pendulum bob, L is the length of the pendulum string, and g is the gravitational field strength (measured in units of m/s^2)?
(A) T = g^(1/2) * L^(1/2) * m^(-1/2).
(B) T = g^(1/2) * m^(-1/2).
(C) T = m^(1/2) * L^(-1/2).
(D) T = L^(1/2) * g^(-1/2).

Correct answer: (D)

Student responses
Sections 70854, 70855
(A) : 11 students
(B) : 3 students
(C) : 3 students
(D) : 7 students

"Difficulty level": 30%
Discrimination index (Aubrecht & Aubrecht, 1983): 0.7