Physics quiz question: Planck length expression

Physics 5A (currently Physics 205A) Quiz 1, fall semester 2007
Cuesta College, San Luis Obispo, CA

Cf. Giambattista/Richardson/Richardson, Physics, 1/e, Comprehensive Problem 1.73

Three of the fundamental constants of physics are:

      The universal gravitational constant, G = 6.7×10-11 m3·kg-1·s-2.
      The speed of light, c = 3.0×108 m·s–1.
      Planck's constant, h = 6.6×10-34 m2·kg·s-1.

What is the combination of these three constants that has the dimensions of length? This length is called the Planck length and represents the scale at which quantum mechanics becomes important.
(A) G(3/2) × c(1/2) × h(-1/2).
(B) G(1/2) × c(3/2) × h(-1/2).
(C) G(3/2) × c(-1/2) × h(1/2).
(D) G(1/2) × c(-3/2) × h(1/2).

Correct answer (highlight to unhide): (D)

The Planck length is in units of m, thus all units of kg and s must cancel out. The speed of light c has units of m/s, and notably does not have units of kg. The universal gravitational constant G has units of kg-1, and h has units of kg, so they must be raised to the same power, in order for the kg units to cancel out. This eliminates choices (A), (B), and (C) as G and h are not raised to the same power, and thus the only remaining choice (D) (where G and h have are raised to the same power) must be the correct answer.

Student responses
Sections 0906, 0907
(A) : 5 students
(B) : 14 students
(C) : 13 students
(D) : 9 students

Success level: 24%
Discrimination index (Aubrecht & Aubrecht, 1983): 0.19

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