Cuesta College, San Luis Obispo, CA
Cf. Giambattista/Richardson/Richardson, Physics, 2/e, Problem 8.5
(Version 1)
[3.0 points.] Four uniform rods with identical masses, and equal lengths are arranged as a long two-bladed fan, or as a shorter four-bladed fan, as shown at right. Which fan blade arrangement has the largest rotational inertia? (A) The two-bladed fan.
(B) The four-bladed fan.
(C) (There is a tie.)
(D) (Not enough information is given to determine this.)
Correct answer: (A)
For a thin, uniform rod of mass M and length L, students were given that the rotational inertia about an axis through its center, perpendicular to the length is given by Icenter = (1/12)·M·L2; while the rotational inertia about an axis at one end, perpendicular to the length is given by Iend = (1/3)·M·L2.
If each individual rod segment has mass m and length l, then the rotational inertia of the two-bladed fan is:
Itwo-blade = 2·Iend (where M = 2·m, L = 2·l) = 2·(1/3)·(2·m)·(2·l)2 = (16/3)·m·l2.
This is larger than the rotational inertia of the four-bladed fan:
Ifour-blade = 4·Iend (where M = m, L = l) = 4·(1/3)·m·l2 = (4/3)·m·l2.
(The four-bladed fan rotational inertia could also be derived from:
Ifour-blade = 2·Icenter (where M = 2·m, L = 2·l) = 2·(1/12)·(2·m)·(2·l)2 = (4/3)·m·l2.)
Student responses
Sections 30880, 30881
(A) : 13 students
(B) : 1 student
(C) : 6 students
(D) : 0 students
(Version 2)
[3.0 points.] Four uniform rods with identical masses, and equal lengths are arranged as a long two-bladed fan, or as a shorter four-bladed fan, as shown at right. Which fan blade arrangement has the smallest rotational inertia?
(A) The two-bladed fan.
(B) The four-bladed fan.
(C) (There is a tie.)
(D) (Not enough information is given to determine this.)
Correct answer: (B)
Student responses
Sections 30880, 30881
(A) : 5 students
(B) : 12 students
(C) : 3 students
(D) : 0 students
"Difficulty level": 64%
Discrimination index (Aubrecht & Aubrecht, 1983): 0.29
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