Physics quiz question: modified Big Ass Fan® rotational speed

Physics 205A Quiz 5, fall semester 2014
Cuesta College, San Luis Obispo, CA

Cf. Giambattista/Richardson/Richardson, Physics, 2/e, Problems 8.5, 8.9

"4900 Silver and Yellow 14 ft. ShopFan"
The Home Depot
homedepot.com/p/Big-Ass-Fans-4900-Silver-and-Yellow-14-ft-ShopFan-F-PF61-1401S34/204316763

The motor of a six-bladed Big Ass Fan® 4900 ceiling fan[*] rotates it at a rate of 10.6 rad/s. Each blade can be approximated as a uniform rod (I_{rod, at end} = (1/3)·M·L²). If three of the six blades were removed (such that three remain) and the motor provides it with the same rotational kinetic energy, the rotation speed would be ________ compared to when it had six blades.
(A) slower.
(B) the same as.
(C) faster.
(D) (Not enough information is given.)

[*] homedepot.com/catalog/pdfImages/ec/ecff159e-ae1c-4085-8960-071af271d74c.pdf

Correct answer (highlight to unhide): (C)

The rotational inertia of a single rod of mass M and length L being swung by its end is:

I_{rod, at end} = (1/3)·M·L².

The unmodified fan has six blades, and will have a total rotational inertia of:

I₆ = 6·I_{rod, at end} = 2·M·L².

Its rotational kinetic energy will be given by:

KE_{rot, 6} = (1/2)·I₆·ω₆².

Removing three (every other) blades from the fan, leaving three remaining will reduce its rotational inertia:

I₃ = 3·I_{rod, at end} = M·L²,

so if the same rotational kinetic energy is supplied to this modified fan, then compared to the unmodified fan:

KE_{rot, 3} = KE_{rot, 6},

(1/2)·I₃·ω₃² = (1/2)·I₆·ω₆²,

M·L²·ω₃² = 2·M·L²·ω₆²,

ω₃ = (2)^1/2·ω₆ = (2)^1/2·(10.6 rad/s) = 14.990663757 rad/s,

such that to three significant figures, the modified three-bladed fan will have a rotational speed of 15.0 rad/s.

Sections 70854, 70855, 73320
Exam code: quiz05mRp4
(A) : 2 students
(B) : 14 students
(C) : 47 students
(D) : 13 students
(No response: 1 student)

Success level: 73%
Discrimination index (Aubrecht & Aubrecht, 1983): 0.24