Cuesta College, San Luis Obispo, CA
Cf. Giambattista/Richardson/Richardson, Physics, 2/e, Problems 8.5, 8.9
The Home Depot
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 (Irod, 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.
(B) the same as.
(D) (Not enough information is given.)
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:
Irod, at end = (1/3)·M·L2.
The unmodified fan has six blades, and will have a total rotational inertia of:
I6-bladed fan = 6·Irod, at end = 2·M·L2.
Its rotational kinetic energy will be given by:
KErot, 6-bladed fan = (1/2)·I6-bladed fan·ω6-bladed fan2.
Removing three (every other) blades from the fan, leaving three remaining will reduce its rotational inertia:
I3-bladed fan = 3·Irod, at end = M·L2,
such that if the same rotational kinetic energy is supplied to this modified fan, then compared to the unmodified fan:
KErot, 3-bladed fan = KErot, 6-bladed fan,
(1/2)·I3-bladed fan·ω3-bladed fan2 = (1/2)·I6-bladed fan·ω6-bladed fan2,
M·L2·ω3-bladed fan2 = 2·M·L2·ω6-bladed fan2,
ω3-bladed fan = (2)1/2·ω6-bladed fan = (2)1/2·(10.6 rad/s) = 14.990663757 rad/s,
such 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