Question

The model of a ceiling fan shown in the figure consists of a uniform solid cylinder, of radius R = 0.067 m and mass MC = 1.8 kg, and two long uniform rods, each of length L = 0.94 m and mass MR = 3.4 kg, that are attached to the cylinder and extend from its center. Ignore the vertical rod that connects the fan to the motor.

(a) Enter an expression, in terms of the quantities defined in the problem, for the moment of inertia of each rod about the rotation axis.
(b) Enter an expression, in terms of the quantities defined in the problem, for the moment of inertia of the cylinder about the rotation axis.
(c) Enter an expression, in terms of the quantities defined in the problem, for the moment of inertia of the whole fan about the rotation axis.
(d) Calculate the moment of inertia, in units of kilogram meters squared, of the whole fan about the rotation axis.

Answer 1

The moment of inertia of a rod about an axis through one end perpendicular to its length is Me^2/3. The moment of inertia of a solid cylinder about its central axis is 1/2MR^2. The moment of inertia of the whole fan is the sum of the moment of inertia of each rod and the cylinder.

Step-by-step explanation:

(a) The moment of inertia of a rod about an axis through one end perpendicular to its length is given by the equation Me^2/3, where M is the mass of the rod and e is the length of the rod.

In this case, the moment of inertia of each rod is MR*(L/2)^2/3.

(b) The moment of inertia of a solid cylinder about its central axis is given by the equation 1/2MR^2, where M is the mass of the cylinder and R is the radius. In this case, the moment of inertia of the cylinder is 1/2MC*R^2.

(c) The moment of inertia of the whole fan can be calculated by adding the moment of inertia of each rod and the cylinder. In this case, the moment of inertia of the whole fan is 2*(MR*(L/2)^2/3) + 1/2MC*R^2.

(d) To calculate the moment of inertia of the fan, substitute the given values into the expressions obtained in parts (c).