An approximate model for a ceiling fan consists of a cylindrical disk with four

Question

An approximate model for a ceiling fan consists of a cylindrical disk with four thin rods extending from the disk's center, as in the figure below. The disk has mass 2.90 kg and radius 0.200 m. Each rod has mass 0.850 kg and is 0.800 m long.

A.

Find the ceiling fan's moment of inertia about a vertical axis through the disk's center. (Enter your answer in kg · m2.)

kg · m2

B.

Friction exerts a constant torque of magnitude 0.111 N · m on the fan as it rotates. Find the magnitude of the constant torque provided by the fan's motor if the fan starts from rest and takes 15.0 s and 17.5 full revolutions to reach its maximum speed. (Enter your answer in N · m.)

N · m

Explanation / Answer

here,

mass of disc , m1 = 2.9 kg

mass of each rod , m2 = 0.85 kg

radius , r = 0.8 m

length of each rod , l = 0.8 m

a)

the ceiling fan's moment of inertia about a vertical axis through the disk's center , I = 0.5 * m1 * r^2 + 4 * m2 * l^2 /3

I = 0.5 * 2.9 * 0.8^2 + 4 * 0.85 * 0.8^2 /3 kg.m^2

I = 1.65 kg.m^2

b)

time , t = 15 s

theta = 17 * 2 * pi = 106.76 rad

let the angular accelation be alpha

theta = 0 + 0.5 * alpha * t^2

106.76 = 0.5 * alpha * 15^2

alpha = 0.95 rad/s^2

torque due to friction, T1 = 0.111 N.m

the magnitude of the constant torque provided by the fan's motor , T = T1 + alpha * I

T = 0.111 + 0.95 * 1.65 N.m

T = 1.68 N.m

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