An electric ceiling fan is rotating about a fixed axis with an initial angular v

Question

An electric ceiling fan is rotating about a fixed axis with an initial angular velocity magnitude of 0.230rev/s . The magnitude of the angular acceleration is 0.893rev/s^2 . Both the the angular velocity and angular accleration are directed clockwise. The electric ceiling fan blades form a circle of diameter 0.790m . What is the tangential speed v_tan(t) of a point on the tip of the blade at time t = 0.204s ? Express your answer numerically in meters per second. What is the magnitude a of the resultant acceleration of a point on the tip of the blade at time t = 0.204s ? Express the acceleration numerically in meters per second squared. Through how many revolutions has the blade turned in the time interval 0.204s from Part A? Express the number of revolutions numerically.

Explanation / Answer

w = -2*3.1416*0.230 - 0.893*2*3.1416 * 0.204 = -2.5898

v = R w = 0.790/2 * 2.5898 = 1.02 m/s

What is the magnitude a of the resultant acceleration of a point on the tip of the blade at time t = 0.204s ? Express the acceleration numerically in meters per second squared.

ar = v^2/R = 1.022971*1.022971/(0.790/2) = 2.64929

at = R = 0.790/2 * 0.893*2*3.1416 = 2.216304552

a = 3.45 m/s2

Through how many revolutions has the blade turned in the time interval 0.204s from Part A? Express the number of revolutions numerically.

= -0.5 * 0.893*2*3.1416 * 0.204*0.204 - 2*3.1416*0.230 * 0.204

= -0.4115593012608

= 0.0655 revolutions

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