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 0.200rev/s . The magnitude of the angular acceleration is 0.893rev/s2 . Both the the angular velocity and angular accleration are directed clockwise. The electric ceiling fan blades form a circle of diameter 0.750m .

Part A:
Compute the fan's angular velocity magnitude after time 0.192s has passed.
Express your answer numerically in revolutions per second.

Part B:

Through how many revolutions has the blade turned in the time interval 0.192s from Part A?

Express the number of revolutions numerically.

Part C:

What is the tangential speed vtan(t) of a point on the tip of the blade at time t = 0.192s ?

Express your answer numerically in meters per second.

Part D:

What is the magnitude a of the resultant acceleration of a point on the tip of the blade at time t = 0.192s ?

Express the acceleration numerically in meters per second squared.

Explanation / Answer

(a)
angular velocity, ? = ?o + ?t = 0.200 + 0.893 * 0.192
angular velocity, ? = 0.371 rad/s

(b)
Revolutions, ? = ?o + ?o * t + 1/2 * ?t2
Revolutions, ? = 0 + 0.200 * 0.192 + 1/2 * 0.893 * 0.1922 = 0.0564 rev
Revolutions, ? = 0.0384 + 0.01646
Revolutions, ? = 0.055 rev

(c)
speed, v = r * ? = 0.375 m * 0.371
v = 0.139 m/s

(d)
tangential acceleration, atan = 0.893 * 0.375 = 0.335 m/s^2
Radial accerelation, aradial = r*?^2 = 0.375 * 0.3712 = 0.0516 m/s^2

Resultant, a = sqrt(0.3352 + 0.05162)
a = 0.339 m/s2

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