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 0.270 rev/s . The magnitude of the angular acceleration is 0.885 rev/s2 . Both the the angular velocity and angular accleration are directed clockwise. The electric ceiling fan blades form a circle of diameter 0.740 m .

Part A

Compute the fan's angular velocity magnitude after time 0.201 s has passed.

Part B

Through how many revolutions has the blade turned in the time interval 0.201 s from Part A?

Part C

What is the tangential speed vtan(t) of a point on the tip of the blade at time t = 0.201 s ?

Part D

What is the magnitude a of the resultant acceleration of a point on the tip of the blade at time t = 0.201 s ?

Explanation / Answer

1 rev = 2 * pi radians

0.27 rev = 0.27 * 2 * pi radians

so,

initial velocity = 0.27 * 2 * pi radian/sec

angular acceleration = 0.885 * 2 * pi rad / sec^2

v = u + at

v = 0.27 * 2 * pi +  0.885 * 2 * pi * 0.201

v = 2.814 rad/sec

angular velocity after 0.201 sec = 2.814 rad/sec

s = ut + 0.5 * at^2

s = 0.27 * 2 * pi * 0.201 + 0.5 * 0.885 * 2 * pi * 0.201^2

s = 0.4533 radian or 0.4533 / (2 * pi) rev

revolutions made in 0.201 sec = 0.072 rev

tangential speed = angular speed * radius

tangential speed = 2.814 * 0.74 / 2

tangential speed = 1.04 m/s

resultant acceleration = sqrt(radial acceleration^2 + tangential acceleration)

resultant acceleration = sqrt((1.04^2 / 0.37)^2 + (0.885 * 2 * pi * 0.37)^2)

resultant acceleration = 3.574 m/s^2

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