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 0.760 m .

A) Compute the fan's angular velocity magnitude after time 0.207 s has passed.

B) Through how many revolutions has the blade turned in the time interval 0.207 s from Part A?

C)What is the tangential speed vtan(t) of a point on the tip of the blade at time t = 0.207 s ?

D)What is the magnitude a of the resultant acceleration of a point on the tip of the blade at time t = 0.207 s ?

Explanation / Answer

Here

angular speed , wi = 0.270 rev/s

angular accelerartion , a = 0.885 rev/s^2

radius ,r = 0.760/2 m = 0.38 m

a) after t = 0.207 s

final angular speed = 0.270 + 0.885 * 0.207

final angular speed = 0.453 rev/s

B)

let the angle is theta

theta = wi * t + 0.5 * a * t^2

theta = 0.270 * 0.207 + 0.5 * 0.885 * 0.207^2

theta = 0.0749 rev

C)

tangential speed = wi * r

tangential speed = 0.207 * 2pi * 0.38 m

tangential speed = 0.494 m/s

D)

tangential acceleration , at = 0.494^2/(0.38)

at = 0.643 m/s^2
magnitude of resultant accelertion = sqrt(0.64^2 + (0.885 * 2pi * 0.38)^2)

magnitude of resultant accelertion = 2.21 m/s^2

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