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 0.230 rev/s . The magnitude of the angular acceleration is 0.906 rev/s2 . Both the the angular velocity and angular accleration are directed clockwise. The electric ceiling fan blades form a circle of diameter 0.770 m .

Compute the fan's angular velocity magnitude after time 0.200 s has passed.

Through how many revolutions has the blade turned in the time interval 0.200 s from Part A?

What is the tangential speed vtan(t) of a point on the tip of the blade at time t = 0.200 s ?

What is the magnitude a of the resultant acceleration of a point on the tip of the blade at time t = 0.200 s ?

Explanation / Answer

w1 = 0.23 rev/s = 0.23*2*pi rad/s = 1.444 rad/s

alfa = 0.906 rev/s^2

d = 0.77 m

r = d/2 = 0.77/2= 0.385 m

A) w2 = w1 + alfa*t

= 0.23 + 0.906*0.2

= 0.4112 rev/s or 2.58 rad/s

B)angular displacemnt, theta = (w2^2 - w1^2)/(2*alfa)

= (0.4112^2 - 0.23^2)/(2*0.906)

= 0.064 revolution

C) v_tan = r*w2

= 0.385*2.58

= 0.9933 m/s

d) a_tan = r*alfa

= 0.385*0.906*2*pi rad/s^2

= 2.2 m/s^2

a_rad = r*w2^2

= 0.385*2.58^2

= 2.56 rad/s

a_total = sqrt(a_tan^2 + a_rad^2)

= sqrt(2.2^2 + 2.56^2)

= 3.38 m/s^2

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