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 0.240 rev/s . The magnitude of the angular acceleration is 0.893 rev/s2 . Both the the angular velocity and angular accleration are directed clockwise. The electric ceiling fan blades form a circle of diameter 0.790 m .

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

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

C)What is the tangential speed vtan(t) of a point on the tip of the blade at time t = 0.207 s ?(in m/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 ?(in m/s^2)

Explanation / Answer

given

w1 = 0.24 rev/s

alfa = 0.893 rev/s^2

A) angular velcoity at t = 0.207s

w2 = w1 + alfa*t

= 0.24 + 0.893*0.207

= 0.4248 rad/s

B) theta = (w2^2 - w1^2)/(alfa)

= (0.4248^2 - 0.24^2)/(2*0.893)

= 0.06878 rev

C) tangential speed at a point on the rim, v = r*w

= (0.79/2)*0.4248

= 0.168 m/s

D) alfa = 0.893 rev/s^2

= 0.893*2*pi rad/s^2

= 5.61 rad/s^2

a_tan = r*alfa

= (0.79/2)*5.61

= 2.22 m/s^2

a_rad = r*w2^2

= (0.79/2)*0.4248^2

= 0.071 m/s^2

total acceleration, a_total = sqrt(a_tan^2 + a_rad^2)

= sqrt(2.22^2 + 0.071^2)

= 2.22 m/s^2

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