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 0.898 rev/s2 . Both the the angular velocity and angular accleration are directed clockwise. The electric ceiling fan blades form a circle of diameter 0.800 m .

a) Compute the fan's angular velocity magnitude after time 0.193 s has passed.

Express your answer numerically in revolutions per second.

b)

Through how many revolutions has the blade turned in the time interval 0.193 sfrom Part A?

Express the number of revolutions numerically.

c)

What is the tangential speed vtan(t) of a point on the tip of the blade at time t = 0.193 s ?

Express your answer numerically in meters per second.

d)

What is the magnitude a of the resultant acceleration of a point on the tip of the blade at time t = 0.193 s ?

Express the acceleration numerically in meters per second squared.

Explanation / Answer

Wi is the initial angular velocity - 0.24 rev/s = 0.24*2*3.142 = 1.508 rad/s

alpha is the angular accelaration = 0.898 rev/s^2 = 5.64 rad/s^2

A) apply wf = wi + (alpha*t)

wf = (1.508)+(5.64*0.193) = 2.6 rad/s = 2.6/(2*pi) = 0.413 rev/s

B) apply theta = (wi*t)+(0.5*alpha*t^2)= (1.508*0.193)+(0.5*5.64*0.193*0.193) = 0.396 rad = 0.063 rev

C) v = r*wf = (0.8/2)*2.6 = 1.04 m/s

D) a = sqrt(ar^2+at^2)

at = (0.8/2)*5.64 = 2.256 m/s

ar = v^2/r = 1.04^2/0.4 =2.704 m/s

a = sqrt(2.256^2+2.704^2) = 3.52 m/s

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