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 0.210 rev/s . The magnitude of the angular acceleration is 0.887 rev/s2 . Both the the angular velocity and angular accleration are directed clockwise. The electric ceiling fan blades form a circle of diameter 0.700 m .

A.Compute the fan's angular velocity magnitude after time 0.202 s has passed.

B.Through how many revolutions has the blade turned in the time interval 0.202 s from Part A?

C.What is the tangential speed vtan(t) of a point on the tip of the blade at time t = 0.202 s ?

D.What is the magnitude a of the resultant acceleration of a point on the tip of the blade at time t = 0.202 s ?

Explanation / Answer

initial angular velocity wi = 0.210 x 2pi rad / s = 1.32 rad/s

angular acceleration = alpha = 0.887 x2pi rad/ s^2 = 5.57 rad/s^2

A) wf = wi + alpha*t

wf = ( 1.32 ) + ( 5.57 x 0.202) = 2.45 rad/s

b) ang;e rotated = wi *t + alpha*t^2 /2

      angle = (1.32 x 0.202 ) + ( 5.57 x 0.202^2 ) /2   = 0.380 rad

revolutions = 0.380 / 2pi =0.06 revolutions.

c) v = w* r = 2.45 x (0.700/2 ) = 0.858 m/s

d) tangential acc. a_t = alpha*r = 5.57 x 0.700/2 = 1.95 m/s^2

centripetal acc. a_c = w^2 r = 2.45^2 x (0.700/2) = 2.10 m/s^2

total acc. = sqrt( a_c^2 + a_t^2) = 2.87 m/s^2

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