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 0.230rev/s . The magnitude of the angular acceleration is 0.903rev/s . Both the the angular velocity and angular acceleration are directed clockwise. The electric ceiling fan blades form a circle of diameter 0.790m. Compute the fan's angular velocity magnitude after time 0.210s has passed Through how many revolutions has the blade turned in the time interval 0.210s from Part A? What is the tangential speed vtan(t) of a point on the tip of the blade at time = 0.210s? What is the magnitude a if the resultant acceleration of a point on the tip of the blade at time = 0.210s ? Compute the fan's angular velocity magnitude after time 0.210s has passed Compute the fan's angular velocity magnitude after time 0.210s has passed Compute the fan's angular velocity magnitude after time 0.210s has passed Compute the fan's angular velocity magnitude after time 0.210s has passed

Explanation / Answer

A) The angular velocity w = wo+ a*t where w0=0.23*2*Pi rad./s
also where 'a' = 0.903*2*Pi rad./s^2
Thus at t=0.210,

w(0.210) = 0.23*2*Pi + 0.903*2*Pi*0.210

=1.444 + 1.1908 = 2.634 rad/sec = 0.42 rev/sec

B) Let angular displacement be 'd'
d(t) = integral {w0 + a*t}dt between 0 and 0.192 s
=w0*t+a*(t^2)/2+c
d(0)=0=c
d(0.231)= 0.23*2*Pi*0.210 + 0.903*2*Pi/2*0.210^2
=0.303+0.125 radians.

=0.428 radians

This is 0.428/(2*Pi)=6.8*10^(-2) revolutions.

C) tangential speed = w*r = 2.634*0.79/2=1.04 m/s (diameter = 0.79,radius = 0.395)

D) resultant accn = sqrt((centripetal accn)^2 + (tangential accn)^2)
=sqrt((w^2*r)^2 + (a*r)^2)
=sqrt((2.634^2*0.39)^2 + (0.903*2*Pi*0.39)^2)

=3.539 m/s^2

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