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 0.260 rev/s . The magnitude of the angular acceleration is 0.881 rev/s2 . Both the the angular velocity and angular accleration are directed clockwise. The electric ceiling fan blades form a circle of diameter 0.750 m .

What is the tangential speed vtan(t) of a point on the tip of the blade at time t = 0.203 s ?

What is the magnitude a of the resultant acceleration of a point on the tip of the blade at time t = 0.203 s ?

Explanation / Answer

initial angular velocity , w0 = 0.260 rev/sec                                             ( w = theta /t,   theta = 2*pie --- in 1 rev)

                                        = 0.260*2*pie = 1.632 rad/sec                        ( pie= 3.14)

@ ( angular acceleration) = 0.881 rev / s2

                                   = 0.881*2*3.14 = 5.53 rad/s2

w ( final angular velocity) = w0 + @ *t

w = 1.632 + 5.53*0.203

w = 2.75 rad/s

a) tangential velocity, Vt = w*r                                    ( r = radius = 0.750/2 m)

                                    = 2.75* 0.750 / 2

                                    = 1.031 m/s

b) resultant acceleration ( aR ) will be the vector sum of centripetal acceleration ( ac ) and tangential acceleration ( at )

ac = w2 * r

      = 2.752 * 0.750 / 2

      = 2.835 m/s2

at = @*r

     = 5.53* 0.750/2

    = 2.073 m/s2

aR = ( ac2 + at2 ) 1/2

    = 3.51 m/s2

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