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 0.300rev/s . The magnitude of the angular acceleration is 0.905rev/s2 . Both the the angular velocity and angular accleration are directed clockwise. The electric ceiling fan blades form a circle of diameter 0.720m .

Compute the fan's angular velocity magnitude after time 0.208s has passed.

express your answer in rev/s

Through how many revolutions has the blade turned in the time interval 0.208s from Part A?

express your answer in rev

What is the tangential speed vtan(t) of a point on the tip of the blade at time t = 0.208s ?

express your answer in m/s

What is the magnitude a of the resultant acceleration of a point on the tip of the blade at time t = 0.208s ?

express your answer in m/s^2

Explanation / Answer

A) The angular velocity w = w0 + a*t where w0=0.3*2*Pi rad./s

also where 'a' = 0.905*2*Pi rad./s^2

Thus at t=0.208, w(0.208) = 0.3*2*Pi + 0.905*2*Pi*0.208

=3.067 rad/s =0.488 rev/s


B) Let angular displacement be 'd'

d(t) = integral {w0 + a*t}dt between 0 and 0.208 s

=w0*t+a*(t^2)/2+c

d(0)=0=c

d(0.208)= 0.3*2*Pi*0.208 + 0.905*2*Pi/2*0.208^2

=0.515 radians. This is 0.515/(2*Pi)=0.082 revolutions.



C) tangential speed = w*r = 3.067*0.72/2=1.104 m/s

D) resultant accn = sqrt((centripetal accn)^2 + (tangential accn)^2)

=sqrt((w^2*r)^2 + (a*r)^2)

=3.957 m/s^2

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