Question

An electric ceiling fan is rotating about a fixed axis with an initial angular velocity magnitude of 0.270 rev/s . The magnitude of the angular acceleration is 0.899 rev/s2 . Both the the angular velocity and angular accleration are directed counterclockwise. The electric ceiling fan blades form a circle of diameter 0.740 m.
A. Compute the fan's angular velocity magnitude after time 0.207 s has passed.B. Through how many revolutions has the blade turned in the time interval 0.207 s from Part A?C. What is the tangential speed vtan(t) of a point on the tip of the blade at time t = 0.207 s?D. What is the magnitude a of the resultant acceleration of a point on the tip of the blade at time t = 0.207 s?

Answer 1

Answer:

A)0.456 rev/s

B) 0.075 rev

C) 1.06 m/s

D) 3.7 m/s²

Explanation:

Initial angular velocity = $\omega _{o}$ = 0.270 rev/s =  1.6965 rad/s

Angular acceleration = α = 0.899 rev/s/s = 5.64858 rad/s/s

Radius = r = 0.74/2 = 0.37 m

Time = t = 0.207 s

A) Final angular velocity is given by the equation

=$\omega_{f}=\omega _{o}+ \alpha t$

=1.6965 + (5.64858)(0.207) = 2.87 rad/s = 0.456 rev/s

B) No. of revolutions is calculated using the equation

$\theta =\omega_{o}t + 1/2 \alpha t^2 = (1.6965 \times 0.207 )+ 1/2 (5.64858)(0.207)^2$= 0.472 rad  = 0.075 revolutions

C) v = r ω =(0.37)(2.87) = 1.06 m/s

D) Centripetal acceleration = $a_{c}=r\omega_{f} ^{2}$ = (0.37) (2.87)² =3.05 m/s²                                

Tangential acceleration =$a_{t}=r \alpha$

= ( 0.37)(5.64858) = 2.08997 m/s²

Resultant acceleration = $a = \sqrt{a_{t}^{2} + a_{c}^{2$

= 3.697 m/s²